Combinatorial Mathematics: International Colloquium Proceedings

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COMBINATORIAL MATHEMATICS

annals of discrete mathematics General Editor

Peter L. HAMMER, University of Waterloo, Ontario, Canada Advisory Editors C. BERGE, Universitt de Paris, France M.A. HARRISON, University of California, Berkeley, CA, U S A . V . KLEE, University o€ Washington, Seattle, WA, U.S.A. J.H. VAN LINT, California Institute of Technology, Pasadena, CA, U.S.A. G.-C. ROTA, Massachusetts Institute of Technology, Cambridge, MA, U.S.A.

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

NEW YORK

OXFORD

NORTH-HOLLAND MATHEMATICS STUDIES

75

Annals of Discrete Mathematics (17) General Editor: Peter L. Hammer University of Waterloo. Ontario, Canada

COMBINATORIAL MATHEMATICS Proceedings of the International Colloquium on Graph Theory and Combinatorics Marseille-Luminy June 1981 Edited by:

C. BERGE, C.N.R.S., Paris D. BRESSON, E.H.E.S.S., Paris P. CAMION, C.N.R.S.-I.N.R.I.A., Paris J.F. MAURRAS, University of Paris XII, Paris F. STERBOUL, University of Lille I, Paris

1983 NORTH-HOLLAND PUBLISHING COMPANY

- AMSTERDAM

0

NEW YORK

0

OXFORD

@ North-Holland Publishing Company, 1983

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 86512 8

Publishers NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK OXFORD Sole distributorsfor the U.S.A. and Canada ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 VANDERBILT AVENUE, NEW YORK, N.Y. 10017

Library of Congress Cataloging in Publication Data International Colloquium on Graph Theory and Combinatorics (2nd : 1981 : University of Marseille-Luminy) Combinatorial mathematics. (Annals of discrete mathematics; 17) (NorthHolland mathematics studies; 75) Organized by the National Centre for Scientific Research, June 14-19, 19x1, at the University of Marseille-Luminy . 1. Combinatorial analysis-Congresses. 2. Graph theory-Congresses, I. Berge, Claude. 11. Centre national de la recherche scientifique (France) 111. Title. IV. Series. V. Series: North-Holland mathematics studies; 75. X2-18765 QA164.157 1981 511’.6 ISBN 0-444-86512-X (U.S.)

PRINTED IN THE NETHERLANDS

PREFACE The International Colloquium on Graph Theory and Combinatorics was held in Marseille from June 14 to June 19, 1981, under the auspices of the National Centre for Scientific Research (C.N.R.S.). The Colloquium - the second of its kind organized by the C.N.R.S. - was a great success. In addition to the invited talks, 95 participants from many parts of the world were invited to present communications in combinatorics: graph theory, hypergraphs, designs, coding. This volume contains most of the papers that were presented: new results or surveys. We wish to thank the organizations that have contributed to the success of the Colloquium: Direction des Recherches, Etudes et Techniques, Ministere des ArmCes (D.R.E.T.), I.B.M. France, Mairie de Marseille, SociCtC MathCmatique de France, International Mathematical Union, UniversitC de Paris XII. The lectures were given at the University of Marseille-Luminy, and the participants had the pleasure of being the first guests of the Centre International de Rencontres MathCmatiques (C.I.R.M.). We wish to express our gratitude to this new centre and particularly to Mrs Anne Litman for her many contributions to making this colloquium a success. We are also indebted to Mrs Martine Ulrich for her thoughtful help at Luminy as well as for her careful preparation of the final manuscript. In addition, it is a pleasure to acknowledge the continuing support of the Ecole des Hautes Etudes en Sciences Sociales (C.M.S.) that helped make possible the organization of the colloquium. Last but not least, we wish to thank the North-Holland Publishing Company for the smooth and efficient job they did in preparing this volume. THE EDITORS

CONTENTS PREFACE

v

J. ABRHAM and A. KOTZIG,Extremal bases of additive permutations

1

and A. GERMA,Ordre minimum d’un graphe D. AMAR,I. FOURNIER simple de diamktre, degrC minimum et connexitk donnCs

7

and A. GERMA,Structure des graphes de D. AMAR, I. FOURNIER connexiti k 2 2 et de stabilitC a = k + 2

11

L. D. ANDERSEN, A. J. W. HILTONand C. A. RODGER, Small embeddings of incomplete idempotent latin squares

19

J. AYEL,Degrees and longest paths in bipartite digraphs

33

CaractCrisation mCdiane des arbres J. P. BARTHELEMY,

39

A. BENHOCINE and A . P. WOJDA,On the existence of D ( n , p ) in directed graphs

47

C. BENZAKEN, Haj6s theorem for hypergraphs

53

C. BERGE,Path partitions in directed graphs

59

J.-C. BERMOND, C. DELORME and J.-J. QUISQUATER, Grands graphes de degrC et diamktre fixes

65

C. BERNASCONI, Combinatorics of incidence structures and BIB-designs

75

K. P. BOGART,Ideal and exterior weight enumerators for linear codes: Examples and conjectures

81

B. BOLLOBAS, The evolution of the cube

91

F. BORIES,J. L. JOLIVET and J. L. FOUQUET,Construction of 4-regular graphs

99

A. BOUCHET,A construction of a covering map with faces of even lengths

119

A. BOUCHETand J. L. FOUQUET,Trois types de dCcompositions d’un graphe en chaines 131

F. BRY. Sur les couplages dans les graphes localement finis

143

P. CAMION,A deterministic algorithm for factorizing polynomials of FI , [XI

149

vi

Contents

vii

A . H. CHAN,R. A. GAMESand E. L. KEY, On the complexities of periodic sequences 159

P. CHARPIN, The extended Reed-Solomon codes considered as ideals of a modular algebra

171

0. CHEVALIER, F. JAEGER,C . PAYANand N. H. XUONG,Odd rooted orientations and upper-embeddable graphs

177

V. CHVATAL and E. SZEMEREDI, Notes on the Erdos-Stone theorem

183

M . COCHAND and P. DUCHET,Sous les pavCs ...

191

E. J. COCKAYNE and FANGZu Yao, Rotation numbers for unions of triangles and circuits 203 G. COHENand P. FRANKL, On cliques and partitions in Hamming spaces

21 1

R. CORDOVIL, Oriented matroids of rank three and arrangements of pseudolines 219

B . COURTEAU, G . F O U R N ~ Eand R R. FOURNIER, Etude de certains paramktres associCs A un code lindaire 225 M . CROCHEMORE, Mots et morphismes sans carrd

235

L. J . CUMMINGS, Strongly qth power-free strings

247

G . DELCLOS, Cyclic codes over GF(4) and GF(2)

253

J. DENES,Some connections between groups and graphs. A survey

259

PH. DELSARTE and PH. PIRET,Algebraic codes achieving the capacity of the binary symmetric channel

265

W. DEUBER,Partitioning subwords of long words

27 1

P. ERDOSand Z. FUREDI,The greatest angle among n points in the 275 d-dimensional euclidean space S. FIORINI and J. LAURI,Edge-reconstruction of graphs with topological properties

285

P. FRANKL, Constructing finite sets with given intersections

289

H. DE FRAYssEIX and P. ROSENSTIEHL, Systkme de refhence de TrCmaux 293 d’une reprksentation plane d’un graphe planaire

G . GIRAUD, Sur un problkme d’Erdos et Hajnal

303

M. H A B ~and B B. PEROCHE,La k-arboricitd IinCaire des arbres

307

viii

Conrents

MARSHALL HALL,JR., Designs and coding theory

319

S. HARARI,Le dCcodage rapide des codes de Reed-Solomon et leur 327 gCnCralisation

M. C. HEYDEMANN, Nombre maximum d’arcs d’un graphe antisymk337 trique sans chemin de longueur 1 W. IMRICH and G. SCHWARZ, Trees and length functions on groups

347

B. JACKSON, Maximal cycles in bipartite graphs

361

H. JACOBand H. MEYNIEL, Extension of Turhn’s and Brooks’ theorems and new notions of stability and coloring in digraphs 365

F. JAEGER,Symmetric representations of binary matroids

37 1

G . 0. H. KATONA,Sums of vectors and Tur6n’s graph problem

377

J. M. LABORDE,L’hypercube infini et la connexitC dans les graphes infinis 383 J. LACAZEand L. BENETEAU, The automorphism group of the smallest non-affine Hall triple system

387

A. LAHRICHI and J. L. SERET,Un algorithme IinCaire pour I’CnumCration des crCneaux d’une suite 393 M. LAS VERGNAS, Le polynbme de Martin d’un graphe eultrien

397

J. LEHEL,7-critical hypergraphs and the Helly property

41 3

A . LENTIN,On certain families of disjoint perfect matchings in K z .

419

R. L6PEZ BRACHO,Dtcompositions en chaines d’un graphe complet d’ordre impair

427

W. MADER,On n-edge-connected digraphs

439

A. R. MAHJOUB, Polytope des absorbants dans une classe de graphe A seuil

443

H. F. MAITSON,JR., An upper bound on covering radius

453

J. F. MAURRAS, Sur une propriktk extreme des plans projectifs finis

459

J. MAYER,Note sur le problkme de Heawood

465

M. MOLLARD, Le nombre d’absorption du n-cube

469

B. MONTARON, Sur une construction de codes sphkriques

473

A. F. MOUYART, Symmetric inseparable double squares

483

Contents

ix

H. M. MULDER,The number of edges in a k-Helly hypergraph

497

U. S. R . MURTY,Projective geometries and their truncations

503

M. W . PADBERCand L. A. WOLSEY,Trees and cuts

51 1

G. PASQUIER,Binary self dual codes construction from self dual codes over a Galois field h r n 5 19

C. PAYANand N. H. XUONG,Sur un thiorbme min-max en thiorie des graphes 527 J. E. PIN, On two combinatorial problems arising from automata theory

535

A. POLIand M. VENTOU, Construction de codes autoduaux de profondeur 1 ou 2 dans A = h[XI,. . . , X,]/(Xi- 1,. . . ,X: - 1) 549

M . PREISSMANN, C-minimal snarks

559

A . RECSKI,On the generalization of the matroid parity and the matroid partition problems, with applications 567

I. G. ROSENBERG, Cycle structure of affine transformations of vector spaces over GF(p) 575 H. SACHSand M. STIEBITZ, Construction of colour-critical graphs with given major-vertex subgraph 581 S. C. SHEEand H . H. TEH,Graphs and inverse semigroups

599

F. STERBOULand D. WERTHEIMER, Colorations extrkmes dans les hypergraphes 605

W. T. TROTTER, JR. and J. A. Ross, Every ?-irreducible partial order is a suborder of a t f 1-irreducible partial order 613 P. VADERLIND, Between clutters and matroids

623

S. A. VANSTONE, A note on the existence of strong Kirkman cubes

629

W. F. D.

DE

VEGA, On the bandwidth of random graphs

633

WERRA,On the use of bichromatic interchanges

639

DE LA

D. WERTHEIMER, Extreme coloring of the edges of a graph

647

J. WOLFMANN, New decoding methods of the Golay code (24,12,8)

65 1

T. ZASLAVSKY, Uniform distribution of a subgraph in a graph

657

List of problems submitted during the conferences

665

This Page Intentionally Left Blank

Annals of Discrete Mathematics 17 (1983) 1-5 @ North-Holland Publishing Company

EXTREMAL BASES OF ADDITIVE PERMUTATIONS Jaromir ABRHAM Depr. of Industrial Engineering, University of Toronto, Toronto, Ontario, Canado, M5S 1 A4

Anton KOTZIG C.R.M.A.. Universite' de Montre'al. C.P. 6128. MontrPal, P.O., Canada, H3C 3J7

Let X = ( x , , x ? , . . . ,x k ) be a vector with relatively prime integer components x , < x z < * . . < X k , k 3 1. Then X is called a basis of additive permutations (A-basis) if there exists a vector Y = ( y l , y 2 , . . . ,y k ) such that both ( y l , y:, . . . , y , ) and ( x , + y l , x 2 y z , . . . ,X k + y k ) are permutations of X . We will write X Y for ( x l + y , , . . . ,X k + Y k ) . The concept of an A-basis was introduced by Kotzig (71.

+

+

All A-bases of cardinalities five and less are described in [ 6 ] and in the appendix of [ 3 ] ; all A-bases of cardinality six together with their additive permutations have been found in [I]. Additional results on additive permutations have been presented, e.g., in [4],[ 5 ] , [8], [9], [lo]. If X is an A-basis, then the following symbols will be used to denote certain parameters of X : 1 XI - the cardinality (number of components) of X ; T ( X ) - the number of positive components of X ; v ( X ) - the number of negative components of X ; w ( X ) - the number of odd components of X ; $ ( X ) - the number of additive permutations of X . The following three theorems give some extremal values of the above parameters.

Proof. For the proof, see [l, Theorem 11. Theorem B. If 1x133 then w ( X ) = O (mod 2 ) and 2 S w ( X ) S $ ( X ( .

Proof. For the proof, see [7, Theorem 31. 1

J. Abrham, A. Kotzig

2

Theorem C. If 1 X

13 3

then $ ( X ) = 0 (mod 2), hence $(XI 5 2 .

Proof. Let Y be an additive permutation of X , and let Y’ be the inverse permutation to Y ; then Y’ is clearly an additive permutation to X.To complete the proof it is enough to show that Yf Y ’ . Since Y cannot be an identical permutation there exists an x in X such that the corresponding element y in Y is different from x. We have y E X, and if Y‘ = Y , x is the element corresponding to y in Y’.Then X + Y contains x + y in two positions, so X + Y is not a permutation of X.

On the basis of the above theorems, we may want to investigate the following extremal bases of additive permutations: (a) A-bases with the minimum/maximum number of positive (negative) components. (b) A-bases with the minimum/maximum number of odd (even) components. (c) A-bases with the minimum number of additive permutations. (Very little is known about the maximum number of additive permutations; however, the results presented in [4] imply that, for X , = ( - r , - r + l , . . . , r - l , r ) , $(X,)++m as r + + m if r = O o r 1 (mod 4).) Let us now summarize what is known about these classes of extremal A-bases. (a) All A-bases of cardinality at least five with exactly two negative elements have been described by the authors in [3] and are listed in Table 1 (which also comes from [3]; the cases listed refer to [3] as well). The reader will observe that, in Table 1, ( X1 = n + 3 with n 5 2 and the subscripts - 2, - 1 , O corresponding to the two negative components and to this zero component of X respectively. Some preliminary attempts at finding all A-bases with 1 X 1 - 3 positive and three negative components have indicated that the number of such A-bases will grow very rapidly with the increase in the cardinality. (b) A-bases with the given number of odd elements have been studied in [ 2 ] where the following theorem has been proved: Let k be an integer, k 2 3, k # 4. Let m be an even number such that 2 S m S $k.Then there exists an A-basis X such that I X l = k , w ( X ) = m . This theorem implies that for any k a 3, k # 4, there exists an A-basis of cardinality k with the minimum (maximum) number of odd/even components. The reader will observe that the A-bases listed in Table 1, which do not depend on the parameters r, s, contain exactly two odd components. For the remaining A-bases listed, r, s can be chosen in such a way that each A-basis obtained contains exactly two odd components. (c) According to Theorem C, $ ( X ) 2 2 if 1x133. For any A-basis of cardinality six or more, listed in Table 1, we have $(X)==2. For X = (-2, - 1,0,1,2) (the only A-basis of cardinality five) Table 1 yields four

3

Extremal bases of additive permutations

additive permutations and two more can be found in [3]. Theorem D below shows that there exists a class of A-bases for which $ ( X ) 3 4. For this theorem, we will need the following definition.

Definition. An A-basis X = (xl,. . . ,xk) is called symmetric if x, = i = 1 , 2 ,..., k. Theorem D. Let X be a symmetric A -basis, 1 X

13 3 .

for

-&+I-,

Then $ ( X ) 2 4.

Table I -2

-1

0

1

2

3

4

... n - 2

n-1

n

1 1 2 1

2 2.2.1. (n24)

X - a.-,-s Y a,,+s Z

Y'

-an.,

0

Z ' -a,-,-s

-an.& 0 a,-, -a.-,-s 0 -a.-,-s an.,+s r s

r

r 0 r S

a,)

2 2 2'

7

Z2 2.2.2.2.

(n33)

0 r X -a*-,- r -an-, Y a.-,+r an-J - a n - , - r s Z -a,-, 0 - a m - , - r a, Y' 0 a..*+r s a. - 4 Z'-an-,-r r s an.,+ r

S

0 S

r

a,,

Please note: (1) Only A-bases with cardinality five or more and exactly two negative elements are listed. (2) If Y is an additive permutation of X , and 2 = X + Y,then Y'is the inverse permutation, and Z' = x + Y'. (3) If r, s are used, they denote relatively prime integers such that O < r < s. In that case, a, denotes Zk(r + s).

J. A brham, A. Kotzig

4

Proof. Let Y be an additive permutation of X . Three cases (in general, not mutually exclusive) will be considered. The term ‘monocycle’ will refer to a cycle of length one. (1) Let Y have at least two cycles of length two or more. Let P be obtained from Y by changing the orientation in one of the cycles. Then Y, and their inverse permutations form a set of four different additive permutations of X . (2) Let Y consist of one cycle, containing all nonzero components of X; if 0 is in X, then 0 may either be in the above cycle, or it may form a monocycle. (In the latter case, we have 1 XI > 5 . ) Both cases being similar, only the case of a unique cycle will be considered. Let Z = X + Y.The symmetry of X together with the relation X + ( - Z ) = - Y imply that - Z also is an additive permutation of X . We will show that Y, - Z, and their respective inverse permutations Y ’ , - Z ‘ form a set of four different additive permutations of X . Since Y ’ # Y, - Z ’ # - Z (see the proof of Theorem C) it is sufficient to show that Y ‘ # - 2. Let us write now X = (xl, x,,, x,,, . . .,x,,) where x,x,,x,, . . . xtk is the cycle of Y . (The reader will observe that - to simplify the notation -the components of X do not appear in their natural order.) Then

Y = (x,,, Xt2’. . . ,x,,, Y’ = (x,,, XI,. . . ,XI,

XI), 2,

x,,.,),

z = x + y = ( X I + x,,, x,, + x,,, . . ,x,, , + x,,, x,, + X I ) , 2 = x + Y ’ = (XI + x,,, xt, + X I , . . . ,x,, , + x,, >, x,, + XI, *

,).

If cC,(X)=2 we have necessarily 2 = - Y ’ , Z = - Y ; taking the last two components into consideration we get x,, , + x,, = - x,, I and x,, + x,,-, = - x , which implies x,,-, = xl, and this contradicts our assumption. (3) Let Y have at least one monocycle (x,) where x, # 0. Let Y be an additive permutation of X , Z = X + Y . If $ ( X ) = 2 then I” = - Z. If y,, y : , z, denote the ith component of Y, Y ’ , Z respectively, then y : = x, (from the definition of the inverse permutation), and y: = - 2x, (from the relation Y ‘ = - Z ) and this is impossible if x, # 0. The statement of Theorem D is stronger than it may seem at first glance: The equality t+h(X)=4 is satisfied for the symmetric A-basis of cardinality 6 presented in [l]. Since no other solution to the equality x ( X ) = 4 has been found yet, a stronger statement might be possible if the case ( X 1 = 6 is excluded. However, the improvement - if any - will be small because of the following fact: For any odd k > 3, a symmetric A-basis X of cardinality k exists such that + ( X ) = 6. It is sufficient to choose (with k = 2r + 1 , r 3 1 )

x = ( - 2‘-l, - 2‘-?, . . . , - 2, - 1,0,1,2,2?,. . . , 2 ‘

-I).

Extremal bases of additive permutations

5

This A-basis has already been mentioned in the conclusion of [2].

Acknowledgement

The research was sponsored by NSERC Grants No. A7329 and A9232 and by an Ontario-Quebec Exchange Program Grant.

References [ I ] J . Abrham and A. Kotzig, Generalized additive permutations of cardinality six, Proc. 11th Southeastern Conf. Combinatorics, Graph Theory, and Computing (Congressus Numerantiuni Vol. 28, Winnipeg, Manitoba, December 1980), 175-185. [?I J. Abrham and A. Kotzig, Bases of additive permutations with a given number of odd elements, Utilitas Mathematica 18 (1980) 283-288. [3] J. Abrham and A. Kotzig, Bases of additive permutations with the minimum number of negative elements, Ars Combinatoria, to appear. [J] J. Abrham and A. Kotzig, Skolem sequences and additive permutations, Discrete Mathematics 37 (1981) 143-146. [5] J. Abrham and A. Kotzig, Cubic graphs and certain pairs of additive permutations, Congressus Numerantium 32 (1981) 89-101. [6] H. Desaulniers, Sur un problkme de Kotzig sur les permutations additives, Rapports de recherche de I’universitk du O u t b e c a Trois Rivikres, No. 11. 1980. [7] A. Kotzig, Existence theorems for bases of additive permutations, Proc. I Ith Southeastern Conf. Combinatorics, Graph Theory, and Computing (Congressus Numerantium Vol. 29. Winnipeg, Manitoba, December 1980), 573-577. 1x1 A. Kotzig and P.J. Laufer, When are permutations additive? Amer. Math. Monthly 85 (1Y78) 363-365. [Y] A. Kotzig and J. Turgeon, Perfect systems of difference sets and additive sequences of permutations, Proc. 10th Southeastern Conf. Combinatorics, Graph Theory, and Computing (Boca Raton, 1979) 629-636. [ l o ] J . Turgeon, An upper bound for the length of additive sequences of permutations. Utilitas Mathematica 17 (1980) 189-196.



ORDRE MINIMUM D’UN GRAPHE SIMPLE DE DIAMETRE, DEGRE MINIMUM ET CONNEXITE DONNES Denise AMAR, Irene FOURNIER and Anne GERMA Equipe de Recherche Associke au C N R S , n o 4.52 A1 Khowarizmi, Bhtiment H90. Y130.5 Orsay. Cedex. France We give the minimum order of a connected simple graph with given diameter, minimum degree and connectivity.

Notations. Soit G u n graphe simple connexe. Pour les definitions des termes suivants voir [l]. On note D le diamirtre du graphe G, 6 le degre minimum de G et n ( G ) I’ordre de G. Si k est un entier vCrifiant 1 k 6, on note % ( D ,6, k ) I’ensemble des graphes G de connexitt 3 k, de diamirtre D et degrt minimum 6. On notera V(G)I’ensemble des sommets de G ; si A C V ( G ) ,on note [ A le cardinal de A . On cherche B dtterminer la fonction f(D, 6, k ) = inf{n(G) G E % ( D ,6, k)} et on montre le thCortme suivant.

I

I

Theorcme. (1) f ( l , S , k ) = 6 + 1. (2) f ( 2 , S , k ) = 6 + 2. (3) S i D 3 3 ef 1 s k s S s 3 k - 1

f(D,s,k)=k(D-3)+26+2. Si D 2 3 et 6 3 3 k

-

1

f ( D , 6, k ) = ( t + 1)(6 + 1) + ~ k , ou t et

E

sont de‘termine‘s p a r : D=3f+&

fEN*

&E{O,1,2}.

Demonstration du thCor&me. ( 1 ) Les graphes de diamktre 1 sont les graphes complets d’oh f ( l , S , k) = S 1. (2) Un graphe G de diambtre 2 n’est pas complet, d’oh si son degrC minimum est 6, n ( G ) > 6 + 2 . On considere le graphe G,, Cgal au graphe complet & + ? privC d’une arkte (u, v ) (Fig. 1). n (G,,) = S + 2 et G,,E %(2,6, k ) d’ou le rksultat.

+

7

D.Amar et al.

8

Fig. 1.

(3) Nous supposons dans la suite D 3 3. (a) Kane et Mohanty ont prouvt [Z] le thtorkme suivant:

Si D

33

et G E %(D, 8, k),

n(G)>k(D - 3 ) + 2 6

+ 2 = gi(D, 6,k).

Cette borne n’est pas toujours la meilleure possible. (b) Nous prouvons ici le lemme. Lemme. Si D

33

et G E %(D,6,k),

n ( G )3 ( t + 1)(S + 1) + Ek oti D = 3 t + ~( t E N * ,

E

= gz(D, 6,

k),

E{0,1,2})

Cette borne n’est pas toujours la meilleure possible et nous montrerons en fait

we

f(o,6, k ) = sup{gi(D, 6, k),g2(D, 6, k)}. Demonstration du lemme. Soit G un graphe dans %(D,c5,k)ou D 2 3 . On consid&-exo et xD deux sommets de G A distance D et {xo, xI, x?,. . . ,x D } u n chemin allant de xO A x D . Pour i E{O, ..., D } on note ”(xi) I’ensemble des voisins de x, n’appartenant pas 5 I’ensemble {xO,xI, . . . ,x D } . Remarques. La distance entre xo et x D Ctant D : (1) Deux sommets du chemin ne sont adjacents que s’ils sont consbcutifs, donc N’(xo)et N ’ ( x D )ont au moins S - 1 sommets. Pour 1 s i s D - 1, “(xi) a au moins S - 2 sommets. (2) Si I i - j 1 a 3, ”(xi) n ~ ’ ( x , )= 0. Soient t E N * et E E{O,1,2} dtfinis par D = 31 + E . En fonction de E on considkre les trois cas suivants:

Ordre minimum d'un graphe simple

Y

Premier cas : D = 3t

n ( G ) a D + 2 + 2 ( 6 - l ) + ( t - 1)(6 -2), n ( G ) a (t

+ 1)(6 + 1).

Deuxibme cas: D = 3r + 1 Soit S = {xo,x l }U N'(xll)et soit T I'ensemble des sommets de G n'appartenant pas B S. IS 12 k, T 12 k, d'oh G 6tant k-connexe, il y a k arctes disjointes entre S et T. L'une d'elles peut ktre d'extrCmite xz; la distance entre xIIet xD ttant Cgale B D les k - 1 autres ont pour extrCmitCs dans T des sornmets u I , u z , . . . , uk-l n'appartenant ni au chernin {XI,, xI,. . . ,x D } ni A N ' ( x , ) pour i > 3, donc en particulier B N ' ( X ? ~pour + ~ )i = 1 , . . . , f

I

n ( G ) ? 3 r + 2 + 2 ( 6 - I ) + ( t - 1)(6 - 2 ) + k - I . n ( G ) S ( r + 1)(6 + I ) + k . Troisibme cas : D = 3t + 2 On minore te nombre de sornmets des ensembles disjoints suivants: {xll, X I , . . . ,X D } ?

"(xtl),

"(x3t+!)>

1

-

N ' ( X ~ , +V, ~ )et , V,, ou V, et V, sont dCfinis comrne suit: Soit SI = {Xll, X I } u "(x,,); s, = {X3,+l,X 3 , + : } u " ( X 3 , + z ) ,

TI = V(G)\Sr; Tz= V(G)\Sz. VI (resp. V2)est I'ensemble des extrimitis autres que x2 (resp. x l , )dans T , (resp. T 2 )des arktes entre SI et TI (resp. S , et Tz). On montre que I V, 13 k - 1 d'oG

n(G)>3t+3+2(6-1)+(f-I)(6-2)+2(k-l), n( G )2 ( t

+ 1)(6 + 1) + 2k.

Demonstration du theoreme (continuke).

g l ( D , S , k ) - g z ( D , S , k ) = ( t - l ) ( 3 k - S -1).

(a) Si k S 6 S 3k - 1, g l ( D ,6,k ) * g..(D, 6,k ) . Le graphe G I (Fig. 2) appartient 2 %(D,6, k ) et il a gl(D,6, k ) sommets d'oh si

D.Amar et al,

10

d

-

I copies

Fig. 2.

k G 6 s 3k - 1, f ( D , 6, k) = g,(D,6, k). G I est form6 d'une chaine de graphes complets. Les sommets de deux complets cons6cutifs sont tous his. (b) Si 6 2 3 k - 1, g2(D,6, k ) 5 g,(D,8, k ) . Le graphe G2 (Fig. 3) appartient B g ( D , 6 , k ) et a g 2 ( D , 6 , k )sommets, d'ob f ( D ,6, k = g m 6, k 1.

I

t-l

fois l a s u i t e : K k , K 6 + l - 2 k ,

seulement s i D 5 3 t + l ou D = 3 t + 2

Kk

1

seu 1ement si D = 3t+2

Fig. 3.

G, est form6 d'une chaine de graphes cornplets, les sommets de deux consicutifs itant tous lies. Ces exemples de graphes terminent la dkmonstration du thiorkme.

References [ 11 C. Berge, Graphes et Hypergraphes (Dunod, Paris, 1973). [21 Kane and Mohanty, A lower bound o n the number of vertices of a graph, Proc. Amer. Math. SOC. 72 (1972) 211-212.

Annals of Discrete Mathematics 17 (1983) 11-17

0North-Holland Publishing Company

STRUCTURE DES GRAPHES DE CONNEXITE k 3 2 ET DE STABILITE (Y = k 2

+

Denise AMAR, Irkne FOURNIER and Anne GERMA Equipe de Recherche Associie au CNRS, n o 452, A1 Khowarizrni, Bitirnent 4Y0,91405 Orsay, Cedex, France We prove that if G is a simple graph of connectivity k 2 2 and stability a exists a longest cycle C, such that the stability of G \ C is no more than 2.

+ 2.

=

k

S

k , G est

there

Chvatal et Erdos ont prouvC le thCorkme suivant [2]. Theoreme. Si G est un graphe simple k-connexe de stabilite‘ hamiltonien.

Q

Et nous avons montrC, avec Haggkvist, le resultat suivant [l]. Theoreme. Soit G un graphe simple, k -connexe, k 3 2, de stabilite‘ Q est un cycle de longueur maximum, G \ C est un complet.

Q

=

k

+ I ; si C

Nous Ctudions ici la structure des graphes k-connexes, k > 2, de stabilitC = k + 2 et prouvons le thCorkme suivant:

Theoreme. Soit G un graphe simple, k-connexe, k 2 2, de stabilite‘ a = k + 2 ; il existe dans G un cycle de longueur maximum tel que G \ C soit de stabilite‘ infe‘rieure ou &ale a 2.

Soit G = (V, E ) un graphe simple; V dCsigne I’ensemble des sommets, E I’ensemble des arktes de G. On suppose que G est k-connexe, k 2 2. et de stabilitC Q = k + 2 . G contient donc un cycle d’au moins k + 1 somrnets. Notations. (1) Si C est un cycle dans G, pour une orientation quelconque de C,si y est un sommet de C, on note y + le successeur de y sur C. (2) Si K est un sous-graphe de G, V ( K )dksigne l’ensemble des sommets de K et 1 V ( K ) /le cardinal de cet ensemble. Demonstration du theoreme. Soit C un cycle de longueur maximum de G. 11

D.Amar er al.

12

ltre e‘tape. ToUte composante connexe de G \ C est de stabilite‘ infe‘rieure ou e‘gale I? 2. Soit H une composante de G \C et soient u, u, w trois sommets de H . Entre H et C, il y a k chemins d’extrimitks a I . a 7 , .. . , f f k sur C. L’ensemble {u, u, w,a,+,1 s i c k } a k + 3 ClCments et n’est donc pas un stable (Fig. 1). Les seules aretes qui ne permettent pas d’obtenir un cycle plus long que C lient les sommets u, u, w. Donc { u , u, w } n’est pas un stable.

22me e‘tape. Si une composante connexe H de G\C est de stabilite‘ 2, alors V ( G ) = V(C)U V ( H ) . On consid2re u et w deux sommets indipendants de H et on suppose qu’il existe un sommet u n’appartenant pas 5 V ( C U H ) . On m6ne k chemins sommets-disjoints de u sur c, d’extrimitks a l ,a ? , .. . , ak {u, u, w , a , ’ , 1 i 4 k } n’est pas u n stable; il existe une arCte liant u ou w et I’un des a : ; on suppose que c’est u a ; (Fig. 2 ) . On considere alors I’ensemble de sommets { u , u, w, a ?’, a :, i > l}. Cet ensemble B k + 3 sommets n’est pas un stable et les seules liaisons non exclues sont ua ’ (Fig. 3 ) ou a ; + a : (Fig. 4).

U

H

Fig. 2 Fig. 1. U

H

C‘

C‘

+

a.

1

Fig. 3.

a.

1

Fig. 4.

Srrucrure des graphes

13

Dans les deux cas on considkre un cycle C‘ aussi long que C. De H on peut mener k chemins sur C dont I’un est I’arite ua T et les k - 1 autres d’extremitis p2, P I , . . ., p k . Soit H‘ la composante de a dans G \ C’: cette composante contient H et on peut mener de H’ k + 1 chemins sommet-disjoints sur C’ d’extrimitks pl,= a l , P I = a +I + , P 2 , ,. , ,&.L’ensemble { u , w, p:,O S i S k } n’est pas un stable et toute arite entre deux de ses sommets permet d’obtenir un cycle plus long que C, d’ou contradiction.

3ime Ctape. On suppose que pour tout cycle C de longueur maximum, G \ C a au moins deux composantes connexes (complttes) On fait les hypoth2ses de maximalit6 suivantes: parmi les cycles de longueur maximum, on choisit un cycle C laissant dans G \ C deux composantes connexes (complktes) HI,et H I telles que

1 V(Hll)I+ 1 V ( H l ) J

soit maximum.

Supposons qu’il existe un sommet u de G n’appartenant pas B V ( C U H,, U HI).De u on mkne k chemins sur C d’extrkmitis a I ,. . . ,(Yk. Soit u un sommet de HI,et soit w un sommet de HI.L’ensemble {u, u, w, a:} n’est pas un stable et on peut supposer que ua: E E (Fig. 5). On distingue deux cas possibles:

ler cas. Pour tout w E HI et tout j # 1, w a : e E. L’ensemble { u , u, w , a :+, a:, 2 s i S k } n’est pas un stable. Les arctes non excluses sont: (i) ua;’ ou aYf’a7 pour un j z 2 (ii) w a :+. Dans les cas (i), on consid2re le circuit C’aussi long que C et Hh et HI les composantes de Ho et HI dans G \ C’ (Figs. 6 et 7).

Fig. 5.

D.Amar et al.

14

+

a.

~j

Fig. 6.

V(H:,)3 V ( H J u { a ;I,

a

Fig. 7.

V ( H ; )2 V ( H , ) ;

d’ou contradiction avec les hypotheses de maximalitk. Dans le cas (ii) on remarque que a ; est lik B tous les sommets de HI,.En effet, s’il existait u’E V(Hl,) tel que u‘a;sf E, I’ensemble {u,u ’ , w,a,+,1 < i S k } serait u n stable B k + 3 sommets. On considere alors {u,u, a;”, a:,2 S i S k}. Cet ensemble n’est pas un stable et on examine les differentes liaisons possibles: (a) Arite ua :++. Si HI,contient 2 sommets u et u ’ , on peut construire un cycle C‘ plus long (Fig. 8) que C, d’ou contradiction. Si HI,= { u } , on construit un cycle C’ aussi long que C; la composante HI de a t’ dans G \ C’ contient H , . Soit H:, la composante de u dans G \C’.

I V ( H 0 ) )+ 1 V ( H I ) (< I V ( H X ) (+ 1 V(HI)J, d’ou contradiction avec les hypotheses de maximalite (Fig. 9). (b) Arite a :at” (Fig. 10) ou ua :++ (Fig. 11). On considere le cycle I’ et la composante X de u dans G \f.I1 y a k + 1 chemins entre 2 et I’ qui aboutissent en Po, PI,.. .,P k sur

r.

Fig. 8.

Fig. 9.

Structure des graphes

15

0

0 a.

1

a.

1

Fig. I0

Fig. I I .

{ u , w, p,+,0 =G i G k } n'est pas un stable. II existe donc une arite up,! ou wp,+ou p : p i . Dans chacun de ces cas, on peut construire un cycle C' de mime longueur

que C, laissant dans G \C' une composante connexe de stabi.litC 2, ce qui est exclu par hypothkse, soit deux composantes H:, et HI, cornplktes, telles que

I V(H:l)I + I V(HI)I' I V(H0)I + I V(H,)J, ce qui est Cgalement en contradiction avec les hypothkses de maximalitk.

2 m e cas. w est lip a l'un des cz:, autre que a ;; on l'appelle a ; . I'ensemble des sornrnets {u, u, w,a;', o f + a , t , 3 s i =G k } n'est pas un stable. Les aretes non exclues sont (Fig. 12) (i) ua;' (ou ua?"), (ii) vat' (ou war'), (iii) a ;La:+, (iv) a ; + a : . i > 2 (ou a:+a:, i > 2 ) . Dans les cas (i) on considtre le cycle C" aussi long que C. La cornposante N,', de u dans G \ C'contient H,, U { aY } (Fig. 13) d'oh contradiction avec les hypothkses

Fig. 12.

Fig. 13.

D.Amar et al.

16

de rnaxirnalitb. Dans les cas (ii), on obtient un cycle C' plus long que C (Fig. 13) d'oG contradiction. Dans le cas (iii) on considere le cycle r et la cornposante H:, de a dans G \ f (Fig. 15). De HI on peut rnener k + 1 chernins d'extrCrnitCs sur C', p,,, . . . . pk. L'ensernble { a ; , w , p',O < i S k } n'est pas un stable. Si wp,+est une arste, on peut construire un cycle C' plus long que C (Fig. 16) (on oriente r pour que at' soit le successeur de a 7 ) .Si a ;p' E E ou si p:p,' E E, on construit un cycle C' aussi long que C, laissant dans G \ C' deux cornposantes H:, et Hi telles que

1 V ( H 6 ) )+ 1 V ( H i ) ( > 1 V(H0))+ 1 V(HI)(. Dans le cas (iv), on construit le cycle C' aussi long que C ; la cornposante H:, de u dans G \ C' contient H,, U { a ; } (Fig. 17); la cornposante HI de w contient H , d'ou contradiction avec les hypotheses de rnaxirnalitb.

V

Fig. 13.

++ a1

Fig. 15.

++ a2

Fig. 16.

Fig.

17.

Structure des graphes

17

Conclusion Nous avons montrt qu'on peut recouvrir V(G) par un cycle d e longueur maximum et un graphe connexe de stabilitt 2 , ou bien par un cycle d e longueur maximum et 2 (ou 1 ou 0) complets.

Corollaire. Nous proposons la conjecture suivante. Soit G un graphe simple de connexite' k 3 2, de stabilite' a. O n peut recouvrir les sommets de G par [ a / k ] cycles de G. Thomassen a prouvC cette conjecture pour k = 2 .

References [ I ] D. Amar, 1. Fournier. A . Germa and R. Haggkvist, Covering of vertices of a simple graph with given connectivity and stability number, Internat. Conf. on Convexity and Graph Theory of Israel (1981). [2] V. Chvatal and P. Erdos, A note on hamiltonian circuits, Discrete Math. '7 (1972) 11 1-1 13.



SMALL EMBEDDINGS OF INCOMPLETE IDEMPOTENT LATIN SQUARES L.D. A N D E R S E N Bofelleden. Kirkebakken SO. 8330 Beder. Denmark

A.J.W. HILTON and C.A. RODGER University of Reading, Whiteknights. Reading, U.K .

1. Introduction

A partial r X s latin rectangle on t symbols al, . . . , a, is an r X s matrix, each of the cells of which may be empty or may be occupied by one of the symbols a,,. . . ,a,, and which satisfies the rule that no symbol occurs more than once in any row or more than once in any column. A n incomplete r x s latin rectangle on t symbols is a partial latin square in which there are no enipty cells; it is a latin square if r = s = t. In a partial latin square of side n on t symbols al,. . . , a,,if cell ( i , i) is occupied by symbol a, for each i, 1 S i S n, then the partial latin square is idernpotent. If A is an incomplete latin rectangle on symbols u l , .. . ,a,, let N A ( a , )denote the number of times that a, occurs in A. Ryser [I31 proved that an incomplete r X s latin rectangle A on a , ,. . . , a,, could be embedded in a latin square of side n on the same symbols if and only if NR(a, )2 r

+s -n

(1 s i s n ).

This result was extended to certain types of partial latin rectangles by Cruse [7], and a simpler proof of Cruse’s theorem was given in [lo] by Hilton. Evans [8] showed that it is a very simple matter to deduce from Ryser’s theorem that a partial latin square of side r on r symbols can be embedded in a latin square of side n if n 2 2 r ; the figure 2 r here is best possible. Cruse [6] showed that analogous results hold for symmetric latin squares. If we consider the situation for idempotent latin squares we find that the known results are less complete and much more complicated. The present authors have recently obtained [ 5 ] a proof of the analogue of Evans’ result, namely that a partial idempotent fatin square of side r on r symbols can be embedded in an idempotent latin square on n symbols, for n 2 2r + 1. However the analogue of Ryser’s theorem remains elusive. It is our findings about such an 19

L.D. Andersen et al.

20

analogue that we wish to present in the present paper. Generally we do not give proofs, as some of these are extremely long; they may be found in [I]. Also we confine ourselves to the case when n - r 2 5 (of course, we assume that r = s ) ; there are substantial complications when n - r = 3 and also when n - r = 3 . I t appears to us that the main obstruction to our efforts lies in the fact that we cannot analyse the case n - r = 3 properly; we have formulated a conjecture [ I I ] about edge-colouring cubic bipartite graphs and, if this conjecture could be solved, we feel we would probably be able to obtain a complete analysis of the idempotent analogue of Ryser’s theorem. Some very closely related results are obtained in [ 2 ] ,[3], [4] and [12].

2. The Ryser-type conditions Lemma 1. Let A be an incomplete latin square of side r on symbols uI, . . . , un.For A to be embeddable in a latin square E of side n on uI, . . . , unin which cr,+,.. . . ,a, occur on the diagonal outside A, conditions (1) and ( 2 ) below are necessary: (1) N A ( a , ) * 2 r - n ( l ~ i ~ r ) , (2) N A ( u , ) * 2 r - n + 1 ( r + l S i < n ) . Remark 1. In the lemma it is not required that A be idempotent. If A is idempotent then E can be taken to be idempotent. Remark 2. The conditions (1) and ( 2 ) are similar to Ryser’s and we shall sometimes refer to them as the idempotent Ryser conditions. Proof of Lemma 1. Suppose that A is embedded in a square E as required. Let E be partitioned as indicated in Fig. 1.

n-r

E

Fig. 1.

Let S(i)=Oif i N A

S r,

S ( i ) = 1 if i

5r

+ 1. Then N D ( u , ) 2S ( i ) ( l G i S n ) . Hence

(a#)= NAUB(ff~)--BUD(U,)+ND(U,) =r -(n -r)

+

N D

(a,)

32r-n+6(i), which proves both (1) and (2). 0

Small embeddings

21

I t is easy to see that the idempotent Ryser conditions are not sufficient to ensure that embedding is possible. The square in Fig. 2 is taken from [12]; i t is of side 3 on symbols I , 2 , 3 , 4 , 5 and satisfies (1) and ( 2 ) , but it cannot be embedded in an exact idempotent latin square of side 5 .

4

Fig. 2

Moreover, whether an idempotent embedding is possible or not does not depend only on the number of times each symbol occurs in the latin square to be embedded. Fig. 3 exhibits two incomplete idempotent latin squares A and B of side 5 on symbols 1 , . . , , 8 and an embedding of A in an idempotent latin square E of side 8. The square B is not embeddable; this will follow from results in Section 4. But B satisfies (1) and ( 2 ) , as N A (i) = NB (i) for 1 6 i 6 8 ( A and B are identical except for cells ( I S ) and ( 3 3 ) ) . A

E

Fig. 3.

If n 3 2 r + 1 then (1) and (2) are always satisfied. On the other hand, if n S 2r (2) will not be satisfied if there is a symbol a,( r + 1 S i S n ) which does not occur in A. This shows that the bound 2 r + 1 in our result (mentioned in the Introduction) is best possible. When we want to extend an incomplete idempotent latin square A of side r


22

on c I ,,.. ,ansatisfying (1) and ( 2 ) to one, A ' say, of side r + 1 satisfying (1) and (2) we can single out certain symbols as being special. Consider for example a symbol a, (1 S i S r ) which occurs exactly 2r - n times in A . By (1) for A ' , a, must occur at least 2 ( r + 1)- n = 2r - n + 2 times in A ' . Thus a, must be used twice in the extension A ' \ A , which is a new row and a new column intersecting in cell ( r + 1, r + 1). We shall call such a symbol a, marginal. Had it occurred 2r - n + 1 times in A, we would have had to use it at least once in the extension. In general, we define the following: A symbol a, is marginal if either 1 =s i zz r and NA ( a r= ) 2 r - n, o r r + l s i s n and N A ( a , ) = 2 r - n + 1 ,

{

nearly marginal if

either 1 or r + 1

i

i

r and n and

NA( a , )= 2r - n NA(a,)= 2r - n

+ 1, + 2.

If we extend A to A ' of side r + 1 and we require that A ' must satisfy the idempotent Ryser conditions, we must use each marginal symbol twice and each nearly marginal symbol at least once, in both cases with the exception of the symbol placed in cell ( r + 1, r + 1); it is easy t o see that this symbol will satisfy (1) with r + 1 for r even if it was marginal, because then it occurs ( 2 r - n + 1) + 1 = 2(r + 1)- n times in A '.

3. The graphs G, and G, Given an incomplete latin square A of side r on symbols a i r .. ,. an, we let G, ( A ) denote the bipartite graph with vertex set { p , , . . . ,p , } U { a ; ., . . ,a:}, where p, is joined to a ;by one edge if and only if a, does not occur in the ith row of A, and there are no other edges. Similarly, we let G, ( A ) be the bipartite graph with vertex set { c l , .. . ,c,} U {a:,.. . ,a:},where c, is joined to CT',' if and only if a, does not occur in the ith column of A, these being the only edges of G, ( A ). Where there is no doubt about which square A we refer to, we shall write G,, and G,.Let A (G) denote the maximum degree of a graph G. Then

dc,(p,)=dGc(c,)=n-r ( l ~ i s r ) , (1 s j

dc,( a ; )= dcc(u;)

S

n).

Now suppose that A satisfies (1) and (2). Then, for example, d , , ( a ; ) = r - NA (a,)s r - (2r - n ) = n - r. In general we get

A (G, ) = d (G, ) = n

- r,

(1 s i s r, u, marginal)

3 &,,(a:)= dG,(a',')= n

- r,

23

Small embeddings

(1 S i

S

r, a, nearly marginal)

( r + 1 S i S n, u,marginal) (r + 1S i

S

3 d G P ( d=) dGc(u’:) = n - r - 1,

3 d C p ( u :=) dcc(u’i)= n - r - 1,

n, U~ nearly marginal)

3 dco(u:)= dG,(a’:)= n - r - 2 .

Fig. 4 illustrates G, and G,.

degree n - r

degree

n - r

Fig. 4.

I t is useful to modify these graphs slightly. Let G * , ( A )be obtained from G, ( A )by adding a new vertex p,,, joined by a single edge to each of CT,+,,. . . ,a,. If we want an embedding as in Lemma 1, then, just as the vertices joined to p , (1 < i S r ) in G, correspond to the symbols that can be placed in the i t h row, so do the vertices joined to p,I in G*,correspond to the symbols that can be placed in cell ( r + 1, r + 1). Likewise, G T ( A )is obtained from G, ( A )by adding a new vertex co joined by a single edge to each of d + ,..., , uz. We have

d c ; ( p , )= dc:(C,)= n

-r

(0 S i S r ) ,

d(G*,)=A(GT)=n-r, u,marginal 0;

3 &;(a:)= d g : ( v ‘ : )= n - r (1 zs i s n ) ,

nearly marginal j &;((T:)

= &:(cT:’)= n - r

-

1 (1 c i c n).

A set of edges in a graph is said to be independent if no pair of them have an end-vertex in common. We shall say that an independent set of edges uses a vertex if it is the end-vertex of an edge of the set. An independent set of r edges in G, must use all the vertices p l (1 S i S r ) . Given such a set, we can extend A to an incomplete latin rectangle of size r X ( r + 1) by placing a symbol u,in cell (i, r + 1) if and only if uj and p , are joined by an edge of the independent set. From the remarks about marginal and nearly marginal symbols in Section 2 we get the following extension lemma.

24

L.D. Andersen etal.

Lemma 2. Let A be an incomplete latin square of side r on symbols u,,. . . , un satisfying the idempotent Ryser conditions. Let r + 1 < j < n. Then A can be extended to an incomplete latin square of side r + 1 on the same symbols with a,in cell ( r + 1, r + 1) satisfying the idempotent Ryser conditions if and only if there is a pair E,, E, of independent sets of edges, E, in G, ( A ) and E, in G, ( A ), with the following properties : ( 3 ) IE, I = IEcI = r, ( 4 ) neither a: nor uy is used by E, U E,, ( 5 ) if a,is marginal, then both u:and a’,’are used by E, U E, (1 s i s n, i # j ) , ( 6 ) if u,is nearly marginal, then at least one of a:and u’i is used by E, U E, (1 s i s n, i# j ) . This lemma is the basis for the proofs of most of our theorems. The sets E,, and E, often arise as colour classes in proper edge-colourings of G, and G,. Observe that, if we consider proper edge-colourings of G : ( A ) and G ? ( A) with n - r colours and let E, be the set of edges of G, of the same colour as the edge joining a;to pll in GE, and also let E, be the set of edges of G, of the same colour as the edge joining a;to cfIin G then E, and E, will satisfy ( 3 ) . (4) and (5). The problem is to make (6) satisfied.

r,

4. Two families of non-embeddable squares

There are of course many incomplete latin squares on n symbols that cannot be embedded in an idempotent latin square of side n. When we use the term non-embeddable we shall refer to such squares that cannot be embedded despite the fact that they satisfy the idempotent Ryser conditions. In this section we define a family (which splits into two families) of nonembeddable squares. The definition is given in terms of G, and G , . We prove that an idempotent incomplete latin square A of side r on u 1 7 .. ,. urn satisfying (1) and ( 2 ) is non-embeddable (we always understand that the symbols to be placed on the diagonal outside A are u,+~, . . . ,urn),if G, has a connected component in which exactly one of the vertices u; corresponds to a nearly marginal symbol, the remaining all corresponding to marginal symbols, and G, has a connected component with vertices of degree less than 3 ( G , ) exactly corresponding to those in the component of G,. This can happen in only two ways, so we make the following more elaborate definitions. The square A is said to be of type I if there is a symbol u k , r + 1 S k S n, such that each of u,+~, . . . , a, different. from u k is marginal, and if in addition there is a nearly marginal symbol u ~1,s 1 s r, such that G, ( A ) has a connected compo. . ,uL}\{uL}in which all the other vertices have degree nent containing { u i ,

Small embeddings n - r (then

25

does not belong to this connected component), and such that G, ( A ) has a connected component containing {di, u:+,, . . . ,uK}\{uE}in which all the other vertices have degree n - r (again, uZ cannot be in this connected component). See Fig. 5. (+;

Example. Consider the square B in Fig. 3. The two graphs G,(B)and G, are shown in Fig. 6. 0

a;‘

51 ’

0

=I 0

degree n-r

degree

C

degree n-r-I

“‘I

1%

0

5”

G (A1

Gc(A)

P

Fig. 5. G , ( A ) and G , ( A )for a type I square A (with k = n and I 5”

1

52 I’

u 3” a4 ” 55 ”

a; a 7I ’ a BI’

Gc(Bl

Fig. 6.

=

r).

r

26

L.D. Andersen el al.

G, ( B )has a connected component with vertex set {pl,p z , uk,uL, a;},G, ( B )a connected component with vertex set {c3,c4, a;, ax, a:}. As symbol us= 5 is nearly marginal and symbols uh= 6, u7= 7 are marginal, B is of type I. The square A is said to be of type I1 if there is a nearly marginal symbol uI and a symbol a k , r + 1 S k, 1 S n, k# 1 such that each of ,...,un different from u k and uIis marginal, and G, ( A )has a connected component containing . . .,(+;}\{a;} in which all the other vertices have degree n - r (then a ; does not belong to the connected component), and G,(A) has a connected component containing . . , u:}\{u:} in which all the other vertices have degree n - r (thus a; is not in it). Fig. 7 illustrates this. With the above description of type I and 11, we shall call ffk the exceptional symbol. If n - r = 2 the exceptional symbol is not unique. We note that a square A of type I or I1 exists only if r is not ‘too small’: if A is of type I, ufmust occur 2 r - n + 1 times in A and so 2 r - n + 1 5 0, i.e., r 5 ( n - 1)/2. If furthermore A is idempotent such that uf occurs on the diagonal of A we must have 2 r - n + 1 2 1 and hence r 5 n / 2 . If A is of type I1 (and n - r > 2), the connected component of G, containing {a’+,, . . . ,uL}\{uL} must contain a further cT‘-vertex since any p-vertex of the component has n - r neighbours in it; the corresponding symbol occurs 2 r - n times, and we get r 2 n / 7 , and r 2 ( n + 1)/2 if A is idempotent. Theorem 1. A n incomplete latin square A of side r on symbols uI.. . .,v,, satisfying the idempotent Ryser conditions cannot be embedded in a latin square of side n on uI,. . . , a,,in which . . ,anoccur on the diagonal outside A if A is of type I or type 11. 0

degree n-r

0

c1

degree n-r a’ a’ r+l

f

deeree n-r-2

0

0

0

0

degree n-r-I ‘J’

n-1

‘J’

0 0

Fig. 7. G , ( A ) and G , ( A ) for a type I1 square A (with k

0

=n

and I

=r

+ I).

C

r

Small embeddings

27

Proof. Let A be as stated and let uk,ul be as in the definition of type I and type 11. Suppose that A is embeddable. Then A has an extension to an embeddable square A ' of side r + 1 in which uk occurs in cell ( r + 1, r + 1). This is obtained from the embedding of A by permuting two rows and two columns. By Lemma 1, A ' satisfies the idempotent Ryser conditions (with r + 1 for r ) . By Lemma 2, there are independent sets of edges E, and E, as described in the lemma. Let H,, be the connected component of Go( A ) containing {u,'+,, . . . ,aA}\{u;};assume that H,, contains exactly p of the vertices { p l , .. . ,p , } . Then H, has p ( n - r ) edges. If H, contains q of the vertices {ul,... ,(+A}, then, since all except {ui,u ; + ~ . .,. ,uA}\{u;}have degree n - r, H, has q ( n - r ) - ( n - r ) edges. Hence q = p + I . E, must contain an edge incident with each of p l , . . . ,p,, so it contains p edges from H,. Thus E,, uses all vertices of H,, but one. a; is not in H,,, so as E, uses all vertices u:for which a, is marginal, the vertex not used must be the only vertex u :of H, for which u, is not marginal, namely a ; . The same argument shows that E, does not use u';, and this contradicts (6) that either E, or E, must use the vertex corresponding to the nearly marginal symbol ul. Hence A is not ernbeddable. 0 We note that it is quite general for squares of type I or I1 that H, as defined in the proof has one more a'-vertex than p-vertices. In the proof of Theorem 1 it was shown that if A is of type I or type 11, then A cannot be extended with the exceptional symbol u k in cell ( r + 1, r + 1 ) . We can show, however, that A has an extension to a square A ' of side r + 1 satisfying . . . ,u n } \ { u kin} cell the idempotent Ryser conditions with any choice of {a,+,, ( r + 1, r t 1).

Lemma 3. Let n - r > 2 and let A be a n incomplete latin square of side r on ul,. .. ,onsatisfying the idempotent Ryser conditions. If A is of type I or type I1 with exceptional symbol c T k , then A can be extended to a latin square A ' of side r + 1 on the same symbols, also satisfying the idempotent Ryser conditions, with any of { u , + ~ ., ..,a,}\{uk} in cell ( r + I , r + I ) . Remark. If A is of type I then A ' will be of type I, and if A is of type I1 then A ' will be of type I or type 11. The proof of Lemma 3 is omitted.

Theorem 2. Let n 3 15. For each r, [ n /2I < r < n - 1, there exists an incomplete idempotent latin square of side r on n symbols satisfying the idempotent Ryser conditions which cannot be embedded in an idempotent latin square of side n.


28

Remark 1. That n be at least 15 is probably not essential, but it makes the proof work more easily. Remark 2. The proof is based on construction of a type I1 square. A very slight change would base it on a type I construction, but the construction would not work for n even, r = nI2. Proof of Theorem 2. Assume first that n is odd and r = ( n + 1)/2. Let B be an u4,.. . ,u,containing in idernpotent latin square of side r on the symbols a, p, u,,, the top left-hand corner the idempotent latin square C, of Fig. 8. Since r = ( n + 1)/2 z=7, B exists by a theorem of Hilton [9] about embeddings of idernpotent latin squares. Let A be obtained from B by substituting Czof Fig. 8 for C,and by replacing the remaining r - 3 occurrences of each of a and p by two occurrences of each of u ~ +. .~.,,u,,-~. This can be done because 2(r - 3 ) = 2(n - r - 2) and because the occurrences of a and p as a graph can be thought of as the disjoint union of even circuits. Construction of G,(A) and G , ( A ) immediately shows that A is of type I1 with exceptional symbol un.See Fig. 9.

c,

c2

=

Fig. 8 .

degree n-r

Fig. 9. n - r = r - 1

=

Small embeddings

29

If n is even and r = ( n + 2)/2 the argument is very similar, using D , and D , of Fig. 10. Here we start with symbols a, @, y, a,,us,.. . ,a, and replace the occurrences . . .~, , In this case it of a, p, y outside D , by 3 occurrences of each of u ~ + corresponds to a proper edge-colouring with n - r - 2 colours in which each colour occurs on three edges of a bipartite graph which is regular of degree 3 . The assumption n 2 15 is due to the fact that we need r 2 9 . For r > [ n / 2 ]+ 1 the theorem now follows from Lemma 2, because extensions of the two squares just constructed must be non-embeddable. 0

5. Extension results First we give a result showing that the squares of type I and type I1 in a way are the only ones that cannot be freely extended to a square also satisfying the idempotent Ryser conditions.

Theorem 3. A n incomplete latin square of side r on a,,. . . ,a,satisfying the idempotent Ryser conditions can be extended to an incomplete latin square of side r + 1 satisfying the idempotent Ryser conditions, with any of a,+l,. . ., anin cell ( r + 1, r + I), if and only if it is not of type I or type 11. The proof is omitted (it is fairly lengthy). Theorem 3 might cause one to conjecture that squares of type I or I1 are the only non-embeddable squares. While we may as well admit that this is not true, we do have another result pointing in the same direction, Theorem 4 below. The best one can hope for now, is that there are only ‘few’ non-embeddable squares that are not of type I or I1 (for example, all the non-embeddable squares not of type I or I1 that we know of at present have n - r S 4). If this is so. Theorem 4 may be a useful tool in the final classification of embeddable squares.

, a

o1

=

Y

a

On

6

Y

O1

On

Y

a

On

a

Oi-+I

5

‘4

0

O

Fig. 10.

n

Or+l

On

u2

On

03

‘n

O3

Or+?

O?

Or+I

O4


30

ThForem 4. Let n - r 3 5 . Let A be an incomplete latin square of side r on (T,, . . . ,a,,satisfying the idempotent Ryser conditions, and suppose that A is not of type I or type 11. Then A can be extended to an incomplete latin square of side r 1 on the same symbols, which also satisfies the idempotent Ryser conditions and is not of type I or type 11, with one of 0;+], . . .,onin cell ( r + 1 , r + 1).

+

Remark. We have not proved the same result with any of u,+,, . . . , un in cell ( r 1, r + 1); this of course would have to be true if A is embeddable.

+

The proof of Theorem 4 is omitted (it is extremely long). Theorem 4 is not true for n - r = 3. It may be true for n - r = 4,but the proof would need several additions to cover this case, and in view of the fact that there are non-embeddable squares with n - r = 3 and n - r = 4 that are not of type I or 11, Theorem 4 by itself would not be adequate for the case n - r = 3. We have not been able to find any non-embeddable squares with n - r = 5 that are not of type I or 11. We conclude with an example of a non-embeddable square with n - r = 4 ( n = 16, r = 12) which is not of type I or type I1 (Fig. 11).

Fig. 1 1 .

Small embeddings

31

References [ I ] L.D. Andersen. Latin squares and their generalizations, Ph.D. Thesis, Reading, 1979. [2] L.D. Andersen, Embedding latin squares with prescribed diagonal, Ann. of Discrete Math. 15 (North-Holland, Amsterdam, 1982) pp. 9-26. [3] L.D. Andersen, R. Haggkvist, A.J.W. Hilton and W.B. Poucher, Embedding incomplete latin squares in latin squares whose diagonal is almost completely prescribed, Europ. J. Combinatorics 1 (1980). 5-7. [4] L.D. Andersen and A.J.W. Hilton, Thank Evans!, submitted. [5] L.D. Andersen, A.J.W. Hilton and C.A. Rodger, A solution to the embedding problem for partial idempotent latin squares, J. London Math. SOC.,to appear. [6] A.B. Cruse, On embedding incomplete symmetric latin squares, J. Comb. Theory, Ser. A 16 (1974) 18-22. [7] A . Cruse, On extending incomplete latin rectangles, Proc. 5th 'Southeastern Conf. on Combinatorics, Graph Theory and Computing. Florida Atlantic University, Boca Raton, Florida (1974) 333-348. [8] T. Evans, Embedding incomplete latin squares, Amer. Math. Monthly 67 (1Y60) 958-961. [9] A.J.W. Hilton, Embedding an incomplete diagonal latin square in a complete diagonal latin square, J. Combinatorial Theory A, 15 (1973) 121-128. [lo] A.J.W. Hilton, Embedding incomplete latin rectangles, Ann. of Discrete Math. 13 (1982) 121-138. [ I 11 A.J.W. Hilton and C.A. Rodger, Edge-colouring regular bipartite graphs, Ann. of Discrete Math. 13 (1982) 139-158. I121 C.C. Lindner and T. Evans, Finite embedding theorems for partial designs and algebras. Collection siminaires de mathimatiques supkrieures. Vol. 56, Les presses de I'Universiti de Montreal, Montreal, 1977. [ 131 R.J. Ryser, A combinatorial theorem with an application in latin squares, Proc. Amer. Math. SOC.2 (1051) 550-552.



DEGREES AND LONGEST PATHS IN BIPARTITE DIGRAPHS Jacqueline AYEL Universite'de Savoie, Faculte' des Sciences er des Techniques. B P 1104. 7.?01I Chambery. France In this paper we give a sufficient condition on the degrees of the vertices of a digraph to insure the existence of a path of length greater than or equal to a given value, when the digraph is bipartite and when the digraph is bipartite and oriented. These conditions are shown to be best possible.

Introduction We use standard terminology (see Berge [2]). A digraph D = ( X , A ) consists of a finite set of vertices X and a set A of ordered pairs (x, y ) of vertices called arcs. In what follows the considered digraphs are without loops, o r multiple arcs. We denote by

T ' ( x ) = { y , y E X / ( x , y ) E A } and

d+(x)=lf+(x)l,

T - ( x ) = { y , y E X / ( y , x ) E A } and

d - ( x ) = (r-(x)I,

d ( x ) = d'(x)+ d-(x). We will say that a digraph is bipartite, if there exists a partition of the vertices of X , X = X I U X , such that for any i = 1 , 2 there is no arc between the vertices of X , . We will say that a digraph is oriented if it contains no cycle of length two. A digraph is strong if for any two vertices x and y there exist both a path from x to y and a path from y to x . When we speak of paths (cycles) in digraphs, we always mean elementary directed paths (or cycles). The length of a path (cycle) is the number of arcs of this path (cycle). First let us recall some results for general digraphs. Theorem A [3]. Let D be a strong digraph of order n satisfying d ' ( x )2 k and & ( x ) k h for every vertex, then D contains a path of length min{h + k, n - 1). (Extremal graphs are characterized.)

Theorem B [4]. Every oriented strong graph of minimum in-degree and outdegree k 3 2 contains a path of length rnin(2k + 2, n - 1). 33

J. Aye1

34

Here we consider bipartite digraphs and give a lower bound of the length of a path in the general case (Theorem 1) and when the digraphs are oriented (Theorem 2). These results are to appear in Discrete Mathematics [l] with only a sketch of the proofs. We give here the complete proof of Theorem 1. The proof of Theorem 2 is similar and we refer to [l]. Theorem 1. Let D be a bipartite digraph of order n satisfying for any vertex x, d ' ( x )3 k and d - ( x ) 3 h ; then D contains a cycle of length at least 2max{h, k}. Furthermore D contains a path of length min{n - 1,2(h + k ) - 2) except if each component D, of D is a component source and sink, and consists of a cycle of maximum length in D, plus vertices joined only to this cycle. Proof. We will use the following immediate lemma. Lemma. Let P be a longest path in a bipartite digraph starting at u and ending at u. I f d - ( u ) * 1 , Dcontainsa cycleof length at least 2d-(u). I f d ' ( v ) 2 1, Dcontains a cycle of length at least 2d'(v).

Let X , and X z be the two parts of D. If x E X I , T - ( x )U T ' ( x ) C X 2 implies Similarly 1 XI1 L rnax{h, k} and thus V ( D )L 2max{h, k } 3 k + h. We can suppose that D is strong. Otherwise D contains a 'component source' (every vertex of this component is not the end vertex of an arc with origin in another component) and a 'component sink'. If there exists a component source and sink, it suffices to apply the theorem for this component. Otherwise by the lemma, as d - ( x )3 h for every vertex of a component source, there exists a cycle of length at least 2h. Similarly in any component sink there exists a cycle of length at least 2k. As there exists a component source and a different component sink with a path between, then we have a path of length 2(h + k ) - 1 in D. Now let C be a longest cycle in D , of length C. By the lemma, C > 2max{h, k } . If D is hamiltonian the theorem is proved since V ( D )5 2max{h, k}. Otherwise let P be a longest path of D - C of length I, starting at x and ending at y. Let

1 X , 13 max{h, k } .

s = l f - ( x ) nC

/ (number of arcs from C to x ) ,

r = ) f + ( y ) nCI

(number of arcs from y to C).

If r = 0 (resp. s = 0) by the lemma applied to P in D - C,D - C contains a cycle of length at least 2h (resp. 2k). Thus if r or s is zero, D contains two disjoint cycles of length at least 2 max{h, k } and 2min{h, k } . As D is strong we

Degrees and longest paths in bipartite digraphs

35

obtain a path of length at least 2 ( h + k ) - 1. If r 2 1 and s 3 1 a calculation using the maximality of C and the fact that D is bipartite shows that if 1 = 0 ,

Ca2max(r,s)

C 3 2r + 2s + 1 - 1 if 1 is odd (Fig. l), C 3 2r

+ 2s + 1 - 2

if 1 is even (Fig. 2 ) .

If 1 = 0, r 5 k , s 5 h implies c L 2max{h, k } and D - C consists of isolated vertices. Otherwise the lemma applied to P in D - C shows that 12 2 ( k - r ) - 1 and 1 2 2 ( h - s ) - 1, the inequalities being strict if 1 is even. Thus we obtain that:

As s

2

2r+2s22h+2k-21-2

if 1 isodd,

2r+2ss2h+2k-21

if 1 is even.

1 the cycle C and the path P give rise to a path of length l+c32(h+k)-3

if 1 isodd,

1 + c 2 2 ( h + k ) - 2 if 1 is even. We will prove that 1 + c 2 ( h + k ) - 2 also for 1 odd. Consider the following assumption (H): 1 is odd, 1 + c = 2 ( h + k ) - 3 and there is no path of length greater than 1 + c in D - C. Then 1 = 2 ( k - r ) - 1 = 2 ( h - s ) - 1 implies k - r = h - s. Put C = {u,,,. . . , and P = {ue, u I , . . . , u,}. As 1 is odd, if uoE X I , ur E X z and I r - ( u , J nPI = IP n X , I = h - s , Ir+(u,)nPI = I P n X I (= k - r . Since 1 + c = 2 ( h + k ) - 3 , we can suppose that the numbering of C is such that for any i E [0, I ] , u , E f - ( u O ) ,and u , E T ' ( s ) :

r+(ur) n c, ~= ~r -+ (~~n ), ,c. )

{W+,, w + ~ ., . ., U , + ~ , -=~ J

{ u ~ -u~~, -. .~. ,,u ~ -

P)

Fig. 1. I odd.

Fig. 2. I even.

J. Aye1

36

As f-~(uo) n P = P fl X 2 , uI E f - ( u o ) and (u", uI, . . . , u l ) is a cycle. Thus ~. . . ,~ul, uo,. ~ .~. ,u z , - l ) is a maximal path of D - C. Consequently r-(uZi) n P = P n X I ;r+(u2i-l) n P = P n X ? . Similarly r-(uzi+l)n P = P nX2; r+(uZi)n P = P 17 X I . This implies that 9 ' = (uo, u l , , , ,,u , } is a complete symmetric bipartite digraph. For any u2i E XII\ P, there exists in CP a path of length I from uZi to u, and thus f - ( u 2 i ) r )C = f - ( u o ) f lC. Similarly, for any uZi+,E x2n P, r+(uZi+,) n c = r+(u,) n c. . ., u = - ~ }c' . is a longest cycle in D and by Put C'={uu, ul,. . . , uI, assumption (H) P' = (uu,uI,.. . , uI)is a longest path in D - C'. Thus 9' = (u",uI,.. . ,u I }is a complete symmetric bipartite graph, and, as before, for any uZiE P' r l XI, f - ( u z i )fl C'= T ( u 0 )f l C', and for any u2i+lE P' n X z , f + ( u z i + l ) nC'= T'(u1)n C'. Put Q = ( ~ ~ , u ~..-,~uo), , . Q is a longest path in D - C. As ( r ~ ~ ( uQ , )I n= h - s = k - r = I T'(vo) n Q I and by maximality of Q, I r+(un) f l C I 2 r and I f - ( u r ) n C l 3 s. There exists a numbering of C, fu&.. . ,U C ' - ~ ) such that for any i E [0, l ] , K : ~ ZT - ( s )and u : E f ' ( u " ) : ( v z i ,u

r+(uo) n c, = T-(ur)nc.

( u ; + ~u, ; + ~. ., . , UI:~,-J = , { u 1, as 1 is odd there exists u ~ , E+ P~ n X z , u2i+l# ut,. Put P'(ua -+u2i+l) the maximal path from u, to uzitlin 9".Its length is 1 - I .

Fig. 3.

Degrees and longest paths in bipartite digraphs

37

For uI-l E XI n P,put P(uf-l+uo) the maximal path from uI-l to uo, in P. If 1 > 1 there is uf-I # u,, and the length of this path is 1 - 1. As f + ( u Z i + ,n) C ' = I " ( u f )n C' and ulelE f'(u,) f l C' then uI+IE ~ + ( u , ~ + As J . f-(u,-J n C = f --(oil) n C and V,-l E f-(uo) flC then u,-~E r - ( ~ ~ Consequently -~). (ull,u,, 9"(u, --* u ? ~ +u~!)+,~.,. . , u = - ~u,~ -P~(, U ,+ - ~uO))is a cycle of length 1 + c 1. Thus if 1 > 1, this cycle is longer than C, the maximal cycle, and the assumption (H) is false. Consequently if (H) is true, 1 s 1 and as 1 is odd 1 = 1. Now we will prove that if (H) is true and 1 = 1 the maximal path of D is hamiltonian. Suppose that D - C U P is not empty. Let P' be the path of D - C U P of maximum length l ' , and 1'6 1. First remark that P = { u o , u l } as {uo,uI} are 2-cycle and there are two maximal cycles in D ; C = { u o , u , , . . .,u,_~}and C' = {u,,, uI, u 2 , .. . , u ~ - ~Furthermore }. as the path (u", u l ) (resp. the path ( u , ~ul)) , is maximal in D - C (resp. in D - C') for any w belonging to D - C U P, T(W)

n {u0,uI, uIl,v l ) = 0,

{Ur,UJ,. . .

7

Uzk-?}

= f'(U1)

and

..

{uc-l, u c - 7 , .

7

U?k-l}

=

nc

r-(uO)n c.

(1) Suppose that I' = 0. Then P' = {wo} and T ( w o ) C{u,, u 3 , .. . , u ~ - n ~ Xi } if w a € Xi with j # i. ( r + ( w , , )3J k , as [{u,, u 3 , .. . , uzk-,} n Xi 1s k - I there is u p ~ f + ( w c with l) ~ ~ ~ { u ~ k - ,..., I , u u ~, - k, } n X , . I f - ( w o ) [ a h h ,as I { u ? k - I , u 2 k , . . . , u,-~} n xi 1 s h - 1 there is u, E f - ( w l l ) with u, E { u ? ,u Z , .. . , u z k - , }f l X i . Thus if wIIE XI (resp. if w,,E X , ) there exists a path (Uptl..

,

.

7

UI), UI,.

. . urn',w ! l ~u p , UIJ,UI, uo+l,.. . t

7

up-1)

(resp. a path (u,+~, . . . , u p - l ,uII,uI, u,, wfj, u p , u p + l , .. . , UO,u I , .. . ,u,-I)) of length greater than 1 + c. This contradicts the assumption (H). (2) Suppose that I' = 1. Put P' = { wo,w As P' is a maximal path of D - C, P' is a 2-cycle and { w , , wo} is a maximal path of D - C. w oE XIand w I E X,. Therefore there is a numbering of C, {u6, u l , . . . , u f ( u ) entraine I’inCgalitC f ( u ) > f(t). On verra que cette dernikre condition sufit B Ctablir certains rCsultats sur les sommets qui rendent f minimum. Notons qu’elle n’est pas iquivalente B la premiere (prendre sur les arites de G la longueur unit6 et pour f, f ( u ) = 1, f ( t ) = 3, f ( u ) = 4). Definition. On dit qu’une fonction f, dCfinie sur S et 9 valeurs rCelles est brunchbe sur G lorsque, pour tout u, t, u E S distincts et tels que ut E U, ut E U, I’inCgalitC f ( t ) - f ( U 1 > 0,

entraine I’inCgalitC

f ( v 1- f ( 0 > 0.

Caractirisation midiane des arbres

41

Ainsi, sif est branchCe sur G, pour tout chemin tll, t l , . . . , t, de G, on aura

dks que

f (to) < f(tl). Pour une fonction ii valeurs rielles f quelconque, on pose

M C f ) = { u / f ( u ) ~ f ( tpour ) , tout t E S } , M l , , , ( f ) = { u / f ( u ) ~ f ( pour t ) , tout t E s tel que ut E U } .

II est clair que Mcf)C MI,,#). Proposition 1. Pour une fonction f, branche‘e sur G , il vient, (i) M u ) = Mldf); (ii) Mcf) est l’ensernble des sommets d’un sous graphe connexe (donc d’un sous-arbre) de G. Preuve. Soit s E Mlc,ccf) et t E M u ) , supposons t# s et considirons le chemin s = s,,,sI,.. . ,s, = t entre s et t. On a alors, soit f(s,)> f(s), soit f ( s l )= f(sll). Dam le premier cas, il vient f ( t ) > f(s), ce qui est impossible, nous devons donc supposer f(s,)= f(so).Lorsque f ( t ) = f(s), s E M U ) . Sinon, on dCsigne par j le plus petit indice tel que f ( s , ) # f ( ~ , + 1~ f(s,) = f(s), ce qui est impossible. Si f ( s , ) > f ( ~ , + il~ )vient , f(s,-l)>f(sl)de qui contredit la difinition de j . Le seul cas possible est donc f ( t ) = f(s). Ce qui dCmontre I’assertion (i). Pour I’assertion (ii) il suffit de considkrer deux sommets s et t dans M u ) et de remarquer que sur le chemin s = sII, sI,. . . , s, = t qui les relie, on ne peut avoir que f ( s ) = f(sl) = . * = f ( t ) . 0

La fonction D, dCfinie dans I’introduction est un cas particulier des fonctions D,. Pour t E S et A C S ( A non vide), on pose D,(t,A)=

( c d(t,s)P)I” SEA

pour p < TX;,

et D,(t,A)= m axd(r,s). ,€A

Proposition 2. Pour tout entier p , fini ou non, et pour tout A C S, la fonction f ( f ) = D, ( t , A ) est branche‘e sur G.

J.P. Burrhilerny

42

Preuve. Soient u, r, u trois sommets distincts de G tels que ut E U et fu E U. Soit A C S. O n cansidkre la partition e n trois classes A,,, A,, A, d6finie par: s E A, si et seulernent si d ( s , u ) < min(d(s, t ) , d(s, u ) } ; s E A, si et seulement si d(s, r ) < min(d(s, u), d(s, u ) } ; s E A, si et seulement si d(s, u ) < min{d(s, u), d(s, f)}. Supposons d'abord p } and V;' = { x L I ...,x ~ - ~ Then }.

The existence of D (n,p ) in directed graphs

40

d , ( x ) + d L ( x ) = d;l;(x)+d;.;(x)+ e(x,-I+x)+e(x--+xq).

If ( x , - , , x ) or ( x , x , ) f f E ( D ) , we have d i i ( x ) + d ; , i ( x ) 2 q - 1. The induction can be applied to D - x, which contains a directed hamiltonian path from x 1 to the same can be extended by adding the edge ( X , . . ~ , X , ) . If ( x ~ . x~ ), and ( x , x,) E E ( D ) , the required path is evident. Theorem. If D has n vertices and e ( D )3 (n - l)(n - 2 ) + 2, then D contains D (n,p ) , 2 s p G n, except D is isomorphic to M (n) or n = 5 and D is isomorphic to N. Proof. The proof is by induction on n 3. Assume that if D' is a subdigraph of D,with n - 1 vertices and e ( D ' ) * (n - 2 ) ( n - 3 ) + 2 then, either D ' is isomorphic to M ( n - 1) or N , or D' contains D ( n - I , p ) , 2 s p n - 1. As the cases n = 3 and n = 4 are simple, assume n 2 5. Let x E V ( D )and D ' = D - x. (1) If D' is isomorphic to M ( n - l), let a be the pendent vertex o f D'and b its neighbour; we have d ( x ) = e(D)- e(D') 2

( n - 1)(n - 2 ) + 2 - ( n - 2 ) (n - 3 ) - 2 = 2(n - 2).

Either (x, a ) P E ( D ) ,then D is isomorphic to M ( n ) which does not contain any D ( n , p ) , for every p , or (x, a ) E E ( D )but, as there exists y E V ( D ) - { a ,b , x } connected to x by an edge, D ( n , p ) is immediate. ( 2 ) If D' is isomorphic to N, we have d ( x ) a 8 , and D ( 6 , p ) exists. (3) Assume that D' contains D(n - l , p ) , 2 S p S n -1. As D ( n , p ) and D ( n , n - p + 2 ) are isomorphic, discussion on D ( n , p ) can be limited to n 2 3p - 2 . Assume that D does not contain D ( n , p ) . (3.1) If n + p - 3 s d ( x ) s 2 n - 4 , then e ( D - x ) = e ( D ) - d(x)

s ( n -l)(n - 2 ) + 2 - 2 n + 3 = ( n - 2 ) ( n - 3 ) i - 2 . By the induction D - x contains

D ( n - l , p ) = [ X I , . . . ,xn-p+l;xr = Y I , . . .

t

Yp = ~ n - p + i ] .

Let us put V ' = { x ,,..., x - ~ }and V"={xZ,..., x " - ~ + ~ }By . Lemma 2, d ; . ( x ) + d $ ( x ) s n - p , moreover, e ( y p - l , y p - + x ) S 1 and e ( x -+y l , y Z ) c 1, but d ( x ) = d,.(x)+ e(y,-,,y,-,x)+

e ( y 2 , .. . , y P - : ! + x )

+d~,~(~)+e(x+y,,y:!)+e(x-,y.~,...,y,-~)

~n-p+2+2(p-3)=n+p-4,

hence, we have a contradiction with d ( x )

n

+ p - 3.

A. Benhocine, A.P. Wojda

50

(3.2) If every vertex of D has degree 2 2n - 3, then D is complete and the antisymmetric edges form a matching; D(n,p ) is immediate. (3.3) If every vertex of D has degree s n + p - 4, then e ( D )s in (n + p - 4), which contradicts the inequality e ( D )3 (n - l)(n - 2) + 2. (3.4) Assume that there exists no vertex x such that n + p - 3 < d ( x ) s 2n -4. Let us consider the partition X U Y of V ( D )defined by X={x~V(D)[d(x)a2n-3}, Y

= {y E

v ( D ) (d ( y ) c n + p -41,

let I X ( = k > O , let

IYI

=I

>o.

Let us partition X as follows:

1

XI = { x E x d (x) = 2n - 2}, XzUX3={xEXJd(X)=2n-3}, Xz induces Kl:21minus the edges of a perfect matching, and X, induces K i f , ~ . Note that the subdigraph induced by X contains a directed hamiltonian cycle C*, with symmetric edges, except if I X z l = 2 (if ) X , (= 2 and XI = X, = P), C* is an antisymmetric edge, if I X, 1 = 2 and XIU X, # 0, C* is a cycle containing at most one antisymmetric edge). Also, it is possible to find Y l = { y l , .. . , y , } C Y and to partition X, in C, so that for every x E C, the edge (x, y , ) is ) O U ,P+i ~s antisymmetric(i = 1, ..., r).Letuschoose C * = P O ( ~ : = ~ ( U , ~ Q , )where a hamiltonian dipath in X I U X,,of which all the edges are symmetric except one if IX I = 2, Q, is a symmetric hamiltonian path of C, ( i = 1,. . . ,r ) , u, is a symmetric edge from the end vertex of to the initial vertex of Q, ( i = 1, ..., r + 1) by assuming that Q o = Qril = P. A vertex y of Y verifies d(y) = (2n - 1) - d(y) 3 2(n - 1) - ( n + p - 4) = n - p + 2, therefore 2 I(n - p 2)/2. Furthermore,

u:=,

e(o)

+

e ( D )= n ( n - 1) - e ( D )=z n (n - 1) - (n - ~ ) ( n- 2 ) - 2 = 2n - 4; this implies 1 S (4n - 8 ) / ( n- p + 2). (3.4.1) If 1 = 1, it is possible to connect the unique element of Y to two consecutive vertices of C,then D ( n , p ) is immediate. (3.4.2) If 2 S 1 s k , we have

1 Y - YII= 1 Y J- r s J X Iu x , ( +

c

(JC,I- 1).

t = I

The second member of this inequality expresses the edge-number of C * - {u?,.. . , u,+~}.It is then possible to replace the edges of the last set by symmetric paths of length 2, with the same ends, containing the elements of Y - YI (we start by replacing the antisymmetric edge of X , if ) X , )= 3). The edges u 2 , . . . , u,+, are replaced by the symmetric paths with the same ends of length 2, containing y,, y l , . . . , yr-l respectively. Thus, we have constructed a

The existence of D(n,p ) in directed graphs

51

directed hamiltonian cycle of D with, at most, two antisymmetric edges. We deduce D (n, p). (3.4.3) If 1 3 k + 1, then 1 > n/2, hence n (n - p + 2)/4 < e ( B )s 2n - 4 and n 2- n ( p + 6) + 16 < 0; inequality verified only if p 2 + 12p - 28 > 0. We deduce 4 p - 4 S 2 n < p + 6 + d p 2 + 1 2 p -28. If p 2 4, we have ( 3 p - lo)’ < p’ + 12p - 28, then p G 6, but p’ + 12p - 28 < ( p + 3 ) ’ , i.e., 2 p - 2 6 n S p + 4 S 1 0 . I f p S 3 , then n S 6 . As n/2 < 1 G (4n - 8)/(n - p + 2) S (8n - 16)/(n + 2) (because p C (n + 3)/2), the cases to study are n = 9 and 1 = 5, n = 7 and 1 = 4, n = 5 and 1 = 3, n = 6 and 1 = 4. Note that the three first cases verify 1 = k + 1. Case (a) 1 = k + 1. Let x, E C,, i = 1 , . . . , r and X i = { x I , . . . ,x,}. We have / X i /= I Y ,1, IX - X i ( = 1 Y - Y ,1 - 1. We put s = k - r and denote t l , t 3 ,..., t z r + , the vertices of Y - Y l and t 2 r r 4 , . . . , f 2 r the vertices of X - X i . The sets E ( X I , YI) and E ( X - X i , Y - Y,) form KT(Xi, Y I ) and K * ( X - Xl. Y - U , ) respectively; furthermore, e(Xl, Y - YI)= 2 ) X I l Y - Y , l (the antisymmetric edges of E ( X i , Y , )are (x,, y,), i = 1 , . . . ,r ) . E ( Y ) # 0 , otherwise (n - I ) ( n - 2 ) + 2 s e ( D ) s 2 ( k + l ) k + k ( k - l ) , which, with n = 2k 1, implies either k = 1 or k = 2. If k = 1, D is isomorphic to M(3) and if k = 2, D is isomorphic to N. Let (a, b ) E E ( Y ) . Let us consider the following cases and subcases and the corresponding hamiltonian cycles C‘ (i = 1,. . . ,9). These cycles are constructed in such a way that all the edges are symmetric except perhaps two. So we show the existence of D ( n , p ) in D for each case by Lemma 1. (a.1) ( a , b ) E E ( YI, Y - Y l ) .We put a = y, and b = t , . r 2 4: C’ = (x,. yj, xz, yl, x3, y q , . . . ,~ ~ - y,-l. 3 , x,-:, y1,xr-,, y2, x,, Y , , tl, f 2 , . . .,t2s,f Z r + l , xl), r = 3: C’ = (xl. y,, x3, yl, x?,y3, tl, t ? , . . . , f Z q , flril,xl), r = 2: C‘ = (x’, y l . x l , y z , tl, t ? , . . . , rZE,t 2 r + l , ~ ? ) . r = 1: C‘= (x,, y l , tl, rz,. . . , tZr,t 2 r + lxl). . (a.2) (a. b ) E E ( Y , ) . We put a = yl and b = y z . r 3 2: C’= (y1, y:, X I , y3, Xz, y 4 , . . . , X,-Z, y , , x , - , , tl, t z , . . . ,l Z \ , t ’ s t l . x,, y,). (a.3) ( a , b ) E E ( Y - Yl). We put a = f l and b = t 3 . r 2 3: C6= (x,, y3, xz, y 4 , . . . ,x,-Z, y,, x,-I, yl, x,, yz, t r , tl, t i , h , . . . , t z s r f z s + l . X I ) . r = 2 : C’ = (x2,y , , xI,y r , r:, tl, t 3 ,t4,. . . ,i Z r t, Z s + l x2). r r = 1: One can assume that E ( Y I ,Y - Y , )= 0, if not we would have one of the preceding cases. One of the edges (y, t‘,), i 2 1, is then symmetric (if not 3k + k(k - 1)/2 s e ( D )6 2n - 4 and while k > (n - 2)/2, we have n = 3 or n =4). s 3 2: Cx= (x,, t , , tj, r4,. . . , t z St.l s t l , r2, yl, x,), s = 1: one of edges (yl,tl) or ( t l , r 3 ) is symmetric in D. We put C’=

I

+

(XI,

yl, ti?, tl,

tbX1).

52

A. Benhocine, A.P. Wojda

Case (b) n = 6 and 1 = 4 . We have e ( D ) 2 2 2 . We note X = { a l , u 2 } and Y = { b l , bl, b l , b,}. As in the case (a), we consider the following subcases and the hamiltonian cycles C‘, i = 10,. , . , 17, having the same property. (b.1) E ( Y ) contains two consecutive symmetric edges, for example bib, and b,b,. We put C“’= (al,b l , b,, b,, a:, b,, a,). (b.2) E ( Y ) contains two non-consecutive symmetric edges, for example bib, and b,b,. Then C” = (al, b , , b2,a2,b,, b4, a l l . (b.3) E ( Y ) does not contain a symmetric edge. Consider four subcases: (i) The subdigraph D ( Y ) induced by Y is a tournament and (a,,b,) antisymmetric, i = 1,2. (ii) D ( Y ) is a tournament, (al, b l ) and (a,, b , ) are antisymmetric. (iii) D ( Y ) is a tournament minus one edge and ( a l ,b l ) is antisymmetric. O n e can assume that the edges ( b , ,b2)and (bl, b,) exist, all edges between X and Y

are symmetric except perhaps one. (iv) D ( Y ) is a tournament minus 2 edges. O n e can assume that ( b l , b , ) € E ( D ) and (bl, b4) E E ( D ) . In all these subcases, the other edges of E(X,Y ) are symmetric. We put C” = (bi, bz, a ] ,b3, ~ 2 bq, , br). (b.4) E ( Y ) contains exactly one symmetric edge b i b 2 .So there exists an edge ( b e b,), , i s 2, j 2 3, for example ( b z ,b J . (i) ( a l ,b l ) antisymmetric: C” = (al, b,, b2,b , , az, b4, al). (ii) a l b l , a r b 4E E ( D ) and (az,b,) antisymmetric (so arb, E E ( D ) ) : CI4= (ai, bi, bZr b3, a2, b,, ail. (iii) a l b lE E ( D ) and ( a l ,b,), ( a 2 ,b,) antisymmetric: C” = (ar, bi, br, b,, ai, b+ az). (iv) alblE E ( D ) and u2b3E E ( D ) . If o n e of the edges ( a l ,b,) or (azb,) is symmetric, we put C’”= (al,bl, b2. 63, ~ 2 b,,, ~ 1 ) . If these two edges are antisymmetric, (61, b4) or ( b 2 , b q )exists in D, if not e ( D ) < 2 2 . We put for example ( b l , b , ) E E ( D ) , then C”= (al, b z ,b l , b4, az,bi, al).

References [ I ] C. Berge, Graphes et Hypergraphes (Dunod, Paris, 1973). [ 2 ] J.-C. Bermond, A. Germa, M.C. Heydeman and D . Sotteau, Chemins et circuits dans les graphes orientts, Proc. Coll. Montrtal, Ann. Discrete Math. 8 (1980) 293-309.


HAJOS’ THEOREM FOR HYPERGRAPHS C . BENZAKEN I.M.A.G., BP 53X, Grenoble, France

We extend the well-known Hajbs’ graph-theoretical theorem to hypergraphs.

1. Introduction

We have proved the following in 111: Any hypergraph H with weak chromatic number x ( H ) > q 3 3 can be obtained from the q-complete graph by means of reductions and compositions. This result lcoks like the well-known Hajbs’ theorem for graphs [3] but, at the same time, is far from it because the composition operation is really distinct from the Haj6s’ construction which starts from two graphs while the composition needs q initial hypergraphs. The version presented here is closer to Hajbs’ theorem both for the basic operations and for the proof method.

2. The Hajos theorem for hypergraphs 2.1. Notation Without loss of generality, we may suppose that the hypergraph H has no isolated vertex and is Sperner. In other words, H is a finite (nonempty) set of finite sets (the edges).

H

= {el,

e,, . . . , e m } such that

i # j .$ e, and e, are uncomparable with respect to inclusion

and with vertex-set: m

VH=

U

ei

i=l

(so H has no isolated vertex). The weak chromatic number x ( H ) of H is the least integer k such that there exists a map (colouring) 53

C. Benzaken

54

c :VH+{1,2,. . . , k } satisfying Vi

lC(e,)la2

(IS 1 denotes the cardinality of a set S ) . If some edge e, is such that I e, 1 s 1 then H is said to be singular and we put in this case x ( H ) = 00. The following partial order between hypergraph:

H s H’

def

e For each edge e E H

there exists an edge e ’ E H‘ with e 2 e’

plays an important role [2] mainly because H S H ’ j X(H)SX(H’).

Recall that a stable (or independent) set of H is a subset S of V Hcontaining no edge of H. In other words S is a stable set of H

{S}P H .

We consider the Sperner simplification aH of a (possibly non-Sperner) hypergraph H, by deleting from H all nonminimal edges of H (with respect to inclusion order) and we adopt the notation:

H v H ’ = u ( H u H‘). 2.2. The basic operations We consider three basic operations, the first two dealing with one hypergraph, the third with two hypergraphs. They are called augmentation, reduction (of a hypergraph) and Hajbs’ construction (from two hypergraphs).

Augmentation. This operation starts with a given hypergraph H and adds to it a new edge S (and possibly new vertices if Sg V H )giving the hypergraph H’ = H v { S } . We note that in general H < H’ except when { S } C H. On the other side, any H’ with H < H ’ is obtained from H by repeated augmentation operations.

Reduction. Given the hypxgraph H = {el,e2,. .. ,em} and a map IZ : V, --., X where X is any finite nonempty set, we call HH = { H e , , 1 7 e z , .. . , H e , } the reduction of H (via n).This operation consists in identifying all vertices of VH having the same image by n. We can define a more elementary concept of

H a j h ’ theorem for hypergraphs

55

reduction (identification of only two vertices) so that the general reduction operation is the composite of elementary ones. Note that in the case of a graph, the corresponding elementary Hajbs’ operation consists only in identifying two nonadjacent vertices. This restriction is not kept here because x ( I 7 H )= 50 as soon as a nonstable set of H is reduced to a singleton by 17.

Hajos’ construction. Consider two hypergraphs HI,H2.By taking a copy of one of them, we suppose that VH, n VH?= { x ) .

Take one edge ei = { x } U ai ( x g a i ) in each hypergraph Hi ( i = 1,2) and let us consider e = el U e2. The following hypergraph HI Da H2 = (HI\{el})U (Hz\{e,})U { e } is defined as the Hajbs’ construction from HI and H z (via e l , e2 and x). This general Hajbs’ construction does not correspond to the graphic one (in the case of a graph) but the graphic Hajbs’ construction may be defined, in terms of the general one (with a further augmentation operation) by

(HI w H 2 )v {al U

4.

2.3. The Hajds’ theorem for hypergraphs

Theorem. Let H be a Sperner hypergraph. Then y, ( H ) 3 q 2 2 if and only if H can be built from copies of the q-complete graph by finitely many applications of the three basic operations. Proof. The if part is easy. By augmentation of H we get H ’ and clearly x ( H ‘ ) a x ( H ) .The same is true for the reduction (see [2]):

x ( H H )3 x ( H ) . For the third operation, let H , and H 2 be such that x(H1)3 q, x ( H z )3 9 and suppose that x(HIw H 2 ) < 9. Thus there exists a k-coloration of HI ~3 H2 with k < q. But the vertices in e, have the same colour (otherwise x ( H , ) < q ) and since e l n e 2 = { x } , the edge e = e l U ez in HIw Hz is monochromatic, a contradiction. The only if part is like the graph-theoretical proof. Firstly we remark that if H is singular (i.e., x ( H ) = 03) then H is generated from any complete graph K , ( q 2 2 ) (reduce K, via a constant map I7 and proceed with augmentation operations). The main proof is by contradiction. Suppose that there exists H with q S X ( H ) < = and such that H is not 9-Hajbs constructible (i.e., not built from K, by applications of basic opera-

C. Benzaken

56

tions). Then V = VH has cardinality at least 9. The finite set 9 of hypergraphs G with V , C V and x ( G ) 3 9 is then partitioned into two nonempty and complementary subsets 2 and 2

X = {G I x ( G ) > 9 and G is not q-Haj6s constructible},

I

S\2 = 2 = {G x(G) 5 9 and G is 9-Haj6s constructible}

(2is not empty because

the complete graph on V is in %). Moreover

G E 2,G ' E 9 and G ' S G =$ G ' E 2, G € % , G ' € S and G S G ' + GI€%. Take now a maximal element of 2 with respect to the order relation G and call it H. H has the following property: if S , and S2 are stable sets of H with S , n Sz# 0 then S1U S2 is a stable set of H. For the proof, it is clear that

HI=HV(SI},

HZ=Hv{Sz}

are q-Hajbs constructible (because Hi > H ) . Take x E SIfl SZ. Take copies H i and HI of, respectively, H , and H r by changing the set W = V \ { x ) into W' and W" so that VH; = w' u {x},

VHZ = w" u {x}

and vH;

n vH-,

= {XI.

We write then

H;=H'v{S;},

H;'=H"v{S;'}.

The Hajbs' construction from HI and H ! (via Sl, Sl: and x ) gives

H : w H ; = H' v H" v { S I u sn}. The reduction of H i w W v i a the map IZ,which acts as identity on x and deletes the quote (or double quote) of vertices in W' (resp. W") gives now

n ( H ;w H ; ) = H v {SI u SZ}. Since H is not 9-Haj6s constructible we must have { S , u SJ $ H,

which means that S , U S , is a stable set of H , proving the announced fact. But now it is clear that H is a complete k-partite graph. Indeed the relation R V X V defined by (x, y ) E R

{x} U { y } is a stable set of H

Hajbs’ theorem for hypergraphs

57

is an equivalence relation. Each of the k classes of this equivalence, is a stable set of H and if x and y belong to two distinct classes then { x , y } is not stable and H being nonsingular, { x , y } E H. Finally, since x ( H )2 q, we have k 2 q and H is the augmentation of a q-complete graph and then is q-Ha@ constructible (a contradiction). 0 Remark. If we replace the third operation H , w Hz by H m H 2 , where

the theorem is no longer true. For example, the hypergraph H = {f}where f has at least three elements cannot be 2-Ha@ constructible.

References [ I ] C. Benzaken, Post’s closed systems and the weak chromatic number of hypergraphs, Discrete Math. 23 (1978) 77-84. [2] C. Benzaken, Critical hypergraphs for the weak chromatic number, J. Comb. Theory, Ser. B 29 (1980) 328-338. 131 G. Hajbs, Uber eine Konstruktion nicht n-farbbarer Graphen, Wiss. 2. Martin-Luther-Univ. Halle-Wittenberg A10 (1961) 116-117.



PATH PARTITIONS IN DIRECTED GRAPHS C . BERGE C.N.R . S . . Paris. France

1. Introduction Let G = ( X , U ) be a (directed) graph with a set U of arcs. A (directed) path p will be defined as a sequence of distinct vertices x i , xz, . . . ,xk, but for convenience, we put p = { x I , x 2 , . . , x t } . A well-known result of Gallai and Milgram [7] asserts that the least number of paths which partition the vertex-set X is at most the stability number a ( G ) (maximum number of independent vertices). For instance, if G is a tournament, i.e., a ( G ) = 1, then one path suffices to cover the vertex-set, and we get the Redei’s result. The Gallai-Milgram theorem has been extended independently by Las Vergnas [8] and Linial [lo]. In Section 2 , we give another extension, which contains both the result of Las Vergnas and those of Linial, and we show some new applications. I n Section 3, we survey recent problems about the covering of strongly connected graphs with paths or circuits.

2. An extension of the Gallai-Milgram theorem Let H be an arborescence; the root of H is the vertex x with d J x ) = 0, and a sink of H is a vertex y with d L ( y ) = 0. If y is a sink, the path of H leading to y , which does not contain any vertex z with d ; ( z ) 3 2, and which is maximal, is called the terminal branch of H attached to y . Let H be an arborescence forest of G, that is a partial graph of G whose connected components are arborescences. We denote by R ( H ) the set of the roots of these arborescences, and by S ( H ) the set of the sinks of these arborescences. So a vertex in R ( H )n S ( H ) is an isolated vertex of H .

Theorem 1. Let H I , be an arborescence forest of G with R ( H , , )= R,, and S ( H o )= S,,. For every arborescence forest H with R ( H ) C R,,, S ( H ) C S,, and 1 S ( H ) I minimum, there exists a stable set which meets every terminal branch of H . 59

C.Berge

60

Proof. We assume that the result holds true for all graphs of order less than n, and we consider a graph G of order n. Let H be an arborescence forest of G with R ( H )C Ro,S ( H ) = So,I S(H)I minimum. If S ( H ) is stable, the theorem is proved. If S ( H ) is not stable, there exists in G an arc ( b , a ) connecting two vertices a and b in S ( H ) . We have a $ZR ( H ) , because otherwise, H ' = H + (b,a ) satisfies R (H') C Ro, S ( H ' ) = S ( H ) - { b } C So, l S ( H ' ) ( < l S ( H ) ( ,a contradiction. So H has an arc incident to a, say ( a l , a ) . Furthermore, d h ( a , )= 1, because otherwise a 1has at least two descendents in H which belong to S ( H ) , and H' = H - (a,, u ) + (6, a ) satisfies R ( H ' )C Rn, S ( H ' ) C S,,, JS(H')J< ( S ( H ) I ,a contradiction. The subgraph (? of G induced by = X - { a } admits H = H2 as an arborescence forest with R (I?)C Ro, S ( Q ) C (So- { a } )U {al}. Now, we shall show that I S ( f i ) l is minimum. Otherwise, there exists in G an arborescence forest H' with R ( H ' )C Ro, S ( H ' ) C ( $ - { a } ) U {al}, IS(H')l S 1 s(rl)l-1. The following cases can occur. Case 1 : alES(H'). Then H" = H' (al,a ) is an arborescence of G which satisfies R ( H " )C Ro, S ( H " ) C So, I S ( H r ' ) ( = ( S ( H ' ) I ~ ( S ( f l ) = J (- S 1 ( H ) I - l , a contradiction. Case 2: a l E S ( H ' ) , b E S ( H ' ) . Then H " = H ' + ( b , a ) satisfies S ( H " ) C So, R ( H " ) C Ro, ( S ( H " ) l= l S ( f i ' ) l c IS(H)I - 1, a contradiction. Case 3: a l E S ( H ' ) , b E S ( H ' ) . Then ~ S ( H ' ) ~ S ~ S ( I ? ) and ~ - 2 ,H " = H ' + ( a l , a ) satisfies R ( H " ) C Ro, S ( H " )C So, IS(H")(s I S ( H ) l - 1, a contradiction. Thus, we have proved the minimality of ( S ( H ) I . So, by the induction hypothesis, there exists in a stable set S which meets every terminal branch of I?. Clearly, S meets also every terminal branch of H = R + (al,a ) .

x

+

Corollary 1 (Las Vergnas [S]). Every quasi-strongly connected graph G contains a spanning arborescence with at most a(G)sinks. A graph is quasi-strongly connected if for every x, y EX, there exists an ancestor common to x and y. Corollary 2 (Linial [lo]). Let M = {plrp 2 , .. . , p , } be a path partition of G ;either there exists a stable set S which meets each of the p,'s, or there exists a path partition M' with S ( M ' ) C S ( M ) , S ( M ' ) # S ( M ) . Corollary 3 (Camion [ 5 ] ) . Let G be a strongly connected graph such that every pair of vertices is linked with at least one arc ('strong tournament'). Then there exists a hamilton circuit.

Path partitions in directed graphs

61

Proof. Let p be the largest circuit in G. If p does not cover the vertex set, every arc going out of p is the initial arc of a path p' which comes back into p (since the contraction of p gives a graph which is also strongly connected). Let a E p n p' be the terminal vertex of p'; the graph obtained from p + p ' by removing the arc ( y , a ) E p and the arc ( y ', a ) E p' is an arborescence with root a and with sinks y and y ' . By Theorem 1, G can be covered by an arborescence of root a with only one sink, either y or y ' . By adding the arc of G which goes from that sink to a, we form a circuit larger than p,which is a contradiction. Corollary 4. If a graph G has a basis B with I B 1 = LY ( G ) , the vertex -ser can be covered by a ( G ) disjoint paths all starting from B. Proof. A basis of G = (X, V )is a set B C X such that each vertex is the terminal end of a path starting from B,and no two distinct vertices in B are connected by a path. A basis always exists, by a theorem of Konig. Consider the sets B,, = B, B,,B2,. . . where Bi is the set of all vertices which can be reached by a path of length i from B and by no path of length smaller than i. Since B is a basis, U B i = X . Consider a graph H , with vertex-set X , obtained by taking for each x E Bi, i 1, one of the arcs of G going from B i - , to x . Clearly Ho is an arboresence forest with R ( H o )= B. By Theorem 1 there exists an arborescence forest H with R ( H ) C B, ( S ( H ) I s a ( G ) . However B is a basis, so R ( H )= B.

3. Path partitions in strongly connected graphs

Let G be a strongly connected graph; as for the similar problem dealing with chromatic numbers (see Bondy [3]), it is reasonable to conjecture the existence of better path partitions. Let us mention the following. Conjecture 1 (Las Vergnas). Every strongly connected graph G with a ( G )2 2 has a spanning arborescence H with 1 S(H)l S a(G)- 1. Conjecture 2 (Bermond [2]). Every strongly connected graph G can be cooered with a(G)circuits.

In this section, we shall apply Theorem 1 to prove the first conjecture for a graph G having the following property: (P)

G has a circuit which meets every maximum stable set.

In fact, many graphs satisfy the property (P).

62

C. Berge

Example 1 (Thomassen [12]). A symmetric graph satisfies Property (P). Let ( y o , y , , y 2 , . . . ,y k ) be the longest path issuing from y o ; all the neighbours of y k are of the type y i with 0 s i < k ; let i be the smallest index 3 0. Then the circuit p = ( y i , y i + , ,. . . ,y k , y i ) contains y k and all its neighbours; therefore every maximal stable set meets the circuit p. Example 2 (Meyniel [ll]). Not every strongly connected graph satisfies the property (P). Consider a graph G with vertices 1,2,. . . ,11, whose arcs are: (1,2), (2,3), (1,8), (7,8), (7,3), (9, lo), (10,l I), (4,l I), (4,5), (9,5); join all the vertices in A = {9,10,11,4,5} to 6 by arcs directed towards 6. Join 6 to all the vertices in B = {1,2,3,7,8} by arcs directed out of 6, also add all possible arcs from B to A, but remove the arc (3,9). This graph G has stability number a ( G ) = 2, but no circuit meets all the maximum stable sets. Theorem 2. Let G = ( X , V ) be a strongly connected graph with CY ( G ) > 1 that satisfies Proper0 (P); then there exists a spanning arborescence H with I S(H)I S a ( G ) - 1. Proof. Let p be a circuit which meets every maximum stable set. Let G be the graph obtained from G by contracting p into a single vertex c. The graph G is strongly connected; so, by Theorem 1, it admits a spanning arborescence fi with root c, and G has a stable set S with 1 S 1 = 1 S ( f i ) l which meets every terminal branch of fi. Clearly, ~ $ S2 ; so, from the definition of p, we see that S is not a maximum stable set of G. So, IS 1 S a ( G ) - 1. We can construct in G a spanning arborescence H with arcs of p and the image of the arcs of fi, and by removing one of the arcs of p so that ( S ( H ) I = l S ( f i ) l S a ( G ) - l . This arborescence H fulfills the conditions of the theorem. Remark. This result shows that the Las Vergnas conjecture is true for symmetric graphs; the same argument shows also that the Bermond’s conjecture is true for symmetric graphs. However, we can expect better results; in fact, Amar, Fournier, Germa, Haggkvist and Thomassen [ I ] have conjectured that the vertex-set of a k-connected graph G can be covered by [(l/k)a(G)]* cycles (eventually reduced to a single vertex). By using Theorem 1, we can also extend well-known properties of tournaments to ‘join’ of graphs. The following result has been proved by Las Vergnas [9], and later by Linial [lo].

Path partitions in directed graphs

63

Proposition 1. Let G = ( X , U ) be a graph, let ( A , B,. . . , K ) be a partition of X such that G,, GB,. . . , have hamilton paths ( a , , a 2 , .. . , a’), ( b l ,b z , . . . , b‘), . . ., ( k , ,k 2 ,. , . ,k ’ ) , respectively. If every pair of vertices in different classes is joined by at least one arc, then G contains a hamilton path starting in { a l ,b l ,. . . ,k l } and ending in {a’,b‘, . . . , k ’ } . Proof. Clearly it suffices to show the result for a partition ( A , B ) of X in two classes. In this case the two hamilton paths of G, and GB constitute an arboresence forest Ho with R ( H o )C { a , ,b l } ,S ( H , , )C {a‘,b‘}. From Theorem 1, this arborescence forest is not minimum, and there exists a unique path H with R ( H )C {al, b J , S ( H )C { a ‘ ,b’}. For strongly connected graphs, the same argument shows the existence of hamilton circuits. More precisely, we can get the following. Proposition 2 (Las Vergnas [9]). Let G = ( X , V ) be a strongly connected graph ; let ( A , €3,. . . , K ) be a partition of X such that G,, GB, . . . have hamilton circuits (or: are reduced to a singleton). If every pair of vertices in different classes is joined by at least one arc, then for every integer I, 3 S 1 S 1 X 1, the graph G contains a circuit of length 1. For a tournament G this result was found by Moon (1969). For a complete proof of Proposition 2, see [9].

References [ I ] D. Amar, I. Fournier and A. Gerrna, Private communication, May 1981. [2] J.C. Bermond, Private communication, February 1978. [3] J.A. Bondy, Disconnected orientations and a conjecture of Las Vergnas, J . London Math. SOC. (2) 14 (1976) 277-282. [ 3 ] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (MacMillan, London, 11~72). [S] P. Camion, Chemins et circuits des graphes complets, C.R. Acad. Sci. Paris 249 (1959) 2 IS1 -2 152. [h] T. Gallai, On directed paths and circuits, in: Erdos-Katona, eds., Theory of Graphs (Academic Press, New York, 1968) pp. 115-118. [7] T. Gallai and A.N. Milgram, Verallgemeinerung eines Graphentheoretischen Satzes von Ridei, Acta Sci. Math. 21 (1960) 181-186. [HI M. Las Vergnas, C.R. Acad. Sci. Paris, A 272 (1971). [9] M. Las Vergnas, Sur les circuits dans les sommes complitttes de graphes orientis, Coll. sur la ThCorie des Graphes, Institut de Hautes Etudes de Belgique (1973) 231-244. [lo] N. Linial, Covering digraphs by paths, Discrete Math. 23 (1978) 257-272. [ I I ] H. Meyniel, Private communication, May 1981. [I21 C. Thomassen. Private communication, June 1981.



GRANDS GRAPHES DE DEGRE ET DIAMETRE FIXES J.-C. BERMOND and C. DELORME Universid Paris-Sud, Informatique, B i t . 490, ERA 452 du C.N.R.S., 91405 Orsay, France

J.-J. QUISQUATER Philips Research Laboratory. A v . van Becelaere 2, B-I I70 Brussels, Belgium The following problem arises in telecommunication networks: given A, the maximurn degree, and D, the diameter. find graphs having the greatest possible number of vertices. We give here two constructions improving earlier results and we present an extended table with the best known values.

1. Introduction

Soit G = ( X , E ) un graphe simple, non orientt, ou X dCsigne I’ensemble des sommets et E celui des aretes. La distance entre deux sommets x et y de G, notCe S(x, y), est la longueur d’une plus courte chaine entre x et y. Le diamttre D du graphe est dCfini par D = m a ~ ~ , . ~ ~ ~ .y ~) .2Le S ( degre‘ x , d ( x ) du sommet x est le nombre de sommets adjacents 2 x et A dCsigne le degrt maximum de G. Nous nous intkressons au probltme suivant (appelC ‘(A, D)-graph problem’ dans la IittCrature): trouver le nombre maximum n ( A , D ) de sommets d’un graphe de degrC maximum A et de diamktre D. Ce problkme se pose dans les rtseaux de communications (ou rtseaux de microprocesseurs): les sommets reprtsentent les stations ou processeurs et les aretes, les liaisons bilatCrales entre ces stations; le degrt d’un sommet est le nombre de canaux bidirectionnels d’une station et le diamhtre reprksente le maximum de liaisons (ou de stations intermediaires plus un) & utiliser pour transmettre un message. Ce problkme semble avoir CtC considCrC pour la premikre fois dans la IittCrature par Elspas [ I l l . La table de Storwick [I91 rtsume les diverses contributions jusqu’en 1970. Rtcemment, ces rCsultats ont Ctk amCliorCs par divers auteurs, e.g. Memmi et Raillard [16], Arden et Lee [I], Leland, Finkel, Qiao, Solomon et Uhr [15] et les auteurs de [4], [ S ] , (91, [lo], [MI. Une borne thkorique sur n (A, D ) a CtC donnCe par Moore (voir [6] ou (71)

n ( 2 , D ) S 2 D+ 1, et pour A > 2 , 65

J.-C. Bermond et al.

66

n (A, D ) (A (A

- l)D - 2)/(A - 2).

I1 existe des graphes, appelCs graphes de Moore, pour lesquels il y a Cgalitt?dans la formule ci-dessus: il s’agit des ( 2 0 + 1)-cycles pour A = 2, des (A + 1)-cliques pour 0 = 1, du graphe de Petersen B 10 sommets pour A = 3 et D = 2 , du graphe d’Hoffman-Singleton B 50 sommets pour A = 7 et D = 2 et, peut-Ctre, d’un graphe i 3250 sommets pour A =57 et D = 2 . La recherche de la valeur exact de n ( A , D ) apparait comme un problkme difficile (voir [3]). On s’est donc int6ress6 i construire des graphes de degrC maximum et diambtre donnCs ayant un grand nombre de sommets. Le but de cet article est de donner la table des meilleures valeurs connues actuellement. Une bonne partie provient d’autres articles, essentiellement [4], IS], [9], [lo] et [18]. De plus, nous donnons deux thkorbmes nouveaux, qui permettent d’amkliorer certaines valeurs. 2. Nouvelles constructions Theoreme 1. Soit G I ,un graphe dont l’ensemble X d e s sommets a nl e‘le‘ments,de diamttre DI et de degre‘ maximum A l . Soit A 3 A l . Soit G2 un graphe a nr sommets, de diamttre D2 et de degre‘ maximum au plus n l A - z x E x d l ( x )ou , dl est le degre‘ dans G I .Alors, il existe un graphe, note‘ GI[G2],ayant n l n 2sommets, de degre‘ maximum A et de diamttre au plus (Dl+ l)(D, + 1) - 1. Preuve. Soit X (resp. Y )I’ensemble des sommets de G I (resp. G2).Le graphe G aura pour ensemble de sommets le produit X x Y. Les aretes sont de deux sortes: (1) Les aretes {(x,y)(x’,y)}avec {x,x’} arCte de GI (on a donc n2 copies de

GI). (2) Pour d6crire l’autre sorte d’aretes, associons B chaque sommet y de G2 une application f,, : r 2 ( y ) + X telle que pour tout x on ait l f ; ’ ( x ) I s A - d l ( x ) . Une telle application existe car Ir2(y)1 G A2 6 ( A - d,(x)). Les arCtes de la deuxikme sorte sont alors du type {(f, ( y ’ ) , y ) , ( f , , ( y ) , y’)}, ou { y , y’} est une arite de G,. Le graphe G ainsi obtenu a n i n 2sommets; son degrC maximum est au plus A (le degrC du sommet (x, y ) est d l ( x ) +I f y ’ ( x ) ( C A ) . Cherchons son diamhtre. Soient (x, y ) et (x’, y’) deux sommets du graphe. I1 existe une chaine de longueur D 6 0; reliant y B y‘ dans GZ;soit y = yo, y l , . . . ,yD = y’ cette chaine. La distance de (f,, ( Y , - ~ ) y, , ) B (f,, ( Y , + ~ )y, , ) est au plus D1,de meme que la distance de (x, y ) B (f,( y l ) , y ) et celle de (x’, y ’) B ( f y ’ ( y D - l ) , y ’). O n a ainsi form6 un chemin de longueur au plus D1(D+ 1) + D entre (x, y ) et (x’, y‘); le diambtre est, donc, au plus D1(D2+ 1) + D2= (Dl+ l)(D2+ 1) - 1. 0

z,,,

Grands graphes de degre‘et diamLtre fixis

67

Exemples. (Les notations utiliskes sont dCtaillCes i la fin de cet article.) En Fig. 1, nous avons represent6 le graphe G = G1[Gz], avec

et G2= K7,A = 3. Les somrnets de K7 sont notCs O , l , ...,6 et t Y ( y ‘ ) = [ z / 2 ] oU 1s z s 6 et y z = y ’ (mod 7). Le graphe G, de diarnbtre 5 , a 28 sornrnets de degrC 3. Dans le cas particulier o i ~GZ est le graphe cornplet, on retrouve une construction due i Quisquater [18]: par exernple, en prenant pour GI le graphe d’Hoffrnan-Singleton B 50 sornrnets, de degrC 7 et de diarnbtre 2, et pour Gz,le graphe Ksl, on obtient un graphe i 2550 sommets not6 HS[K,,], de degrC maximum 8 et de diarnbtre 5. Ce dernier graphe a aussi Ct6 t r o u d par Uhr (voir

+

~51). D’autres rCsultats sont donnCs h la Table 3: par exemple, soient GI le graphe d’Hoffrnan-Singleton et G2 le graphe P L obtenu cornme graphe quotient du graphe d’incidence du plan projectif P d’ordre 149; le graphe G,a 1492+ 149 + 1 = 22351 sornrnets, de degrC A, = 150, et est de diambtre 2. Soit A = 10, a l ( A - 7) = 150 = A,. Le graphe G est de degrC 10, de diarnbtre 8 et a 1117550 sommets: cette valeur est B comparer avec le graphe B 19305 sommets de la table de Storwick [I91 ou avec le graphe B 745104 sornmets de Bermond, Delorme et Farhi [4]. Si on prend pour GI le graphe B 4680 somrnets de notre table 3, de degrC 12 et diambtre 4, on peut montrer que ce graphe a 27820 arttes. Soit A = 13 et Gr = Ks,o,. On a A, = 5200 et n ,A - 2 d ( x ) = 60840 - 55640 = 5200. Le graphe G a donc 24340680 sornmets et est de diamktre 9.

(0.2)

Fig. 1.

J.-C.Bermond et al.

68

D’autres exemples figurent encore dans la Table 3. Pour cela, nous avons besoin de connaitre les valeurs de t = n l A l -C,,,d,(x) pour certains graphes: par example,

t ( q = q + 1,

?(a:) =

q 2 +

1,

pour GI,le graphe B 715 sommets, de degrt 11 et de diambtre 3, on a t(GJ = 585, pour GI, le graphe B 910 sommets, de degrt 14 et de diambtre 3, on a t(Gl) = 2340.

Remarque. Le choix de la fonction f permet bien souvent de construire plusieurs graphes diffCrents ayant les mEmes p a r a m h e s . Corollaire. Lorsque le degre‘ A

+ m,

alors lim inf(n(A, 5)A -7 3 4‘/Y.

Preuve. Prenons pour graphe G I , le graphe P i associC B un plan projectif de diamktre 2, de degrC q + 1 et ayant q 2+ q + 1 sommets: un tel graphe existe si q est une puissance de premier. Posons q + 1 = 4A/5. Pour G 2 ,prenons le graphe complet sur A ( q 2 + q + 1)/5+ 1 sommets. On a bien ( q 2t q + 1)A - 4 4 ( q 2 +q + 1)/5. Le graphe GI[G2]est de degrt maximum A, de diametre 5 et a A(q’+q + 1 ) ’ / 5 + q 2 + q + 1 = ( 4 4 / 5 5 ) A s + ~ ( A ) Asommets ’ oh 0 E ( A )- 0 quand A -+ X I . Donc, lim inf(n ( A ,5)A - 5 ) 3 44/5‘. Notons que la meilleure borne connue auparavant Ctait 1/32 [lo]. Theoreme 2. Soient G I et G ; deux graphes ayant pour ensembles de sommets respectivement X et X ‘ , auec 1 X I = n l et 1 X’l = n I, de degre‘s maxima A l et A et de diamkres D , et D I, respectiuement. Soient A > A I et A > A I. Soit G2un graphe biparti, ayant pour ensemble de sommets Y U Y’ auec Y, Y’ stables et I Y 1 = nz et 1 Y’I = n;,de diamitre D 2et tel que le degre‘ d’un sommet de Y soit compris entre n , et n , A -C,,,d,(x) et le degre‘ d’un sommet de Y’ soit compris entre nI et n : A - ~ + , . d ~ ( x ’ )(oLi d l (resp. d : ) dksigne le degre‘ duns G I (resp. Gi)). Alors on peut construire un graphe note‘ (GI,G I)[G2] ayant n l n z+ n In; sommets, de degre‘ maximum A et de diamitre au plus D,+ [D2/2]sup (Dl,D : ) + [D2/21inf (D,, D :). Preuve. L’ensemble des sommets est constituC par la reunion disjointe de X x Y et X’X Y’. Les arCtes sont de deux sortes: Les ar&s {(u,y ), (u,y )} oh y est un sommet de G2et {u,v } une arkte de GI ou GI. On a donc n2 copies de G I et n ; copies de GI.

Grands graphes de degre‘ et diamktre fixis

69

A chaque sommet y de Y, associons une application f, : r2(y)+ X telle que pour tout x de X , 1 S If;’(x)( S A - dl(x). De m&me , B chaque sommet y ’ de Y ’ , associons une application f:: r;(y’)+ x ‘ telle que pour tout x ’ de X ’ , 1 c If;! I S A - dI(x’). De telles applications existent d’aprbs les conditions sur les degrCs de G2. Les ar&tes de la deuxikme sorte sont les arites IVY( y ’ ) , y), Vy4y),y‘)) o i ~(y, y’) est une arkte de G2. Le graphe ainsi form6 a nln2+ nln: sommets, de degr6 maximum A car le degrt d’un sommet (x, y ) est d ( x ) + I f ; * ( x ) I . Cherchons son diamktre. Soient (x, y ) e t (x’, y’) deux sommets du graphe. S’il existe dam G2une chaine entre y et y ’ de longueur D S D2- 1, on pourra trouver une chaine de longueur au plus [ ( D + 1)/21 sup(D,, D I)i i ( D + 1)/2J inf(DI,D ;) + D en traversant [ ( D + 1 ) / 4 copies du graphe G I ou GI contenant x, [ ( D+ 1)/2] copies du graphe GI ou GI ne contenant pas x et en utilisant D ar&tesde la deuxikme sorte. Si les sommets y et y ’ sont h la distance D2 dans G2,le sommet y admet au moins un voisin y l , tel que x = f, (yl) et tel que yl est A distance au plus Dz - 1 de y’, car G2 est biparti. Donc, le sommet (fyl(y), yl), qui est adjacent 5 (x, y), est h distance au plus [D2/21sup(Dl,D:)+ [D2/2] inf(D1, D : ) + D , - 1 de ( x ’ , ~ ’ ) , ce qui prouve le thkorkme. 0

Exemples. La Table 1ci-dessous donne des exemples d’application quand G,est biparti complet. Table 1

KaxCli P x F, K4

K , X P,d

K5 K, X P2d K6 HS K.5 HS K, HS p: -

H: H4 r H 5r H,r Hd Hur HI,?

H,,r

15 40 4 24 5 24 6 50 6 50 7 50 91

-

364 2134

1817 -

39223

74906

132869 -

354323 804401

392 n; -

-

-

-

-

-

-

-

15 40 12 24 15 48 24 100 30 150 35 200 283 465

11340 30240 43744 131232 156340 562824 1176690 5883450 2696616 14981200 5580498 33217250 121296802 447391446

5 6 7 9

10 11 13 15

9 10 9 10 9 10 9 10 9 10 9 10 -

-

On peut aussi donner des exemples avec des quadrangles B deux paramktres + v)(l + u v ) et n : = (1i u)(l + u u ) (voir Payne [17] et Thas [201). Q,,,, avec n2 = (1


70

Table 2 GI

G/

G,

n,

n/

K, K,

P HS

Q,,,, Q,,,

1 1

10 50

n,

n:

280 112 1720 2752

n

A

D

1400 154800

4 8

8 8

Table 3 Plus grands (A, D ) graphes connus (Octobre 1981)

A

D

2

3

4

5

6

7

8

9

10

P 10

c,x4 20

LFQSU 34

AL 56

H,r 128

H,idr 158

Y 244

Y

3

340

Y 536

4

K,x5 15

PX4 40

C,X 19 95

H: 364

H,r 731

H,idr 837

Thm. 2 1400

K,x8 24

15x4 60

Q,r

5

174

H:d 532

H,r 2734

H,idr 2988

O,,dr 5004

Thm 2 11340

Thm. 2 30240

6

K,xH 32

21x5 105

Q,r 317

H:d 756

H,r 7817

H,x4 10920

H4x6 16380

Thm 2 43744

Thm. 2 131232

tIS 50

24x5 120

Qsdr

Q,x4 1248

H,dr 8998

H,x4 31248

24[P;,] 54168

Thm. 2 156340

Thm 2

7

P;

352

C,,, 1932

C,,X

C,(S,9)

2560

562824

57

HSx4 200

Q7r 807

HS[K,,) H,r 39223 2550

H,idr

8

40593

Thrn. 2 154800

H,x35 273420

C,(5,Y) 1310720

PAd 74

Q: 585

Qd

9

1178

HS[K,,,,] H,r 71906 5050

H,x4 156864

HS[P:,d] 480250

Thm 2 1176690

Thm. 2 58834.50

10

p: 91

QLd 650

Q,r 1629

Qk[K,51] HS[PI,,] HS[K,,,] H,r 7550 132869 380835 1117550

Thm. 2 2696616

Thm. 2 I4081200

11

Pid 94

QLd 715

Qix5 2925

P:[K,,,] H,dr Q;[K,,,,] HSIP{V,] 11388 142494 723060 1990050

Thm. 2 5580498

Thm. 2 332172.50

p:, 133

QLd 780

0; x 8

P.lK19,1 H , , r QJK18211 PL[pL71

12

4680

17563

LVLQ C, (X5) 10077696 85887453

Plid 136

Q Ad

13

845

Q:x9 5265

P:[K,,,I 25844

14

PI, 183

Q;d 910

650 x 9 P / , [ K 7 J H,,r 910[K,~,,I P3P&ll 4680[K9,,,1 C, (1,101 37107 804481 2130310 12694773 46243080 282475244 5850

15

P:,d 186

354323 1065285

3778261

H,,dr 715[K2,,,,l P;[P;xldl 4680[K5,,,,J Thm. 2 394616 1414440 7211386 24340680 121296802

PI, x 8 Q ; x 13 P j l [ K I 1 , ]H,,dr H I ,x 4 PIIIPiINd]4680[K,,,,,] Thm. 2 54796 892062 3217872 22303302 68145480 447391446 1064 7605

Grands graphes de degre‘ et diamitre fixis

71

3. Notations

3.1. Graphes particuliers

K,, :

graphe complet n sommets. C” : cycle a n sommets. P: graphe de Petersen (10 sommets, A = 3 , D = 2 ) . HS: graphe d’Hoffman-Singleton (50 sommets, A = 7, D = 2). LFQSU: graphe de Leland, Finkel, Qiao, Solomon et Uhr [15]. AL: graphe de Arden et Lee [1]. Pq : graphe d’incidence du plan projectif d’ordre q, de degrC q + 1, ayant 2(q2+ q + 1)sommets et de diambtre 3: il existe si q est une puissance de premier. Qq: quadrangle gCnCralisC d’ordre q, de degrC q + 1, ayant 2(q4- I)/(q - 1) sommets et de diambtre 4; il existe si q est une puissance de premier (Benson [2]). hexagone gCnCralisC d’ordre q, de degrC q + 1, ayant 2(qh- l)/(q - 1) H, : sommets et de diambtre 6 ; il existe si q est une puissance de premier (Benson [2]). P;: graphe quotient de P, par une polarite, graphe de degrC q + 1, ayant q 2+ q + 1 sommets et de diambtre 2. Q ;: graphe quotient de Q, par une polaritb de degrC q + 1, ayant (q4- l)/(q - 1) sommets, de diambtre 3; un tel graphe existe s’il existe une polaritC, c’est-a-dire pour q = 22p+1(en particulier, ici, pour q = 8: voir Delorme [9]). graphe quotient de H, par une polaritC, de degrt q + 1, ayant H;: (4‘- l)/(q - 1) sommets, de diambtre 5 ; un tel graphe existe s’il existe une polariti, c’est-&dire pour q = 3’,+’ (en particulier, ici, pour q = 3 : voir Delorme [9]). O,,, : octogone gCnCralisC (Yanushka [21]). C,(m,k ) : graphe de type C, dCcrit par Delorme et Farhi [lo]: ce graphe a un degrC maximum A, avec rn (A /2)k sommets et de diambtre donnC en [lo]. Cette famille de graphes est en fait la mCme que celle introduite par de Bruijn [8] et retrouvke par Golunkov [I21 et Lam [13]. LVLQ: famille de graphes dCcrite par Quisquater [18] et obtenue a partir d’une construction de matrices circulantes gCnCralisCes due A Lam et van Lint [14]. Ces graphes ont un sommet de plus que les graphes C,(1,D ) . C,(m,k): graphe de type C,,(associC un plan projectif) dCcrit dans Delorme et Farhi [lo]. Chaque graphe a un degrC maximum A, avec m(A2/4 - A/2 1)’ sommets, oii A/2 - 1 est une puissance de premier.

+

72


3.2. Opkrations de constructions Y: 1:

d: r:

X:

Gi[G2]: Thm. 2:

construction de Y-graphes (voir Quisquater [HI): ces graphes interviennent pour le degre 3. insertion de sommets sur certaines ar&tes(voir 99 de Delorme [9]). operation decrite aux 997, 8, 9 de Delorme [9]; le cas le plus simple consiste B dCdoubler des sommets. operation qui consiste B remplacer un sommet de degrC A par un graphe complet Kd (voir Quisquater [18]). plusieurs graphes sont obtenus en utilisant un produit spCcial dtfini par Bermond, Delorme et Farhi [4],[5]. Nous les notons G X r, oh G dCsigne le premier graphe et r, le nombre de sommets du deuxikme. Lorsque G est un graphe de notre table (sans nom special), nous le notons par son nombre de sommets. graphe obtenu en appliquant le ThCorbme 1. graphe obtenu en appliquant le ThCorbme 2.

Note ajoutd a I’epreuve

Depuis octobre 1981 de nouveaux (A, D ) graphes ont CtC trouvCs. Une nouvelle table a CtC publike dans Information Processing Letters 15 (1) (1982) 10-13. References [1] B.W. Arden and H. Lee, A multi-tree-structured network, Proc. Fall COMPCON 78 (1978) 201-210. [2] C.T. Benson, Minimal regular graphs of girths eight and twelve, Canad. J. Math. 18 (1966) 1091- 1094. [3] J.-C. Bermond and B. Bollobls, Diameters in graphs: a survey, Proc. 12th Southeastern Conf. on Combinatorics, Graph Theory and Computing, Baton Rouge, 1981, to appear. [4] J.-C. Bermond, C. Delorme and G. Farhi, Large graphs with given degree and diameter 11, to appear. [ 5 ] J.-C. Bermond, C. Delorme and G. Farhi, Large graphs with given degree and diameter 111, in: B. Bollobiis, ed., Proc. Conf. on Graph Theory, Ann. Discrete Math. 13 (1982) pp. 23-32. [6] N. Biggs, Algebraic Graph Theory (Cambridge University Press, London, 1974). 1974). [7] B. Bollobls, Extremal Graph Theory, London Math. SOC.Monograph No. 11 (Academic Press, London, 1978). [8] N.G. de Bruijn, A combinatorial problem, Koninlijke Nederlandse Academie van Wetenschappen Proc. Ser. A 49 (1946) 758-764. [9] C. Delorme, Grands graphes de degri: et diamitre donnks, B paraitre. [lo] C. Delorme and G. Farhi, Large graphs with given diameter I, to appear. [ l l ] B. Elspas, Topological constraints on interconnection limited logic, Proc. 5th Symp. on Switching Theory and Logical Design, IEEE S-164 (1964) 133-137. [ 121 Y.V. Golunkov, Automation program realization of substitutions of symmetric semigroups 11, Kibernetika 5 (1975) 35-42.

Glands graphes de degre' et diamitre jixe's

73

[13] C.W.H. Lam, On some solutions of A Ir = d l + AJ, J. Comb. Theory, Ser. A 23 (1977) 1 6 1 4 7 . [I41 C.W.H. Lam and J.H. van Lint, Directed graphs with unique paths of fixed length, J. Comb. Theory, Ser. B 24 (1978) 331-337. [15] W. Leland, R. Finkel, L. Qiao, M. Solomon and L. Uhr, High density graphs for processor interconnection, Inform. Process. Lett. 12 (1981) 117-120. [16] G. Memmi and Y. Raillard, Some new results about the (d, k) graphs problem, IEEE Trans. Comput. to appear. [ 171 S.E. Payne, Affine representations of generalized quadrangles, J . Algebra 16 (1970) 473-485. [18] J.-J. Quisquater, New constructions of large graphs with fixed degree and diameter, to appear. [19] R.M. Storwick, Improved constructions techniques for (d,k ) graphs, IEEE Trans. Comput. 19 (1970) 1214-1216. [20] J.A. Thas, Combinatorics of finite generalized quadrangles: a survey, Seminar of Geometry and Combinatorics, State University of Ghent, Belgium, Sept. 1980, to appear. [21] A. Yanushka, On order in generalized polygons, Geometriae Dedicata 20 (1981) 451-458.



COMBINATORICS OF INCIDENCE STRUCTURES AND BIB-DESIGNS Carlo BERNASCONI Zstituto di Matematica - Universita, Via Pascoli, 06100 Perugia, Italy We give an exposition of basic properties of incidence structures and designs (BIB-designs). Some properties relating internal and external structures, duality, and complementary structures are stated. We add some basic constructions for BIB-designs.

1. Introduction

The main intention of this communication is to extend the existing state of knowledge of the basic combinatorics of incidence structures and BIB-designs. In particular, for general incidence structures, we state some properties relating internal and external structures, duality and complementary structures (see Section 2). In Section 3 we show how the above properties can be used to simplify and to give a better visualization of some properties of incidence structures and designs. In Section 4 we give a short survey of designs obtained with simple set-theoretic techniques. 1.1. Basic definitions and notations We refer to Dembowski [6, Chapters 1 and 21 for the following definitions and notations. Incidence structure S = S(9,93, I),points, blocks. Dual incidence structure S *. Complementary structure VS. Symmetric incidence structure (when 1 9 1 = 193 I). Internal and external structures with respect to a block B : SB,S B .Internal and external structures with respect to a point p : S,, S p . Design D. We use the following usual notations for designs: 93: block set, 1% I = 6. 9:point set, (91 = 0. For any block B : [B]= k. For any point p : [ p ] = r. For any distinct points p , q : [ p , q ] = A. B = complementary set in 9 of the set of points incident with block B. When a set or a parameter (for instance 9 or b ) refers to a given structure S, we shall use the notation 9 ( S ) , b ( S ) , and so on. Embeddable design E (this means: there exists a (symmetric) design D such that E = D B ) . 75

C. Bernasconi

16

2. Some properties relating internal and external structures,

dual structures, and complementary structures We state three basic theorems which relate the constructions indicated in the title of this section.

Theorem 1. Let S be an incidence structure and S * its dual structure. Then with the notations given in Section 1 we have SB

= [(s*)b]*,

s,

=

sB = [ ( S * ) b ] * ;

[(s*)p]*,

s p

= [(S*y]*.

Here b is the point of S * corresponding to block B of S and P is the block of S corresponding to point p of S. We call B, b and p , P ‘dual’ elements and use the notations b = B * , P = p * .

Proof. We give an outline of the proof for the first identity. Points of [ ( S * ) b ] *=blocks of ( s * ) b =blocks of S * incident with point b = B * = points of S incident with block B = points of S s . Blocks of [ ( s * ) b ] * = points of ( s * ) b = points of S * different from b = blocks of S different from block B =blocks of SB. The incidences are preserved by internal, external and dual structures. This then proves the above identity. As ( S * ) * = S, we have (SB)*

= (S*)br

(S,)* = ( S * ) P ,

(sB)* = (L$*)~; (SP)*

=(S*)P.

Theorem 2. Let S denote an incidence structure and %S its complementary structure. Then with the notations given in Section 1 we have the following identities :

%[(%s)’], s B= % [ ( % s ) B ] ;

SB

=

s,

= %[(%S)p],

s p

= %[(%S),].

Proof. We give a sketch of the proof for the first identity. Points of % [ ( % S ) ” ]= points of ( % S ) B = points of VS non-incident with B =points of S incident with B = points of Ss ( = points of B ) . Blocks of % [ ( % S ) ” ]3 complementary, in B, of blocks of (%S)E. Let B, be the blocks of S , B, # B. The blocks of (%S)” are of the form B, n B = B - B,. Then: blocks of % [ ( % S ) ” ] B

n B, blocks of

SB.

Combinatorics of incidence structures and BIB -designs

I7

This completes the proof. As

%(W) = S, we have

Theorem 3. Let S be an incidence structure and %S, S * its complementary and dual structures, respectively. Then we have

(%S)* = % ( S * ) . Proof (sketch). We show that there is a natural isomorphism between (%S)* and ( e ( S *). The scheme below is self-explanatory. Structure S VS

points P P

blocks BB

@S)* S* %(S*)

B

P P B BZp

B B

incidences PZB PXB BXP BIP BBP

The above diagram shows the 1-1 correspondence between points of (%S)* and % ( S * ) and between blocks of (%S)* and % ( S * ) , which, moreover, preserves the incidences. 3. Applications of Theorems 1, 2 and 3

The almost primitive nature of Theorems 1 , 2 and 3 of the previous section (in the sense of their logic) does not allow us to derive new deep results in the theory of incidence structures from them. However, they do allow us to simplify the exposition and proof of some basic results and characterizations concerning incidence structures, designs and finite geometries. We give some examples. (a) An embedding theorem for BIB-designs. The use of Theorem 1 allowed us to visualize the fairly elementary nature of the following theorem by Hall and Connor characterizing embeddable designs, which is repeatedly applied in [S].

Theorem (Hall-Connor [S]). Let E = E (P, 93,I ) be a design with r = k + A. Then E is embeddable if and only if there exists a system Y of sets of blocks in E such that (a) ( ( p )fl21 = A for every p E P and 2? E 9, (b) 1%’ n 2’1= A - 1 for any two distinct 2,%’ E Y, (c) every block is contained in exactly A sets 2 E 9.

C. Bernasconi

78

We now describe the most elementary condition one expects in a design E in order for it to be embeddable, and show that the theorem above is equivalent to such a condition. Let E be a design, with r = k + A. Let (b, u, r, k, A ) be the parameters of E. If there exists a design F with parameters (b,r, r - 1, A, A - l), then by considering E and F on the same block set, one could, conceivably, embed E in a symmetric design obtained by ‘collecting together’ the blocks of E and F to get blocks of size k + A, and by considering one additional block B made of all the points of F (which are k + A in number). The structure one gets in this way would be a symmetric hesign D, such that E = D o , F = Do. The only missing property is that every pair of points, one chosen in E and the other in F, are not in general incident with A blocks (while this is in fact true for pairs of points both in E or both in.F). Thus, in the most direct way, we have: E is embeddable if and only if there exists a design F with the above parameters such that the following property holds:

(A)

’trP E W E ) , ‘trq E P(F), Ip,41 = A.

The existence of such a design F with property (A) is indeed equivalent to the conditions given in the theorem of Hall-Connor. It is not difficult in fact to show that the system Y of the Hall-Connor theorem can be given an incidence structure in the most natural way: BI2f iff B E 2 f , such that: (i) properties (b), (c) hold for system Y iff the resulting structure S ( % , Y , I ) is dual of design F ; (ii) property (a) holds for Y iff (A) holds for F. This result was suggested by the light that Theorem 1 sheds on the structure (Do)*= ( D * ) bas , the substantial equivalence of design F and system Y is not really so evident, but it becomes apparent in the case where E is embeddable since then, supposing E = D’, system Y corresponds to (I&)*. (b) A theorem on internal and external structures in projective spaces (see [ 11) can be simplified and improved by use of Theorem 2. We refer to 15, Section 21 for details. (c) Using the properties shown in Theorem 2 it is possible to undertake a study and characterization of smooth designs (see [ 5 ] ) : in particular smooth designs D, such that (D,)* is a multiple of a symmetric design, are considered there. These are finite regular ‘locally symmetric’ spaces which generalize fin. reg. loc. projective spaces (see [7]). (d) Theorem 2 may be used to transform an external structure into an internal one, thus obtaining, for example, internal structures (with respect to a block) with no repeated blocks. Furthermore, as Theorem 1 proves useful when applied to self-dual structures, it is to be expected that some applications of Theorem 1 to self-complementary

Combinatorics of incidence structures and BIB -designs

-

79

structures (i.e. CeS S ) may be obtained. Affine Hadamard designs have such a property. Theorem 3 is expected to give results when combined with Theorems 1 and 2.

4. Some basic constructions for designs

Here we give a list of more or less well-known designs. Some of these were first obtained with matrix techniques, while their most natural structure is set-theoretic, as is shown below. Here, we give a summary exposition; for more details (like proofs, and parameters of listed designs) we refer to [ 3 ] . Let D be a symmetric design. Then the following four designs can be obtained from D. (1) Intersection design YD. Point set: 9 ( D ) (i.e., same point set as 0 ) . Block set: {B, r l B,, V B , , B, unordered, B,, B, E B ( D ) } . This design was obtained in [113 in the form of T ( D * ) . (2) Union design QD. Point set: P ( D ) . Block set: {B, U B,, V B , , B, unordered, B,, B, E B(D)}. This appeared in [lo]. ( 3 ) Symmetric difference WD. Point set: P ( D ) . Block set: {B,A B, ( = (B, U B , ) - (B, n B,)), VB,,B, unordered, B,, B, E B(D)l. See [9]. Using incidence matrix methods, W ( D * )and % [ Q ( D * ) ]were obtained in this reference. (4) Difference design VD. Point set: P ( D ) . Block set: {B, - B,, V B , , B, ordered, B,, B, E BAD)}. See [ 3 ] . Constructions 5 and 6 below are obtainable from designs whose parameters satisfy some restrictions. (5) Composition design D x D’. Let D be a design with parameters (b, u, r, k , A ) and D’ be a design with IP 1 = k and parameters ( b ’ ,k, r’, k ‘ , A ’). Let us consider for every block B of D the design on points of B which is isomorphic to D’. We obtain, for every block B of D, b‘ blocks each of them incident with k’ points. The composition design D x D‘ has the same points as D and as blocks the set of bb’ blocks obtained in that way. This is an important and useful technique. Sometimes it is not easy to realize

C. Bernasconi

80

when a given design has a composition structure, i.e., when it is given by the composition of two well-known designs (cf. [ 2 ] ) . For some non-trivial applications of construction 5 , see [3], [4].

(6) Addition design D

+ D’.

Let D, D’ be designs with the same number of points u and with blocks of the same size k : D ( b , u, r, k,A), D’(b’,u, r’, k, A’). Suppose D, D’ are given on the same point set. Take the incidence structure D + D‘ with the same points as D, D’ and with blocks the blocks of D and the blocks of D‘. Here we give an application of construction 6 to geometric designs (see [4] for details). Consider a finite projective geometry PG(d, 4 ) . In [4] it is proved that any set of all isomorphic k-tuples of lines is the block set of a design on the points of PG(d, 4). By construction 6 it is possible to show that it does not matter whether such k-tuples are isomorphic or not. If we take all k-tuples of lines of PG(d,q), then we get the block set of a design which is the addition design obtained by adding all the designs whose blocks are isomorphism classes of k-tuples of lines.

References [ I ] A. Baartmans, K. Danhof and S. Tan, Quasi-residual quasi-symmetric designs, Discrete Math. 30 (1980) 69-81. [2] M. Barnabei and A. Brini, A class of PBIB-designs obtained from projective geometries, Discrete Math. 29 (1980) 13-17. [3] C. Bernasconi, Constructions of designs (1980). [4] C. Bernasconi, Geometric designs (1981). [ 5 ] C. Bernasconi, Some basic properties of incidence structures and designs (1981). [ 6 ] P. Demhowski, Finite Geometries (Springer, Berlin, 1968). [7] J. Doyen and X. Huhaut, Finite regular locally-projective spaces, Math. Z. 119 (1971) 83-88. [8] M. Hall and W. S. Connor, An imbedding theorem for balanced incomplete block designs, Canad. J. Math. 6 (1954) 35-41. [9] K. N. Majindar, Coexistence of some BIB designs, Canad. Math. Bull. 21 (1) (1978) 73-75. [lo] E.J. Morgan, Construction of balanced incomplete block designs, J. Aust. Math. Soc. A 23 (1977) 348-353. 1111 S.A. Vanstone, A note on a construction for BIBD’s, Utilitas Math. 7 (1975) 321-322.


IDEAL AND EXTERIOR WEIGHT ENUMERATORS FOR LINEAR CODES: EXAMPLES AND CONJECTURES Kenneth P. BOGART Dartmouth College, Hanover, NH

037SS,U.S.A.

1. Introduction

Several years ago Gordon 141 and the author began exploring the use of certain algebraic structures related to codes with the hope that elementary invariants of these structures could be used to distinguish between inequivalent codes. We found examples that lead us to believe that exterior - or Grassman - algebra provided a natural context for the study of codes. At this stage of the research major theorems are elusive, but suggestive examples abound. The purpose of this paper is to present recent examples of the author on weight enumerators of ideals and subalgebras of exterior algebras generated by codes and the conjectures these examples suggest. In this paper, an ( n , k ) code C over a q element field F is a k-dimensional subspace of the vector space F" of n -tuples. The weight w ( x ) of a vector x in F" is the number of nonzero entries of the vector and the weight enumerator of a code C is the polynomial (over the integers)

=

2(

i -0

number of vectors of weight i in C

)Y

n-i

We say two codes are equivalent if there is a weight-preserving linear transformation between them. It is a result of MacWilliams that this means that one code can be obtained from the other by permuting the standard basis for F" and perhaps multiplying the standard basis elements by nonzero scalars [l]. Section 2 contains a brief summary of the relevant parts of exterior algebra, developed in an elementary way. Most textbook treatments of exterior algebra assume that the relevant field has characteristic not equal to 2 or in places that it has characteristic 0. Since our fields will often have characteristic 2, we give here

82

K. P. Bogart

a brief elementary development of the relevant aspects of exterior algebra. With the exception of Theorem 13, the results of Section 2 appear in or are direct consequences of the treatment in Bourbaki [2]; the author believes the present development is more elementary. Since working out this approach, the author has found equivalent theorems spread among the books by Chevalley [3], Greub [5], Marcus [8] and Bourbaki [2], however, only Bourbaki’s treatment is independent of characteristic; the interested reader should have little difficulty in translating the results given in Section 2 (with the exception of Theorem 13) into corresponding results in Bourbaki. The author’s intent in presenting this expository section is to make the methods of exterior algebra accessible to a wider audience.

2. Notation and concepts in exterior algebra , Given an n-dimensional vector space V over a field F with base { x ~ } identify V with the homogeneous polynomials of degree 1 in the algebra FB[xl,x2,. . . ,x,] in non-commuting variables x I , x 2 , . . . , x n . By the exterior algebra of V associated with our basis we mean the algebra FR[xI, x?,. . . , x,,]/Z where I is the ideal generated by squares of polynomials of degree 1. The algebras we get from different bases are isomorphic; we assume our basis has been fixed and use the symbol E ( V ) to stand for the exterior algebra (without referring explicitly to the chosen basis in our notation.) Since x?, x f and (x, + x , ) ~ are all zero, x,x, + x,x, = 0, so x,x, = - x,x,. Note that if F has characteristic 2, then

E ( V) = F[xi, x Z , . . . ,x,]/(x:, x : , . . . , x i ) , the ordinary ring of polynomials modulo the ideal generated by squares of the indeterminants. As a vector space over F, E ( V ) has as its basis the (cosets of) polynomials x,,x,>. . * xtv with i , < i z < . . . < i , . If S = { i l , i z , . . . ,i , ) we denote x,,x,, . . . xts by xs. The algebra E ( V ) is graded; the subspace spanned by the monomials xs in which S has size m, denoted by Em(V), consists of the elements of degree m. Theorem 1. If v, = 2 a,.x, for i = 1 , 2 , . . ., m, then the coefficient of xs in the product v 1 v 2 - *v,* is 0 unless the size of S is m, and otherwise is det(Ms) where Mqis the matrix whose columns are the columns of the matrix A = (a,,)indexed by elements of S.

Ideal and exterior weight enumerators for linear codes

83

Proof.

....im

sizem j&.

in S

size m

u

of S

this formula proves the theorem. 0

Corollary 2. A set of vectors in V is independent if and only if its product in E ( V ) is nonzero. Corollary 3. A vector in V is in the annihilator of v t v 2 * is in the subspace spanned by the vi’s.

* *

v, ( # 0 ) if and only if it

Proof. The matrix whose rows are the independent vectors ui as well as a vector v has rank m if and only if v is a linear combination of the vi’s. 0 Lemma 4. If the independent vectors v l , v2,.. . ,vk are in the annihilator of z E E ( V), then there is an element y E E ( V) such that z = v Iu2 * * uky. Proof. Extend v I , u z , . . . , v k to a basis by adding vectors z = csvs where us = v,,vi2. . . vts

Vk+l

through v , ~ Let .

if S = {i,, iz, . . . ,is}.

Since z . u, = 0 for i s k, each i less than or equal to k is in S for each S with cs # 0. Thus each S contains {1,2,. . . , k}, and so v l v z . . * v k is a factor of z while the other factor is

y = C esuS-(1.2....,k).

0

Given a k-dimensional subspace C of V, its vectors generate a subalgebra of E ( V). Since this subalgebra has dimension at least 2k and since the squares of all elements of C must be zero, this subalgebra is isomorphic to E ( C ) and thus is a graded algebra in which the space of elements of degree k is one dimensional. This yields

Theorem 5. For any bases Bt and Bz of C, the products scalar multiples of each other.

nL,EH, v and notH, u are

K.P.Bogart

84

Thus the vector of coordinates of one of these basis products in the space Ek ( u ) of all elements of degree k in V is determined up to a nonzero multiple, i.e., is a ‘projective coordinate vector’. Such a coordinate vector is called a Plucker coordinate vector for C. We write

for some basis B of C so that Ps stands for the “Plucker coordinate of C relative to xs”. We let E(R,S ) = 0 if R n S # 0 and otherwise we let E(R, S ) equal the sign of the permutation that permutes the list consisting of R in increasing order following by S in increasing order into a single list in increasing order. It follows that for arbitrary elements u and w of E ( V ) , if u = x S a S x S ,u = C T b T x T ,

The following lemma, which appears for characteristic zero in Van der Waerden [9], is useful in removing standard assumptions that the characteristic of F is not 2. Lemma 6 . If C has dimension k and Plucker coordinates P,, then for each k - 1 element set R, the vector

is in C. Proof. Let u I ,u z , . . . , UI, be a basis for C and let M be the matrix where rows are the coordinate vectors for the uz’srelative to the basis x I ,xZ,.. .,x, of u. If i!? R, p R u ( # ) is the determinant of the matrix MRu(t) consisting of the columns of M indexed by R U {i}. We expand this determinant along the ith column of MRuil), letting M, denote the matrix obtained by deleting row j and column i from MRui,, and letting u , ( i ) denote the ith coordinate of u,. We obtain k

E(R,{i})PRU(i)=

( - l)“juj(i)det(M,). j= I

If i E R, let M, be the matrix obtained by deleting row j from the submatrix of M whose columns are indexed by R. In this case, both sides of equation (1) above are 0, the right-hand side being an expansion of the determinant of a matrix with 2 equal columns.


85

Thus

=

2 ( - l)k+j det(M,) 2

j=1

uj (i)ei =

i=l

9( - l)k+idet(M,)ui

j=l

so we have realized l7, as an explicit linear combination of our basis vectors for

w. 0 We now assume there is a bilinear form B on V for which x l , x2,. . . ,x, is an orthogonal basis; in the application to coding theory these vectors will form an orthonormal basis. We let ai = B ( x i , x i ) and let as = n,,,a,, and a, = 1; aN = ai. For each u = I:bsxs in E ( V) we define

ny==,

using S' to denote the complement of S in N. This 'star' operator was apparently introduced by Hodge in the case of fields of characteristic 0 (see [6] or [8]), was reintroduced as a ( u ) by Chevalley [3] for fields of characteristic not 2, and is intimately related to 4 ( u ) used by Bourbaki in 111, 11.1 [2]; in this last case 4 actually relates elements in the exterior algebras of V and its linear dual. There does not appear to be an explicit treatment of the star operator for characteristic 2, though implicitly the results that follow can be derived using Bourbaki's 4. Lemma 7. For any u E E ( u ) , u * * = aNu. Lemma 8. The map x + x * is a weight-preserving linear isomorphism of E ( u )

onto E " - k ( u ) . Lemma 9. e ( R , ( R U { i } y ) =E ( R , R ' ) E ( R , { ~ } ) .

-

Lemma 10. Let x E V be orthogonal to u1u2

* uk

E V. Then

x(uIu1'*'2)k)*=o.

Proof. Suppose uIu2*

Vk

=

c Psxs. For each subset R

of N of size k - 1,

is in the space spanned by the ui's, so it is orthogonal to x. Thus if x = I::=,c,ei, using angle braces to denote the inner product,

K.P. Bogart

86

However

= 0.

A result of major interest in coding theory is our next theorem.

Theorem 11. If { w l ,w2, ..., wk} is a basis for the subspace c of V, then ( w I w 2 . . . w k ) * is equal to u l u 2 - - * u n -where k { v I u 2 * ~ * v , -isk }a basis for the orthogonal complement C of C.

-

Proof. By Lemma 6, u(w,wz* * W k ) * is 0 for all u E C. Thus C is a subspace of the annihilator in E ( V ) of (wlw2* * w k ) * . By Lemma 4, if vl, uz,. . . ,u “ - ~is a basis for C,then for some y, uIu2 ’ ’ *

Un-ky

= (w1w2 ‘ . . wk)*.

Since both u I u 2 * *U*n - k and ( w I w 2 - *w*k ) * have degree n - k , y must be a scalar, and so may be “absorbed” into one of the u i k 0

Theorem 12. The mapping (x, y ) = (x y *)*/aNis an inner product on E, ( V )for which the basis of sets xs with S of size i is an orthogonal base. Further this extends by linearity to an inner product on E ( V ) which agrees with the original inner product on V, and in which Ei and E, are orthogonal if i# j . Proof. Note that if S # T but both have size i, then S n T’ is nonempty so xs (xT)* = 0. Thus (ZPsxs)(X?sys)* = ( Z P ~ Q ~ E ( S , S ~ ) U ~This ) X N .is clearly a symmetric bilinear map from Ei ( V )to En ( V ) ;applying the * map again gives a scalar and dividing by aN normalizes the value so the inner product agrees with the original one in u. 0


87

Theorem 13. The orthogonal complement of E ( C )in E (V) is the ideal generated by C', the orthogonal complement of C in V.

Proof. Let C have {ul, u,, ...,u k ) as a basis. By Lemma 10, a vector u is in C' if and only if u ( u 1 u 2 ~ ~ . v k =O. ) * But then u is orthogonal to any subset of { U I , U ~ , . . ,,Uk), SO U ( U , I U , 2 . . * U,,,,)*=O. Thus if U1, Uz,. . . , UJ E C ', and Z = zlul + z2u2+ * * + z,uJ has degree m, then

-

z (o,, Ut2 . . * Utm )* = 0

--

because each u, ( a , u r 2 . v,_)* = 0. Therefore each homogeneous element of the ideal generated by C ' is orthogonal in E ( V) to all elements of E ( C )of the same degree, and it is orthogonal by definition to the elements of E ( C ) of different degree. Therefore each element of the ideal generated by C" is orthogonal to each element of E ( C ) . Now suppose {u1,u2,..., u n - k ) is a basis for C ' and extend it to a basis {u,,u,, . ..,u,) of V. The intersection of the ideal generated by C' and E ' ( V) will contain each monomial in the vectors u, except for the monomials that contain none of u l , u 2 , ...,un-k. There are (t) such monomials, so the dimension of J ( C L )n E ' ( u ) is at least

Therefore the dimension of I ( C " ) is at least

2(3-(3=2"-2k,

i =(I

and since this is the dimension of E(C)', Z(C') must equal E ( C ) . 0

3. Applications to coding theory

We regard an (n, k ) code as a k-dimensional subspace C of the vector space V = F" with standard basis x i , x 2 , . . . ,x , and as before use E ( C )to stand for the subalgebra of E ( V ) generated by C and Z(C) to stand for the ideal of E ( V ) generated by C. We define the exterior weight enumerator f E ( = ) ( y , zto) be the weight enumerator of E ( C ) regarded as a code in E ( V) with the monomials in the x, 's as a basis. The ideal weight enumerator of C is the weight enumerator of the ideal generated by C. In light of Theorem 13 and the MacWilliams theorem [7], studying ideal weight enumerators is essentially the same as studying exterior weight enumerators of dual codes, so we restrict our attention to exterior weight enumerators.

Proof. The weight enumerator of the code direct sum of codes is the product of their weight enumerators, and the representation of E ( C )as the sum of Ei(C) is a code direct sum. 0 Note that Eo(C)is the field F, so its weight enumerator is (q - l)y + z and E,(C)is C itself and so its weight enumerator is the ordinary weight enumerator of C. One other factor of fE(c) is easily interpreted; namely fEk(c)(y,z). Since Ek(C) must be the one-dimensional space spanned by a basis product, it also has q - 1nonzero vectors in it, all having the same weight. In light of Theorem 1, this weight is the number of k element sets of columns of a generator matrix for C that have a nonzero determinant. Thus this weight is the number of information sets of C. Since this is the weight of a Plucker coordinate vector for C, we call it the Plucker weight of C, denoted by PW(C). Summarizing, we have,

Already this information allows us to distinguish between some codes with the same ordinary weight enumerator. For example, the two codes given by the generator matrices below have weight enumerator (z2 +

GI=

[:,

: : :::I9

0 0 1 1 0 0

[

1 1 0 0 0 0

G2= 0 1 1 0 0 0 0 0 1 1 1 1

1

However the code generated by GI has Plucker weight 8 and the code generated by G 2 has Plucker weight 10, and so the codes have different exterior weight enumerators, so they are not equivalent. Similarly, it is possible to give examples of codes with different weight enumerators and the same Plucker weight; once again such codes will have different exterior weight enumerators. There are no known examples of inequivalent binary codes with the same exterior weight enumerators. Both the ordinary weight enumerator and Plucker weight are examples of matroidal invariants of codes; namely invariants that depend only on which sets of coordinate positions are information sets. For arbitrary fields, it is not the case that two codes with the same information sets are equivalent, though two such binary codes are identical. Theorem 16. Two binary codes with the same information sets are identical. Proof. For two binary codes to have the same information sets, they must have


89

the same Plucker coordinate vector. However, Corollary 3 shows that two subspaces with the same Plucker coordinate vector are identical. 0 The generator matrices G3 and G4 below are generator matrices' for MDS codes over an extension field of GF(4) in which u is a cube root of A, and A' + A + 1 = 0. Note that 1, a, and u2 are linearly independent over GF(4).

1 0 0 1 1 0 1 0 1 A 0 0 0 1 A+l

1 A+l], A

[

G4= 0 1 0 1 A 0 0 1 1 A+l

u2

Generator matrices for the 'second exterior powers' E2(C3)and E2(C4) are

1 0 1 A A+101 1 1 0 0 O A + l A 0 1 l A + l A 0 0 0 0 1 1 1 A A+l 0 0 0 0 0 1 l A + l A l A A + l 1 1.

G:

=

']. 1

1 0 1 A cr 0 1 1 1 0 0 0 A + 1 1 + u 0 1 l A + l u 2 0 0 0 0 1 1 1 A 1+u2 A+1+u2 0 0 0 0 0 1 1 A + l u2 1 A u 1 u + u 2 A u + u + A a 2

The sum of the first two rows of the matrix G: has weight 13, but no vectors in the code spanned by G: have weight 13, so the codes C3and C, spanned by G, and G, cannot be equivalent. This example might lead us to conjecture that exterior weight enumerators distinguish between inequivalent codes over all fields. However Gs and G, below are generator matrices for two MDS codes Cs and C,over the same extension field of GF(4) above.

1 0 0 1

1

1 0 0 1

1

0 0 1 1 h+l

By direct computation, it is possible to check that E ( C s )and E ( C 6 )have the same weight enumerators. The matrices G: and GL are generator matrices for the codes E2(Cs)and E2(C6): 1 0 1 A 0 1 1 A+l 0 0 0 0

0 1 1 O O A + 1 0 0 0 1 1 1 1 A + 1 1 A 1

l O l u 0 1 1 O O a + l 0 1 1 u2 0 0 0 1 1 u 2 + 1 ] . 0 0 0 0 1 1 u z 1 ff u + u 2

Now the generator matrices G'; and G: for E2(E2(C5)) and E2(E2(C6)) will each have 90 columns, so they are not listed here! Of these columns they will share 50,

K.P. Bogart

90

while the 40 columns that involve 2 by 2 determinants of submatrices of GI, or GA using columns 4 , 7 , 9 or 10 will be different. In fact, any column of GI: derived from 2 columns of {4,7,9,10}turns out to be a multiple of (1, A, A + l)', while any column of GZ derived from 2 columns of {4,7,9,10}turns out to be a multiple of (1, u,u2)1. Thus when we compute the Plucker coordinate vector from E2(E2(C5)) the determinant

[; ; : ] = o 1 1 A+l

arises often and the corresponding Plucker coordinate of E2(E2(Ch)) is the determinant

[p p

;2]=1+,+,,0.

is zero, the correspondHowever, whenever a Plucker coordinate of E2(E2(C6)) ing Plucker coordinate of E2(E2(Cs))is zero. Thus the Plucker weight of E2(E2(Ch)) is higher than that of E2(E2(C5)), so that E ( C s ) and E(C,) have different exterior weight enumerators, and therefore Cs and C, are not equivalent. The examples given here suggest that we may well be able to distinguish between inequivalent codes by computing weight enumerators (or perhaps Plucker weights?) of iterated exterior algebras of the codes. In the case of binary codes, the author has neither been able to prove nor give a counterexample to the conjecture that the exterior weight enumerator of a code determines the code up to equivalence.

References (1J K. Bogart, D. Goldberg and J. Gordon, An elementary proof of the MacWilliams on equivalence of codes, Inform. Control 37(1) (1978) 19. [2] N. Bourbaki, Elements of Mathematics, Algebra, Part 1 (Hermann, Paris, 1974) Chapter 111. [3] C. Chevalley, Algebraic Theory of Spinors (Columbia University Press, New York, 1954). [4] J. Gordon, Application of the exterior algebra to graphs and codes, Ph.D. Dissertation, Dartmouth College, 1977. (51 W. Greub, Multilinear Algebra (Springer-Verlag, Berlin, Heidelberg, New York, 1967). [6] W.V.D. Hodge and D. Pedoe, Methods of Algebraic Geometry (Cambridge University Press, London, 1947). [7] F.J. MacWilliams and N. Sloane, The Theory of Error-Correcting Codes, Part I, North-Holland Mathematical Library Vol. 16 (North-Holland, Amsterdam, 1977) Chapter 5. [8] M. Marcus, Finite Dimensional Linear Algebra, Part 2 (Marcel Dekker, New York, 1975). [9] B. Van der Waerden, Einfiihrung in die Algebraische Geometrie (Dover, New York, 1945).


THE EVOLUTION OF THE CUBE BCla BOLLOBAS Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England

Given a natural number n, let C" be the graph formed by the vertices of the n-dimensional cube. Thus if the vertices of the cube are the 0-1 sequences ( E , , E ~ , .. . , E , , ) then two sequences are adjacent in C" if they differ in exactly one coordinate. We call C" the n-cube. Our aim is to study random subgraphs of C". Let C; be a random subgraph obtained by choosing the edges of C" independently of each other and with probability p , what is the probability that Cn.pis connected? Burton [3] showed that if p < 5 then a.e. Cf:is disconnected and if p > fthen a.e. Cf:is connected. (For the standard concepts and results concerning random graphs see [2; Ch. VII].) The problem is particularly interesting because of a curious 'double jump' at p = i: Erdos and Spencer [ 5 ] proved that the probability of C ; being connected tends to e-' for p = 4. This double jump is reminiscent of the double jump in the size of the largest component of a random graph encountered by Erdos and RCnyi [4]. We shall prove a slight extension of this result: by concentrating on a small neighbourhood of p = f we shall show the existence of a continuous probability distribution. Not unexpectedly, it turns out that the obstruction to connectivity is the existence of an isolated vertex: if p = p ( n ) is such that P (Ci has an isolated vertex)+c for some c then P ( C ; is connected)+c as well. The proof of our result is very close to the proof given by Erdos and Spencer [5].In particular, we need a result first proved by Harper [6] (see also Bernstein [l] and Hart [7]) about the maximal number of edges spanned by k vertices of the n-cube. By identifying a 0-1 sequence ( E , , E ~. ., .,E,,) with the integer ~ , 2 ' - 'the , cube C" has vertex set {0,1,. . . ,2" - 1) and k is joined to 1 if their binary expansions differ in exactly one place. We state the theorem of Harper 161 as our first lemma since it gives a property of the cube which is crucial in our calculations.

x:=,

Lemma 1. The maximum number of edges spanned by a set of k vertices of C" is equal to the number of edges spanned by {0,1,.. . , k - l}, namely 91

B. Bollobds

92

where h ( i ) is the sum of the digits in the binary expansion of i. For any set S of k vertices there are at least k ( n - [logzk ] ) edges joining S to V(C")\S. Now we come to the main result of the paper. Theorem 2. Let A > O be a constant and p = p ( n ) = 1-fA""(l+o(l/n)). Then limn-= P(C;is connected) = e-*. Proof. Denote by X = X ( C ; ) the number of isolated vertices in C;.We shall first show that

where P ( A ) is the Poisson distribution with mean A, and so lim P(C;has no isolated vertex) = e-A. "A=.

(3)

As usual, (2) follows if we show that for every fixed r 2 0 the rth factorial moment of X tends to A '. Counting very crudely indeed, a set of r vertices is incident with at least r(n - r ) edges and at most rn edges. Furthermore, there are at most

sets of r vertices incident with fewer than rn edges. Hence, with ( a ) b= a ( a - 1 ) ( a - 2 ) . . * ( a - b + l), we have (2"),4"" S E , ( X ) S (2"),q'" + 2"(7-l)rnqr . ( n - r . ) This gives

~E,(X)~(2q)"'{l+2-"rnq-'~}. The condition on p is equivalent to Iim (2q)" = A, n-rn so E,(X)+A'

and ( 2 ) holds.

The evolution of rhe cube

93

Now we turn to the essential part of the proof. We shall show that the probability that C; contains a component with at least 2 and at most n / 2 vertices tends to 0. This and (3) will show that the probability of C; being disconnected and not containing an isolated vertex tends to 0, so the theorem holds. In fact we shall prove somewhat more than we need. Let Ce, be the family of s-subsets of V = V(C") spanning a connected subgraph in C".

Lemma 3. Let p = 1 -$(log n)"". Then the probability that for some S E Y S , 2 s s S n / 2 , no edge of C: joins S to V \ S tends to 0. Proof. Given a set S C V denote by b ( S ) the number of edges joining S to V\S. (We think of b ( S ) as the number of edges in the edge boundary of S.) Then the lemma claims that

By ( 1 ) we know that

b ( S ) > s ( n - [log2s]). In order to prove (4) we partition the range of the summations into several parts. (i) Set s, = L2"/*/n2~ and suppose that 2 s s < sl. Our first estimate is very crude:

1 YS1 s 2 " n ( 2 n ) .. . ( ( s - 1 ) n ) = ( s - l ) !nS-'2". Hence

Since q" = 2-" log n, this gives

Take the logarithm of n s ' l s ! times the right-hand side: 2s log, s + s log2n - (s - l ) n + s log, log n.

If n is sufficiently large, this expression is at most 0 for every s in our range. This is perhaps easiest seen by considering s =s n and s n separately. Hence

and so

B. Bollobds

94

2 p= o(1).

s=2

Let us turn to the range s I< s S 2"-'. Here we shall partition % into two sets. Let %; consist of those sets S in Ce, for which

b ( S ) s s ( n -log2s+log2n),

' 1%;. and set %,+ = K (ii) Let us estimate our double sum in the range sI< s G 2"-' and S E (e;. Clearly

Hence

Our final task is to show that as s runs over [ s l l ,[sll+ 1,. . . ,2"-' and S E %?:, our double sum is o(1). Here we have to be more careful. We shall need the following lemma.

Lemma 4. Let G be a graph of order u and suppose that A ( G )G A , 2e ( G ) = vd and A + 1 s u S u - A - 1. Then there is a u-set U of vertices with

Here, as customary, A ( G ) is the maximum degree, d is the average degree and T ( U )= { x E G :x y E E ( G ) for some y E V } . Proof. What is Av(u - ( N (U ) l ) ,the average of u - IN(V)l as U runs over all u-sets of V(G)? Let d l ,d 2 , .. . , d , be the degree sequence of G. Then

Av(u - 1 N ( U )I) =

2 ('

i=l

- dl U

')/ ( :)

,

since a vertex of degree d , does not belong to N ( U ) for exactly

(u-:-l) choices of a u-set U. The function (t) is convex for x s u so

c:=,d, = ud implies

The evolution ofthe cube

95

Consequently for at least one u-set U we have

1 N ( u)la u - tl

exp

(-+)

+ u (1

-i)

Proof of Lemma 3 (continued). (iii) Let us estimate the double sum over the range s I S s G s2 = [2"/(log n)4J and S E %':. Then b ( S )S s ( n - log2 s +log, n ) , that is in H = C " [ S ]the average degree is at least log, s - log, n. Set

By Lemma 4 there is a set U C S such that N ( U ) ,the set of vertices of H within distance 1 of U, satisfies

This shows that the sets in %': can be selected as follows: first we select u vertices of C" then [ s / 3 ] - u neighbours of these vertices and then [ 2 s / 3 Jother vertices. Hence

and so

B. Bollobds

96

Write s = 2@"so that

p G 1- 4 log,log,/n. Then the sum above is at most

(c log ny22sn(l-@)/3-sn+s@n = (c log n)s2-sn(l-~)~3 9

where c is a positive constant. By our assumption on p, if n is sufficiently large, this is at most (log n)-'14. Hence S.

(iv) Finally suppose that [2"/(log n)4] = sz + 1 G s s 2"-' and S E U:. Then in H = C " [ S ]the average degree is at least log2s - log, n > n - 2 log, n. We shall find an even smaller subset U of S that 'fixes' many vertices of U. First we look for a subgraph of H with large average degree. Denote by T the set of vertices of H with degree at least n - (log, n)2 and set t = 1 T I . Then tn + ( s - t ) ( n - ( l o g 2 n ) 2 ) a s ( n -210gzn),

so

Let HI be the subgraph spanned by T : HI = C " [ T ]= H [ T ] . Consider the following crude estimate of the size of HI:

e ( H , ) a e ( H ) - (s - t)n

a ? s ( n -2Iog,n)--

2s n. log2 n

Consequently the average degree in HI is at least 4n an--. 5n n -210g2n -log2 n logz n Set u = [2"/n '/*I. By Lemma 4 there is a u-set U in T whose neighbourhood in HIis large:

The evolution of the cube

97

The existence of such a u-set restricts severely the number of s-sets S E %':. Indeed, the number of neighbourhoods NH( V )belonging to a u -set above is at most

where the summation is over all (k,,k2,... ,k,) with ki d (log2n)', for at most (log,n)' of the n neighbours of a vertex x E U do not belong to N H ( U ) . Consequently with w = [8s/log2n j we find that in our range

< p', where & (s)

= o(s)

{

-s n

- log2s - log2log n

+h s log2log n n

1

.

AS s s 2"-', we have & ( s ) s o ( s ) - - s { n -(n-l)-log210g, = o(s) - s

+-n -n1

log2log n}

{: I

1 - - log2log n d - s / 2 .

Consequently

as required. This concludes the proof of Lemma 3 and so the proof of Theorem 2

is complete.

0

References [I] A.J. Bernstein, Maximally connected arrays of the n-cube, SIAM J. Appl. Math. 15 (1967) 1485-1489. [2] B. Bollobls, Graph Theory An Introductory Course, GTM Vol. 63 (Springer, New York, Heidelberg, Berlin, 1979). [3] Y.D. Burtin, On the probability of connectedness of a random subgraph of the n-cube, Problemy Pered. Inf. 13 (1977) (in Russian). [4] P. Erdos and A. Rinyi, On the evolution of random graphs, Mat. KutatB Int. Kozl. 5 (1960) 17-60. [5] P. Erdos and J. Spencer, Evolution of the n-cube, Comput. Math. Appl. 5 (1979) 33-39. [6] L.H. Harper, Optimal assignments of numbers to vertices, SIAM J. Appl. Math. 12 (1964) 131-135. [7] S. Hart, A note on the edges of the n-cube, Discrete Math. 14 (1976) 157-163.

-



CONSTRUCTION OF 4-REGULAR GRAPHS Francette BORIES and Jean-Loup JOLIVET Universitt! du Maine. France

Jean-Luc FOUQUET CNRS, Paris, France We show that the class of 4-regular, simple, connected graphs can be generated by three extensions from K,,and by two extensions from K , and two other graphs of order 6 and 11. These constructions yield a proof of the existence of an edge-partition of a 4-regular graph in three linear forests.

0. Introduction Nous donnons deux constructions de la classe (e4 des graphes simples, 4-rkgguliers, connexes en utilisant quatre extensions. Ce type de problhme a CtC CtudiC prCcCdemment par Johnson [5] dans le cas des graphes cubiques; on en trouvera une Ctude dCtaillCe dans la thbse de Fouquet [2] et dans [3] et [ 4 ] .

1. Definition des extensions Soit G = (X, E ) un graphe de (e4 d’ordre n. (1) Soient [ a , b ] et [ c , d ] deux arktes de G formant couplage, % , ( G , [ a , b ] , [ c , d ] est ) le graphe d’ordre n + 1 obtenu B partir de G en supprimant les arEtes [a,b ] et [c, d ] et en ajoutant un nouveau sommet a et les arktes [ a , a ] ,[ q b ] , [ a , c ] ,[ a , d ] (Fig. 1). (2) Soient [a,a‘], [ b , b ‘ ] , [c,c’] trois arCtes de G formant couplage, g 2 ( G ,[a,a ’ ] ,[b, 6’1, [c, c ‘ ] ) est le graphe d’ordre n + 2 obtenu B partir de G en supprimant les arEtes [ a , a ’ ] , [b,b’], [ c , c ’ ] et en ajoutant deux nouveaux sommets a et a ’ e t les arEtes [a,a’], [a,a ] , [a,b ] , [ a ,c ] ,[ a ’ ,a ’ ] ,[a’, b ’ ] ,[a‘, c’] (Fig. 2). (3) Soit e un sommet de G dont les voisins sont notes a, 6, c, d, g3(G,e ) est le graphe d’ordre n + 3 obtenu h partir de G en supprimant le sommet e et en ajoutant une clique d’ordre 4 de sommets a,p, 7, S et les arCtes [a,a ] ,[b, p ] , [ c , y I , [ d , S l (Fig. 3). 99

100

F. Bories ei al. a

Fig. 1.

Fig. 2.

Fig. 3.

(4) Soit [a,b ] une artte de G, g5(G,[a,b ] ) est le graphe d’ordre n

+ 5 obtenu

A partir de G en supprimant I’artte [a,b ] et en ajoutant une clique d’ordre 5 de sommets a, p, 7, 6, E moins I’artte [a,p ] puis en ajoutant les arttes [a,a ] et M,bI (Fig. 4). Tout graphe obtenu 2 partir d’un graphe de q4 par I’une de ces quatre extensions est Cvidemment un graphe de Ce4.

Construction of 4-regular graphs

101

Fig. 4

2. Definition des reductions

Nous noterons Ri ( i = I, 2 , 3 , 5 ) les riductions, opkrations inverses des extensions 2% difinies ci-dessus. Plus prkcisiment, si G = (X, E ) est un graphe de 0, d’ordre n. (1) Nous dirons que G est R,-rkductible s’il possbde un sommet a de voisins a, b, c, d, tels que le graphe R,(G,a, [a,b ] ,[ c , d ] ) ,d’ordre n - 1, obtenu ii partir de G en supprimant le sommet a et en ajoutant les aretes [a,b ] et [c, d ] ,soit un graphe de Le sommet a sera alors dit R,-rkductible.

w4.

Lemme 1. Avec les notations ci-dessus, pourque le sommet a soit RI-rkductible,il sufit que [a,b ] et [ c , d ] ne soient pas dans E et que les sommets a et d appartiennent ci une mime chaine ne passant pas par a. Preuve. Le graphe R l ( G ,a, [a,b ] ,[c,d ] ) est 4-rigulier par construction. Si de plus, [a,b ] et [c, d ] ne sont pas dans E, ce graphe est simple et si les sommets a et d appartiennent h une mkme chaine ne passant pas par a, la connexitk est prkservke (Fig. 1).

(2) Nous dirons que G est R,-rkductible s’il possbde deux sommets adjacents a et a‘ de voisins respectifs a, b, c et a ’ , b‘, c’ tels que le graphe R,(G, a, a’,[a,a’],[b,b’])d’ordre n - 2, obtenu h partir de G en supprimant les sommets a et a ‘ et en ajoutant les arktes [a,all, [b,6’1, [c,c‘],soit un graphe de 0 4 .

L’arkte [a,all sera alors dite R,-rkductible. Lemme 2. Avec les notations ci-dessus, pour que l’arite [ a , a ’ ] soit R,rkductible, il sufit que [a,a ‘I, [b,b’], [c, c’] ne soient pas dans E et que a, b, c (resp. a ’ , 6’. c ‘ ) appartiennent ci une mime chaine ne passantpas par a (resp. a’).

102

F. Bories et al.

Preuve. Analogue A celle du lemme 1 (Fig. 2).

(3) Nous dirons que G est R,-rt?ducrible s’il admet comme sous-graphe une clique d’ordre 4 de sommets a, p, y, 6 ayant comme voisins respectifs (distincts de a, p, y, 6 ) a, b, c, d tels que le graphe R3(G,a,p, y, a), d’ordre n - 3, obtenu A partir de G en supprimant les sommets a, /3, y, 6 et en ajoutant un nouveau sommet e et Ies aretes [e,a ] , [e,b ] , [e,c ] , [e,d ] , soit un graphe de Vd. Lemme 3. Avec les notations ci-dessus, pour que G soit R,-rkductible, il sufit que les quatre sommets a, b, c, d soient distincts deux d deux. Preuve. Immediate (Fig. 3).

(4) Nous dirons que G est Rs-rt?ductible s’il admet cornrnet sous-graphe une clique d’ordre 5 de sommets a, p, 7, 8, E rnoins I’arCte [.,PI tels que, en disignant par a (resp. b ) le quatrikme voisin de a (resp. p), le graphe R,(G, a, p, y, 6, E ) , d’ordre n - 5, obtenu en supprimant les sommets a, p, y, 6, F et en ajoutant I’arite [a,b ] , soit un graphe de V,. Lemrne 4. Aoec les notations ci-dessus, pour que G soit R,-rt?ductible, il sufit que l’arite [ a ,b ] ne soit pas dans E. Preuve. Immediate (Fig. 4). Lemme 5. Tout sommet d’articulation d’un graphe de

V.,est R,-rkductible.

Preuve. Soit G = ( X ,E ) un graphe de V, et soit a un sommet d’articulation de G, le sous-graphe G,+{,) engendrC par X - { a } a exactement deux composantes connexes reliCes chacune A a par deux arites (en effet, un graphe simple 4-rigulier n’a pas d’isthme, sinon la somme des degrCs de chaque coniposante serait de la forme 4(nl - 1) + 3 qui est impair). Soient a, b, c, d les voisins de a, a et c ttant dans I’une des composantes et b et d dans I’autre, le graphe R , ( a , [ a ,b ] , [ c , d ] est ) alors dans V4 d’aprts le Lemme 1 (Fig. 5).

Fig. 5


103

Lemme 6. Tout sommet a non R ,-re‘ductible d’un graphe de %4 appartient a une clique K,, O M le sous-graphe qu’il engendre avec ses voisins est G3,:ou (a,b, c ) esi stable. Preuve. Soit G = (X, E) un graphe de g4et soit a! un sommet de G qui n’est pas R,-rkductible. DCsignons par r ( a )= { a , b, c, d } I’ensemble des sommets adjacents B a. D’aprds le Lernme 5, le sous-graphe engendrt par X -{a} est connexe, et si a n’est pas R,-rkductible, l’une des deux arites [a,b ) ou [c, d ] au moins est dans E, par exemple [c, d]. L‘une des deux arctes [ a ,d ] ou [ b , c ] au rnoins est dans E, par exemple [ a , d ] . L’une des deux arites [ a , c ] ou [ b , d ] au rnoins est dans E. G contient alors cornme sous-graphe partiel I’un des deux graphes suivants (Figs. 6 et 7). Dans le premier cas, a appartient 2 la clique (a.a, c, d ) . Dam le deuxicrne cas, si ( a , b, c ) est stable, on obtient le graphe G 3 , 2Sinon . a! appartient B une clique

K4. Lemme 7 . Tour graphe de Ce4 admettant G,, rkductible.

comme sous-graphe est R I -

Preuve. Avec les notations du lemme 6, disignons de pius par b , et br les voisins de b diffirents de a! et d . G Ctant 4-rCgulier, [ b , ,a ] et [ b z ,d ] ne sont pas des ar6tes de G, a et d &ant de plus adjacents, b est R,-rkductible d’aprks le Lemme 1 (Fig. 8).

Fig. 6.

Fig. 7. a

d

Fig. 8.

F. Bories et al.

104

Lemme 8. Tout graphe de est R ,-riductible.

n ‘admettant pas de clique Kd comme sous -graphe

Preuve. Soit G un graphe de %4 n’admettant pas de clique K4 comme sous-graphe. Si G admet un sommet non R,-rCductible, il admet le graphe G . x ~ comme sous-graphe d’aprbs le Lemme 6 et il est R,-rkductible d’aprks le Lemme 7.

3. PremPre construction de Thhorerne 1. L a classe %, des graphes simples, 4-rkguliers, connexes, est 8,. engendrie ci partir de la clique K s d ’ordre 5 au moyen de trois extensions 85.

Preuve. Nous allons montrer que pour tout graphe G = (X, E), d’ordre n, de %, il existe i ( = 1’3’5) tel que G soit R,-rkductible. Si G possbde un sommet d’articulation ou si G n’admet pas de clique K , comme sous-graphe, ou si G admet G3.*comme sous-graphe, il est R,-rkductible d’aprbs les Lemrnes 5 , 7 et 8. Supposons donc que G ne possbde aucun sommet d’articulation, n’admette pas de graphe G3.2comme sous-graphe et admette une clique K , comme sous-graphe. Nous dCsignerons par a, 6 , c, d les sommets de cette clique et par ( a , ,b,, c,, d , ) les sommets (distincts de a, b, c, d ) adjacents respectivement 2 a, b, c, d. Si G est diffkrent de K 5 , a l , b,, c I , d l ne sont pas tous confondus. ler cas: a l , b l , c,, dl distincts deux i deux. G est R&ductible d’aprbs le Lemme 3. 26me cas: cl = d l , a , # cl, 61 # c,, a l et bl distincts ou non (Fig. 9). DCsigons par c2 et d 2 les sommets distincts de c et d adjacents Q c , . c I Ctant distinct de a l et b , , n’est adjacent ni 2 a, ni 2 b donc cz (resp. d 2 )est distinct de c et d (mais peut etre confondu avec a , ou b l ) .On en dCduit que [c, c:] et [ d , d , ] ne sont pas dans E, donc que c1 est R,-rkductible d’aprbs le Lemme 1. 3 i m e cas: b, = c1 = d , et a l # b , (Fig. 10). c2

b

C

d

a

Fig. 9.


105

C

b

E

Fig. 10.

DCsigons par b, le sommet, diffirent de b, c, d, adjacent a b,. G n’ayant pas de sommet d’articulation, bz est distinct de a l . Si [ a , , b z ]n’est pas dans E, G est R,-riductible d’aprks le Lemme 4. Si [ a , , b 2 ]est dans E, soit I’un des deux sommets a , ou bz est R,-rCductible. soit a 1 et bz appartiennent B une clique K4 d’aprks le Lemme 6; on est alors ramen6 pour cette clique au ler ou au 2kme cas. On a ainsi montr6 que tout graphe de q42 l’exception de K , est R , , ou R,, ou R5-rCductible. En faisant subir 8 un graphe G de Z4des riductions R I , R,, R, toutes les fois que c’est possible, on obtient la clique K , . RCciproquement, en faisant subir 8 K , les extensions inverses on reconstruit G. Tout graphe de V, s’obtient donc partir de K , au moyen des trois extensions 8,, 8,, g5. Remarque. On ne peut pas engendrer la classe (ela partir de K , au moyen de deux seulement des extensions prCcCdentes, en effet: (1) Nous noterons G6 le seul graphe de Ce., d’ordre 6, qui est obtenu en effectuant l’extension EP, sur un couplage quelconque de deux aretes de K 5 (Fig. 11). partir de K , au moyen d’un nombre G , et tous les graphes obtenus quelconque d’extensions 8, n’admettent aucune clique K , comme sous-graphe, donc ne sont ni R,-rCductibles, ni R5-r6ductibles. (2) Tous les graphes obtenus B partir d’un graphe quelconque de (e4en faisant I’extension 8, en chaque sommet ne sont ni I?,-rkductibles, ni Rs-rCductibles.

F. Bories et al.

106 e

f

Fig. I I .

(3) Tous les graphes 4-rCguliers obtenus en remp1ac;ant une arete sur deux d’un cycle pair par une clique K s moins une arCte ne sont ni R,-rtductibles, ni R 3-r&ductibles.

4. Deuxieme construction de %.,

Notation. Nous disignerons par 8, + gj ( i = 1,3,5; j = 1,3,5) une extension sur G obtenue en effectuant une extension 8, sur G, puis une extension %, sur le graphe obtenu, et par R, R, la reduction inverse. Nous dirons de plus que G est R, + R,-riductible si on obtient un graphe de %., en effectuant sur G une reduction R,, puis sur le graphe obtenu une reduction R,.

+

D’aprts le Thtorkme 1, tout graphe G de (e4est: soit K5;soit obtenu 2 partir de Ks au moyen d’une extension 8,,ou % ou 8,; soit obtenu ti partir d’un autre graphe G’ de (e., au moyen d’une extension Gp, + gj (i = 1,3,5 et j = 1,3,5). Lemme 9. Si un graphe de rkductible ou c’est G,.

(e4 a

tous ses sommets R,-rkductibles, il est R I + R , -

Preuve. Soit G = ( X ,E ) un graphe de V, dont tous les sommets sont R,rkductibles. Remarquons tout d’abord que si G admet deux sommets ayant au plus un voisin commun, on obtient un graphe de V4 en effectuant deux reductions R , successivement en ces deux sommets et G est alors R , + R , reductible.


107

De plus, G ne peut contenir de clique K4. Or, les extensions 8,et 8, introduisent des cliques K4. Donc G est soit G6,soit obtenu B partir d'un autre graphe G' de V, au moyen d'une extension:

8,+ 8 , (cas l), ou

8,+ 8 , (cas 2), ou

8, + 8 , (cas 3).

Dons le premier cas: G est R , + R,-rkductible. Dans les cas 2 et 3, G ne contenant pas de clique K4, I'extension 8 , pratiquke utilise au moins une des arites de chaque clique K4 introduite par I'extension 8, ou $ 5 . Dam le deunitrne cas: Soit G' = (X', E'). Avec les notations de la Section 1, G = 8,(8,(G',e ) , [a, b ] ,[ c , d ] ) . Soit G" le sous-graphe de G ' engendri par X I - { e } . Si G" admet au moins six sommets, le sommet cy introduit par I'extension 8 , est adjacent dans G, a au plus deux somrnets de G" et ces deux somrnets ne peuvent itre adjacents, dans G, i tous les autres sommets de G". Un des sornmets de GI' a donc au plus un voisin commun avec a et G est R , R,-rkductible (Fig. 12). Si G" a cinq sornmets, G' est G6et G" est le graphe ci-contre (Fig. 13), d a m

+

G'

Fig. I2.

Fig. 13.

108

F.Bories et al.

lequel aucun couple de sommets adjacents n'est adjacent aux trois autres. U n des sommets de G" aura donc au plus un voisin comrnun avec a et G est R , + R ,-rCductible. Si G " a quatre sommets, G ' est K , et la seule extension g 3 +8, de G' supprimant les cliques K4 est comme dans Fig. 14.

Fig. 14.

Le graphe obtenu a tous ses sommets R,-rCductibles, et il existe deux sommets (notis a et b sur la figure), non adjacents, ayant un seul voisin comrnun a. G est donc R I + R,-rCductible. Dans le troisitme cas: Si G ' a au moins six sommets, un raisonnement analogue 2 celui fait dans le deuxikrne cas (pour G"ayant au moins six somrnets), montre que G est R , + R,-rCductible. Si G ' est K , , la seule extension 8, + 8,de G' supprimant les cliques K4 est comme dans Fig. 15.

Fig. 15.

Comrne dam le deuxikme cas, le graphe obtenu a deux sommets ( a et b sur la Fig. 15) ayant au plus un voisin commun et est R 1+ R1-rCductible. Lernme 10. Si un graphe de g4admet comme sous-graphe le graphe G.3.2difini dans le Lemme 2, il est R , + R,-rkductible. Preuve. On utilise les notations du lernme 6. DCsignons par b 1 et b2 (resp. c , et c,) les sornmets adjacents ?I b (resp. c ) et distincts de (Y et d. On a montrC dans le Lemme 7 que b est R1-r6ductible (Fig. 16). Dans le graphe rCduit, cI (resp. cz) peut 2tre confondu avec b , ou bl, mais ne peut itre adjacent B la fois B (Y et B d. De mCme (Y (resp. d ) ne peut Ctre adjacent B la fois ?I c , et cz, c est donc R,-riductible dans ce graphe rCduit, d'aprks le Lemme 1.


109

Fig. 16.

Lemme 11. Tout graphe de V4 n’admettant pas de clique K4 comme sous-graphe et posse‘dant un sommet non R,-riductible est R I + RI-re‘ductible. Preuve. Soit G un graphe de Ce4 sans K4 et soit a un sommet non RI-rkductible de G. D’aprks le lemme 6, G admet comme sous-graphe le graphe G3.?donc est R I + R I-rCductible d’aprks le Lemme 10. Notation. Nous disignerons par GI, le graphe (Fig. 17).

11 sommets reprksenti ci-contre

Fig. 17.

T h e o r h e 2. La classe %., est engendre‘e 6 partir de K S , G6 et GI, au moyen d’extensions et + 8,.

Preuve. D’apr6s ies Lemmes 9 e t 11, tout graphe de Ce, sans clique K , est R , + RI-rCductible ou est Gh. Soit G = ( X , E ) un graphe de V4 admettant comme sous-graphe une clique K4de sommets a, b, c, d. Nous dksignerons par al. bl, c , , d , les sommets (distincts de a, b, c, d ) adjacents respectivement h a, 6, c, d. Si G est diffirent de K s , a , , b l , cI, d l ne sont pas tous confondus. f e r c a s : a , , b l ,cI, distinctsdeux u deux, c I e t d , distinctsou non (Fig. 18). L’arete [a, a l l n’appartient h aucun triangle. L’ensemble {b, c, d } des voisins de a autres que a l est lie I’ensemble {a?,a : , a ? } des voisins de a l autres que a par

F.Bories et al.

110

Fig. I8

au plus les aretes [b, b , ] , [c, c , ] , [d, d , ] parmi lesquelles deux au plus ont un somrnet commun. 6, c, d itant de plus adjacents deux a deux, I’arCte [ u , a l ] est R?-rCductible d’aprks le Lemrne 2. Zkme cas: ul = bl et c I = d , . Si les sommets a l et c I sont non adjacents et ont a u plus un voisin commun, en effectuant successivernent deux rtductions R , en u l et en c I , on obtient un graphe de ( e 4 (Fig. 19). Si a, et c , ont deux voisins communs, ces deux rkductions donnent encore un graphe de (e4,car u,b, c, d ne sont adjacents A aucun de ces voisins (Fig. 20). Si u Iet cIsont adjacents, dksignons par az le q o a t r i h e somrnet adjacent a a , . Si a? et c I ne sont pas adjacents I’ensemble {a,6, c l } des voisins de a l distincts de a? est lit par une arCte au plus B I’ensemble des voisins de a2 distincts de a l (Fig. 21). L’arCte [a,, a,] est donc Rt-rCductible. b

C

d

a

Fig. 19.

Construction

111

of 4-regular graphs

Fig. 20.

Fig. ? I .

1

Si a z et c , sont adjacents, a , est R,-rCductible et a2est point d’articulation du graphe rtduit donc est R,-rkductible dans ce nouveau graphe (Fig. 22). 3tme cas: b , = c1 = d , et a , # bl (Fig. 23). Si a , n’est pas adjacent ?i b , , I’arite [a, a , ] est R&ductible. Sinon, a , est point d’articulation et est R ,-rkductible. DCsignons par G’ le sous-graphe engendrt par X - { a , b, c, d, b , , a,}. Si tous les sommets de G’ sont R,-rkductibles dam G, on obtient un graphe de %,‘ en effectuant successivement deux rkductions R , en a , et en un sommet de G‘ non adjacent ?i a , et ayant au plus un voisin commun avec a,.

F.Bories et a!.

112

Fig. 22. C

d

b

a

Fig. 23

Si G ’ contient un sommet qui n’est pas RI-rCductible dans G, d’apr6s le Lemme 2, soit G contient G3.*et est R1+R1-rtductible (Lemme 9), soit G contient une clique K , et d’aprbs l’etude ci-dessus, il est R,-rkductible ou R , + RI-rkductible sauf si G = G I , . On a ainsi montrC que tout graphe de %, 2 I’exception de K , , G,. G I , est R,-rCductible on R I + R,-rkductible, donc, par un raisonnement analogue Q celui fait dans la dtmonstration du Thtorbme 1, que tout graphe de Ce, s’obtient a partir de K,, G6 et Gll au moyen des extensions g2 et $, +

Remarque. On ne peut pas engendrer la classe 5 partir de K s . G , et G I ,avec I’une seulement de ces deux extensions, en effet: (1) Le carri d e tout cycle de longueur suptrieure ou igale a 5 (obtenu en ajoutant au cycle toutes les aretes entre deux sommets a distance deux) n’est pas R,-rCductible car chaque ar&te appartient a un triangle.


113

(2) Tout graphe qui n'est pas R,-rkductible, n'est pas non plus R , + R I rkductible et on a montrk B la suite du Thkorkme 1 qu'il existait une infinit6 de tels graphes. Corollaire. La classe Cej est engendrke a partir de K s au moyen des extensions 8 , et gZ.

v4

Preuve. Le ThCorbme 2 montre que est engendrke B partir de K5,G6 et GI, au moyen des extensions 8, et g2.Or G6et GI, sont R,-rkductibies, on obtient donc la classe Ce4 2 partir de K,. On retrouve ainsi le rksultat dkmontrk par Toida [91.

5. Application

Theoreme 3. Pour tout graphe G = ( X , E ) de Ce4, il existe une partition de E en trois forits F , , F2, F1forme'es chacune de chaines e'lkmentaires. Preuve. La clique K s posskde cette propri&tCcomme en tkmoigne la Fig. 24:

Fl = { [ a ,b l , [b, cl. [ c ,dl, [ d , e l ) ,

F2 = { [ e ,a ] ,[ a , d ] ,[ d , b ] ) , F3 = ([b, e I , [e,c I, [c, a 11. Nous montrerons maintenant que chacune des extensions 8,, 8, et 8, pratiqute sur un graphe G de g4posskdant cette propriCtC de partition des arstes, la conserve, ce qui Ctablira bien le thiorkme annonc6. Nous noterons G' = 8, (G). (A) Pour $, pratiquke sur deux arCtes [a, b ] et [ c ,d ] de G.

a

C

Fig. 24.

F. Bories e: al.

114

(1) Si [a,b ]E Fl et [c, d ] E F2 on pose F;=FI-{[a,b]}+{[~,a].[b,~]},

Fi= F , - { [ c , d ] } + { [ c , a ] , [ d , a ] } , F : = F.,. ( 2 ) Si les deux arktes [a,b ] et [c, d ] sont toutes les deux dam la m&me forkt ( F , par exemple) et appartiennent a une mime chaine p = [ . . . , a, b, . . . , c, d, . . . ] de F1 (Fig. 25): Forrnons FI = F , - { [ a , b ] ,[c, d ] }+ { [ a a, ] ,[Q, d ] } . Si b et c sont chacun adjacents B une arite de F3 et de F2 dans G on considkre

F: = F2 + {[ a,b ] } ,

F; = F3 + {[a, c 1).

Si b et c sont chacun adjacents B deux arZtes de F7 (resp. de F 2 )o n fornie:

F ; = F , et

FS

=

F, + { [ a ,b ] ,[a.c]}

(resp. FA = F , et F ; = F3 + { [ a b, ] ,[ a ,c ] } ) . Si b est adjacent pose:

B deux aretes de F3 et c Q une arZte de F7 et I’autre de F,. on

F ; = F , + { [ a c]}, ,

F1 = F2 + { [ a b, ] } ,

Si b est adjacent a deux arZtes de F2 et c a deux de F , on prend alors:

Fi = F2 + { [ ac, J} et

/

\

F ; = F3 + { [ a b. ] } .

/

/

lJ Fig. 25.

Construction of ®ular graphs

115

(3) Si les deux arttes [a,b] et [c, d ] sont dans F,, sur des chaines tlimentaires distinctes. Dans G, {a,b} et {c, d } ne sont pas connectis par une chaine de F , , on peut donc affecter dans F ; , deux arktes parmi {[a,a ] , [a,b 1, [ a ,c ] , [a. d ] } sans creer de cycles de FI. Prenons les deux arCtes [a,a ] et [a,b ] par exemple. L’une des forits F2,F3 n’a qu’une arCte adjacente a c : F2 par exemple; on affecte alors [ a , c ] dans Fi. Si d n’est adjacent qu’a une arCte de F, on prend F : = F3 + {[a,d]} sinon on considbe F : = F, et F ; = F2+ { [ a c, ] , [a,d]}, le seul cas d’impossibilite Ctant celui oil c et d sont sur une m&mechaine Y de F2 (Fig. 26).

Fig. 26.

On considkre alors une autre affectation pour F ; :

FI = F,- { [ a , b ] ,[c,d ] )+ { [ a a, ] ,[ a ,c]}. II est clair que la mtthode prickdente convient alors, car d et b ne peuvent itre sur une mime chaine de F2 si b n’est adjacent qu’a une seule arkte de F:. (B) Pour ‘& pratiquke en un sommet e, auquel on substitute une clique ayant quatre sommets a, P, 7, 6. Deux configurations sont possibles, aux symttries pres: (1) e est adjacent ti deux ar&tesde F, [a,e ] et [ d ,e ] par exemple, et Fz aux deux autres (Fig. 27). On pose:

FI = F , - { [ u , e ] , [ e , d ] } + { [ a , a ] , [ a , P ] + [ P , ~ 1 . [ ~ , S ] , ( S , d 1 } , F;= F ~ - { [ b , e ] , [ e , c ] } + { [ b , ~ ] , [ P , S ] , [ S . ~ ] , [ ~ , ~ l , [ ~ , ~ l } . F:= F,.

F. Bories et al.

116

€3 _____)

d

C

Fig. 27.

(2) e est adjacent ii deux ar&tesde F1[a,e J et [d, el par exemple une dam F2 :[e, b ] et I'autre dans F3 : [e, c ] (Fig. 28). On conserve la mime for&tFI que prCcCdemment, on prend:

Fi = F2 - {[b,ell + {[b,p], [p, 61, [a, a ] ,[a,?]I, F ; = F3 - {[c, el) + {[c,

rl).

(C) Pour gSpratiquke sur une arCte [ a , b ] de F, par exemple (Fig. 29).

>7(

b

1

6

E3

d

h

a

'

/

Y

'\

d

C

C

Fig. 28. F

, €5

L

\

\

\

Y

Fig. 29.

\


117

Remarque. Le Thkorkme 3 a dkji dCmontrC par Peroche [S] en utilisant d’autres mithodes.

6. Problcmes ouverts Conjecture 1. La classe Ce4. i I’exception de G I , ,est engendrCe Q partir de K , et G, au moyen d’extensions g2 et + ‘8,. Cet CnoncC est Cquivalent au suivant: + ou g2 est Rz-rCductible ou R I + R I Toute extension de G I I par reductible en un graphe d’ordre 11 different de G I , . Remarques. G I I n’est pas R,-reductible, ni R I + RI-rkductible, mais R , riductible en Glo(Fig. 30), qui n’est pas R,-rCductible mais est Rz-rCductible en Gs (Fig. 31), qui n’est pas R,-riductible mais est R,-rkductible en G6.

G10

Fig. 30.

‘8

Fig. 31.

F.Bories et al.

118

-

Conclusion. GI, admet une dCcornposition unique au rnoyen de R , et R,: Gii = 8t(%?($z(gI(GF)))) G 6

G,

Conjecture 2. Tous les graphes de (e, sans clique d’ordre 4 sont obtenus A partir de G,au moyen d’extensions 8,. Problkme. En pratiquant des extensions 8,et 8, qui prkservent la planariti, peut-on engendrer la classe des graphes planaires de (e4?

Notons qu’une construction des graphes planaires 4-rkguliers connexes a CtC donnCe par Manca [7] au moyen de quatre extensions a partir de G6. Cette construction a CtC rectifike par Lehel [6].

Bibliographie [l] C. Berge, Graphes et Hypergraphes (Dunod, Paris, 1970). [2] J.L. Fouquet, These d’etat: Contribution A I’ttude des graphes cubiques et problbmes hamiltoniens dans les graphes orientbs, Juin 1981. [3] J. L. Fouquet, J.L. Jolivet et M. Rivibre, Graphes cubiques d’indice 3, graphes cubiques isochromatiques, graphes cubiques d’indice 4, J. Comb. Theory, Ser. B, It paraitre. [4] J.L. Fouquet, J.L. Jolivet et M. Rivibre, Construction de classes de graphes cubiques, Coll. MathCmatiques Discrktes: Codes et Hypergraphes (1978), Cahiers du Centre d’Etudes et de Recherche Optrationeelle 20 ( 3 4 ) (1978) 373-403. [ 5 ] E.I. Johnson. in: 0. Ore, ed. The Four Color Problem (Academic Press, New York, London, 1967). [6] J. Lehel, Generating all 4-regular planar graphs from the graph of the octahedron, J . Graph Theory, It paraitre. (71 P. Manca, Generating all planar graphs regular of degree four, J. Graph Theory 3 (1979) 357-364. [8] B. Peroche, On partitions of graphs into linear forests and dissections. Equipe graphes et graphique, Rappt. Rech. CNRS, no 2, Juin 1980. [9] S. Toida, Construction de quartic graphes, J. Comb. Theory, Ser. B 16 (1974) 124-133.


A CONSTRUCTION OF A COVERING MAP WITH FACES OF EVEN LENGTHS A. BOUCHET Dipartement de Mathimatiques. Uniuersiti du Maine, Roure de Laval. 72017 Le Mans. Crdex, France

Let G be a graph which can be embedded into an orientable surface S. the faces of which have even lengths I , , I,. . . . . I q . Let m be an odd positive integer. Consider the graph G,,, constructed from G by replacing each vertex x by m vertices 1,.x,,. . . ,x, and joining two vertices x, and y, by an edge of G,,,, if, and only if. x and y are joined in G. We prove that C,,, can be imbedded into an orientable surface 3, the faces of which have the same lengths I , , I.. . . . , I,. The same holds in the nonorientable case.

1. Introduction

Let us consider a map (G, S ) defined by a graph G which is 2-cell imbedded in some closed 2-manifold S (called briefly a surface). A covering of (G, S ) consists of a new map ( G , i ) and a covering with possible branchpoints of S by S verifying the two following properties: (i) G is projected onto G ; (ii) the branchpoints occur only at the vertices of G. Thus, if n is the degree of the covering, each face of G is evenly covered by n faces of G with a same length. Let GI,,,,be the graph obtained from G by replacing each vertex x by an independent set of m vertices x , , xz,. . . ,x,, and defining an edge [x,, y,] if, and only if, [x, y ] is an edge of G. Our purpose in this paper is to construct a covering map (G, such that G = G(,,,).The degree of such a covering is necessarily equal to m', and we shall see that the orientabilities of S and 9 are the same. Such a type of construction has already been achieved in some cases where G is triangularly imbedded in S , giving thus a genus imbedding for G(,,,);the papers [l], [3], [ 5 ] ,[6] deal with this problem. In particular, the method developed in [6] can be used for imbeddings which are not triangular. In this paper, we shall use the m-valuations introduced in [l] to prove the following theorem.

s)

Theorem. If (G,S ) is a map such that each boundary of a face has an even length 119

A. Boucher

120

and m is a posifive odd integer, then there exists a covering map (G,S ) such that G = G(,,,). As an immediate corollary we shall obtain the following. Corollary. Let G be a graph which can be 2-cell imbedded into a surface S with faces of even lengths and m be a positive odd integer. The graph G(,,,, can be 2-cell imbedded with the same face's lengths into a surface 2 which has the same orientability characteristic as S. This corollary is interesting when the imbedding of G is quadrilateral; it yields then a genus imbedding of G,,,,).

2. Definitions and elementary properties Our terminology about graphs and their imbeddings into surfaces agrees with [9] and 1121. References [8] and [lo] may be consulted for further topological concepts, and [7] gives a good insight on these subjects. A simplicial complex K will be abstractly defined, finite, with a dimension lower or equal to 2 . The basic definitions about such a simplicial complex can be found in [lo] but in order to avoid ambiguity, we point out that a q-dimensional simplex is merely a set of q + 1 vertices. Hence K is a graph if the dimension of K is equal to 1. V ( K ) , E ( K ) and T ( K ) are the sets of vertices, edges and triangles respectively. I K is the space of K . An edge with endpoints x and y will be denoted [x, y ] , a triangle with vertices x, y and z will be denoted [ x , y , z ] . If x E V ( K ) ,the link of x is the graph L(x,K ) defined by:

I

I[X? Y 1 E E ( K ) ) , E ( L (x, K ) ) = { [ Y , z ] I [ x , y , E T ( K ) J . V ( L ( X ,K ) ) = IY

21

If each link is an elementary cycle and the 1-skeleton K ' is a connected graph, then K will be called a triangulation; in this case each edge will be a common side of exactly two triangles. K is a triangulation if and only if I K 1 is a closed 2-manifold (briefly called a surface here). Our constructions will produce intermediate simplicial complexes where each edge belongs to exactly two triangles but with links which are not necessarily connected. Such a simplicial complex K will be called a pseudo-triangulation, and the number of cycles in a link L(x,K ) will be called the multiplicity of x. A cone around x E V ( K ) is a simplicial complex generated by the triangles determined by x and the edges of a cycle in L ( x , K ) ; thus the number of cones around x is the multiplicity of x.

Construction of a covering map with faces of even lengths

121

3. One simplicial complex above another

Let K be a simplicial complex (the dimension22 is allowed for this definition). A complex above K is a pair consisting of a simplicial complex K and a simplicial mapping p : K -+ K preserving the dimension of simplices. The projection p will often not be expressed, and the projection of a simplex 6 will often be denoted u. The preceding notions were studied in general by Tucker [12] in 1936 as branched coverings with folds. We limit ourself to triangulations (or pseudotriangulations) and graphs for imbedding purposes. If G is a graph, then for a graph G above G, the set of vertices above a vertex x E V(G) is an independent set of G. Therefore a projection has in this case the defining property of a homomorphism [ 4 ] . We shall more particularly consider the graph G(,,,)where rn 3 1 is an integer. It is constructed by replacing each vertex x of G by m independent vertices x t ,x 2 , . . . , x , and defining an edge [ x , , y j ] if and only if [x, y ] E E ( G ) . Now let us consider a pseudo-triangulation K above another pseudotriangulation K, 6 € E ( k ) , ;' and ? the triangles containing P. The projected triangles t' and t" both contain the projected edge e, but these triangles may be either distinct (Fig. la) or equal (Fig. lb). In the first case e' is a regular edge, in the second case it is a fold. If there is no fold K is a branched covering of K . From now onwards we shall consider only this last case and denote K + K such a branched covering. Then if the projection p is interpreted as a continuous map from the space lk I into I K 1, every point which is not a vertex of K is evenly covered by p. This makes well the pair (1 K 1, p ) a covering of I K I with possible branchpoints occurring at the vertices of K . Finally let us suppose that K is a triangulation. Let us consider a cone around a vertex iE V(k); this cone is determined by some cycle in L ( i , K).

Fig. la.

Fig. Ib.

A. Bouchet

122

There is a single cone f around x because K is a triangulation; this cone is determined by the cycle C to which L ( x , K ) is reduced. The cycle is projected onto C ;therefore if a point is turning once along then its projection turns a certain number k of times along C. This number k will be called the degree of the cone (Fig. 2 represents a cone of degree 2 ) .

c,

c

-4

N

C

Fig. 2

4. Method for proving the theorem

We describe here the steps for proving the theorem; the lacking details will be taken up in the sequel. For simplifying the exposition we shall suppose that the graph G is simple; the existence of loops and/or multiple edges requires a more sophisticated formalization without changing anything essential. Thus we start with the given map (G, S ) and we have to construct a covering map (G, 9)such that G = G(,,,)under the assumption that the boundaries of the faces of (G, S ) have even lengths and m is a positive odd integer. 1.I . Derioing a centered triangulation K from ( G ,S )

The first step is to transform a topological object, the map (G, S ) , into a simple combinatorial object, a triangulation K . For that we place a center inside each face and we join it to the vertices on the boundary. The triangulation K which is so constructed verifies two simple properties which will be fundamental for the sequel. 4.1. I . K is a centered triangulation By this we mean that a subset ( e ( K ) C V ( K )is distinguished and such that

Construction of a cooering map with faces of euen lengfhs

123

each triangle contains one, and only one, vertex in % ( K ) .The vertices of % ( K ) are obviously those which have been added inside the faces; thus they will be called the centers of K . In the sequel, we shall also use centered pseudotriangulations; they are defined similarly. 4.1.2. K is an even triangulation By this we mean that each vertex of K has an even degree. This is obvious for the central vertices because their degrees are equal to the lengths of the faces inside which they have been placed. The other vertices were initially the vertices of G and we notice that after joining the central vertices to the vertices of G, we add inside each angle of the map ( G , S ) a new edge; thus the degrees of the vertices of G are doubled, and they are even in K.

4.2. Defining an m -valuation on K for constructing a branched covering K + K It is in that step that we shall use again the techniques which we introduced in a preceding work [l].An rn-valuation of K is a map 4 : T ( K ) + Z / m Z . The expansion of K by 4 is the simplicia1complex k of dimension 2 defined by:

v [ K ] = V ( K )x ( z / m Z ) ; [(x, i).(y,j)] E E @ ) [(x, i ) , ( y , j ) , (2,k)] E

[x, y ] E E ( K ) ;

~(k)

@ t=[x,y,z]ET(K)

and

i+j+k=4(t).

This definition implies that the pair consisting of K and the map ( x , i ) H x is a branched covering += K . Therefore K is a pseudo-triangulation; more precisely it is a centered triangulation with the set of vertices %(k)above V(K). It is clear also that K' = K;,,,,.The definition of K implies that the subgraph of K ' spanned by V ( K ) \ % ( K ) is isomorphic to G ; therefore the subgraph of K' spanned by V(K)\%'(K)is isomorphic to G = G(,,,). Let us consider x E V ( K ) .The degree of x is even; let it be equal to 2d and let us denote by t l , tr, . . . ,f 2 d the triangles of K incident to x ordered following a local orientation around x (Fig. 3). Put

The element & ( x ) depends on the local orientation around x and on the first triangle f , . However if we change this orientation and/or t , , the single other value which can be obtained is - & ( x ) . Therefore we can speak unambiguously of the order k of c ( x ) in Z / m Z and of its index which is equal to m / k .

124

A. Bouchet

Fig. 3.

4.2.1 Theorem. Let f E V(R ) be projected onto x E V ( K ) . The degree of each cone around f is equal to the order of 4 (x). The multiplicity of f is equal to the index of i(x ). This theorem will be proved in Section 6.1,

4.2.2 Definition, The m-valuation 4 is a good m-valuation if the order of & ( x ) is equal to 1 when x E % ( K ) , equal to m if X G %(K).In other terms i ( x ) is a generator of Z/mZ when x!Z %(K);it is null if x E % ( K ) . The following corollary is then obvious.

4.2.3 Corollary. Let 4 be a good m-valuation of K and f E V(K). The degree of every cone around f is equal to 1 i f f E %(K).The multiplicity of is equal to 1 if

i e qK).

x

Now we shall see that a solution to our problem of constructing the map (G,3) will be found if we can determine a good m-valuation of K.

Construction of a covering map wirh faces of euen lengths

12s

4.3. Splitting K into a triangulation Kspeand defining a branched covering Kspe+K

Splitting the pseudo-triangulation K is constructing a new simplicia1 complex

Kspt in the following way: (i) V(Ksp8) is the set of the ordered pairs (f,s) such that f E V(k) and s is a cone around f - in order to reduce the notations we shall write isinstead of

(x’, s); (ii) = {[G ~ 1[i, 191E E ( K ) ,[f, 91E s n t ) ; (iii) T ( K ~ , ~ ! = { [ ~ ~ , ~ , , ~ , ] ([ x[’*, ~~, , z ~~I, E n~ s t nIuE}T. ( K ) , Let x’ E V ( K )with a multiplicity p. Intuitively the vertex f has been split into p vertices replacing f in each of the p cones around x’ (Fig. 4 represents the case where p =2). We notice that KSpeis a centered pseudo-triangulation such that

I

E(Kspp)= (2%X’ E g ( K ) } . 4.3.1 Proposition. If K is the expansion of K by a good m-valuation then KSpris a centered triangulation and there exists a branched covering Kspe + K. Moreover this branched covering is such that the cones around the centers of KSpeare of degree 1 (in other terms, the centers of K are not branched points).

This proposition will be proved in Section 6.2.

4.3.2 Proposition. If K is the expansion of K by a good m-valuation then KSp, and K have the same orientability characteristics.

This proposition will be proved in Section 6.3.

Fig. 4.

A. Bouchet

126

5. Existence of a good m-valuation of K We shall use again the techniques of [l].

5.I . Diagonal components of K Let us consider first an arbitrary triangulation K which is not necessarily even. The diagonal graph of K, denoted DK,is defined in the following way:

V ( D K ) = V(K); [u, w ] E E ( D K )

3[x,y ] E E ( K ) :[ x , y, u ] and

[ x , y , w ]E T ( K ) .

An edge of DK will be called a diagonal edge and a connected component of

DK will be called a diagonal component. A diagonal component will be said to be even (resp. odd) if its number of vertices is even (resp. odd). We proved in [ l ] that there exists either 1 , 2 or 3 diagonal components, and we characterized the triangulations with a given number of diagonal components. The reader will refer to this paper, and more particularly to the results about the double (Dd)and simple (D6) diagonal components contained in Section 3; these results imply immediately the following proposition which is relative to our even centered triangulation K . Let us recall first that the given graph G is obtained by deleting % ( K )from K ' ; thus V ( G ) = V(K)\%'(K). 5.1.1

Proposition. The uertex-set V ( G ) is a diagonal component of K if G is not a bipartite graph; otherwise each of the two partite classes of G is a diagonal component of K.

5.2. Chains and chain -groups We recall now the elementary definitions about chains and chain-groups introduced by Tutte. Let E be a finite set and R be a commutative ring. A chain on E over R is a mapping f : E + R. If x is an element of E, the value f ( x ) is called the coefficient of x in f . I f f and g are chains on E over R and if A is an element of R , the sum f + g is the chain on E over R satisfying

Cf+g)(x)=f(x)+g(x) for each x E E, and the product A f is the chain on E over R satisfying

( A f ) ( x )= A f ( x ) for each x E E.

Construction of a covering map with faces of even lengths

127

A chain-group on E over R is a class of chains on E over R that is closed under the operations of addition and multiplication by an element of R.

5.3. The triangle chain-group T Consider again the even centered triangulation K and an integer m 2 2. We shall be working with chains on V ( K ) over the ring Z / m Z . In order to abbreviate the terminology they will simply be called chains and the chain-group on V ( K )over Z / m Z will be called simply a chain-group. To each vertex x we associate the chain f satisfying ? ( x ) = 1 and f ( y ) = 0 if y # x ; so if f is a chain we have the equality

f

=

c

xEV(K)

cxf,

where each c, is the coefficient of x in f . To each m -valuation C#J we have associated in Section 4.2 a chain defined up to the sign. It is proved in [ 11- equality (E3) in Section 4 - that all these chains constitute a chain-group denoted f.The following lemma is proved in [ 11 where it is numbered (4.5).

4

5.3.1 Lemma. Let ( v , w ) be an ordered pair of distinct vertices of K which belong to a same diagonal component. There exists a chain ci ( v , w ) E T such that ci ( v , w ) = 6 + EKJ with E = k 1.

We are now in position for proving the existence of a good m-valuation. 5.4

Proposition. Every even centered triangulation K can be provided with a good m-valuation if m is a positive odd integer.

Proof. The idea is to determine 4 as a linear combination of chains given by will be of the form Lemma 5.3.1. This chain

with each value C, equal to + 1, - 1, + 2 or - 2. It will actually be derived from a good m-valuation because 1 and 2 are generators of Z / m Z when m is odd. will be made by distinguishing the two main cases The determination of indicated by Proposition 5.1.1.

4

A. Bouchel

128

5.4.1

V ( G )is a diagonal component and it is even. We partition V ( G )into pairs & = ci(ui,w , ) . All the coefficients C, are equal to 5 1 in that case.

( u t , w , ) and we put

5.4.2 V ( G )is a diagonal component and it is odd. The cardinality of V ( G )is at least 3 ; so we consider three different vertices x , y , z. We partition V ( G ) \ { x , y , z )into pairs (u,, w i ) and we set 6 = ~ C ( u , , w i ) + c i ( x , y ) + f f ( x , z ) . The coefficients are still equaI to ? 1 with the exception of the coefficient of X which is equal to 2.

5.4.3 V ( G )is partitioned into two diagonal components D and D’. If D (resp. D ’ ) is even we partition it into pairs ( u , , w , ) (resp. ( u : , w : ) ) and we set 8 = C ( u , , w e ) (resp. 8 ’ = z.dr(v:,w : ) ) . If D (resp. 0’)is odd then its cardinality is at least 3 ; so we choose three distinct vertices x, y, z in D (resp. x ’ , y ’ , z ’ in D’), we partition D \{x, y , z } (resp. D’\{x‘, y’, 2’)) into pairs, and we set 8 = C c i (u,, w, ) + ci (x, y ) + ci ( x , z (resp. S’ = C ci ( u : , w :) + ci ( x ’ , y ’) + ci (x I, z’)). Finally we set & = 8 + S’.

c

6. Proof of the main theorem

To prove the theorem stated in Section 1 there remains to give the lacking proofs of Section 4. 6.1. Proof of meorem 4.2.1

The degree of x is even, say 2d. Let xl, xz, . . . ,x ? d be the successive vertices on the cycle C = L ( x , K ) and let us set f, = [ x , x,, x , , , ] for each i = 1,2,. . . , 7 d (see Fig. 3), and counting i + 1 (mod 2 d ) . Let us consider a sequence of successive vertices f , ,izr . , . , i 2 d r i i on a cycle t of L (i ff), , and suppose that the vertices of that sequence are respectively projected onto x I ,x z , . . . ,x z d , x ;. Let us put i = ( x , i ) and i, = (x,, k). Then it is easy to verify the successive following equalities: i2 = (xz, 4 ( t l ) - i - k),

i,=(~,,4(t~)-~(ti)+k), i4 = ( x 4 ,4 (1,) - 4 ( f 2 )

+ 4(tl)- i - k 1,

Construction of a covering map with faces of even lengths i s = (xs,

4

(r4) -

4 (ti) + 4(tz)

- 4(ti)

129

+ k 1,

i; =(x,,i(x)+ k).

To illustrate the situation, we can suppose that is coiled around C (see Fig. 2). The preceding equalities show that after turning once around x, the component of iI is increased by &(x). Thus after turning around x a number of times equal to the order of i ( x ) we meet again, and for the first time, the vertex XI. This fact means well that the degree of every cone above x is equal to the order of &(x), and as a direct consequence that the multiplicity of 1 is equal to the index of &(x). 0 6.2. Proof of Proposition 4.3.1 Each link of K ,, is reduced to a single cycle; thus ESP[will be a triangulation if its 1-skeleton K,',e is connected. Each vertex X'P %(E)is not split because its multiplicity is equal to 1 by Corollary 4.2.3; therefore G = G ( mis) a connected subgraph of Kip,. The other vertices are the centers of Kspp.Each of these centers is single in each triangle which contains it; therefore it is joined to some non-center. Since every non-center is a vertex of G, it proves well that Kf,,is connected. H x' can be extended to a simplicia1 mapping from K S p into r The function p :is K preserving the dimension. The image by p of every pair of distinct triangles sharing a common edge of ESP/ is a pair of distinct triangles of k. Therefore the pair (Kspt,p)is a branched covering of K. Its composition with the branched covering K+ K gives a branched covering k,,e +K. The last sentence of the proposition is a direct consequence of Corollary 4.2.3. 0

6.3. Proof of Proposition 3.3.2 We defined in [2] a transversal subgraph of a graph H as a graph spanned by a set of vertices T C V ( H )such that every edge of H has at least one endpoint in T. For example, the graph G is a transversal subgraph of K ' . A direct consequence of Theorem (VI.5) and Corollary (VI.8) of [2] is that Kspt and K will have the same orientability characteristic if we can find a subgraph T of K&( which is 1-1 projected onto G. We know that G(,,,) is a subgraph of k,',,and that it is projected onto G ;therefore it is sufficient to take in G(,,,)a copy of G which will actually be 1-1 projected onto G. 0 And the main theorem is finally proved. 0

130

A. Boucher

References [ I ] A. Bouchet, Triangular imbeddings into surfaces of a join of equicardinal independent sets following an eulerian graph, in: Theory and Applications of Graphs, Proceedings. Michigan 1976, Lecture Notes in Mathematics 642 (Springer-Verlag, Berlin, New York, 1978). 121 A. Bouchet, Covering triangulations with folds, in: Proceedings of the 4th Internat. Conf. o n the Theory and Applications of Graphs, Kalamazoo, 1980, t o appear. 131 A. Bouchet, Constructing a covering triangulation by means of a nowhere-zero dual flow. submitted to J. Comb. Theory, Ser. B. [4] F. Harary, Graph Theory (Addison Wesley, Reading, MA, 1971). 151 B. Jackson, Generative m-diagrams and embeddings of K,,,,, unpublished. [6] B. Jackson, T.D. Parsons and T. Pisanski. A duality theorem for graph imbeddings, J. Graph Theory, to appear. [7] G.A. Jones and D. Singerman, Theory of maps on orientable surfaces. Proc. London Math. Soc. (3) 37 (1978) 273-307. [XI W.S. Massey, Algebraic Topology: An Introduction (Springer-Verlag, New York, 1077). [Y] G. Ringel, Map Color Theorem (Springer-Verlag, New York, 1974). 10) E.H. Spanier, Algebraic Topology (McGraw-Hill, New York, 1966). I I ] A.W. Tucker, Branched and folded coverings, Bull. Amer. Math. SOC. 42 (1936) 850-862. 121 A.T. White, Graphs, Groups and Surfaces (North-Holland. Amsterdam, 1973).


TROIS TYPES DE DECOMPOSITIONS D’UN GRAPHE EN CHA~NES Andrt BOUCHET Universite‘ du Maine. France

Jean-Luc FOUQUET C.N.R.S., Paris, France Let G be a graph with m edges and i odd vertices. Let I , , I,, . . . ,Is be positive integers. We shall say that G is universally decomposable into chains if a necessary and sufficient condition for finding a decomposition of G into chains of lengths I , , I,, . . . , Is is I, + 1, + . . . + Is = M and s 2 i/2; G is (k)-universally decomposable if I,# k (1 S i S s). We prove that a cubic graph is (2)-universally decomposable into chains if, and only if, it has an hamiltonian path. Let G be a decomposition of G into chains; G is called a factorization into chains if each vertex of G is an endpoint of two chains in G. The graph G is defined to be (k)-universally factorizable in a similar way as above. We prove that a 4-regular graph is (0)-universally factorizable into chains if it has an hamiltonian cycle. Finally, if each chain of the decomposition of G is of the same length k, we shall say that G is P,-decomposable. We thus study this last concept and relared problem.

1. Introduction Nous prtsentons ici quelques problbmes de dtcomposition de graphes en chaines qui peuvent itre considtrts d’un certain point de vue comme des gCntralisations de problkmes eultriens. En fait, ces questions ont t t t introduites pour I’ttude du nombre achromatique des arbres [4]. Nous considtrons les chaines simples d’un graphe G = ( X , E ) fini, non orientt, avec des boucles et/ou ar&tes multiples. Rappelons qu’une chaine simple est dtfinie comme une suite tventuellement vide d’arites deux a deux distinctes e l ,e 2 , .. . , e l telle qu’il existe une suite de sommets xo,xI,x2,. . . , x i vtrifiant ei = [xi-l, x i ] pour tout i = 1 , 2 , . . . ,1. La valeur de 1 est la longueur de la chaine; si I = 0 alors la chaine est rtduite au sommet xo. Notons bien que ces chaines ne sont pas tltmentaires. Une dtcomposition en chaines de G est un ensemble de chaines simples C = {Cl, C2,.. .,Cs}tel que chaque ar&te de G appartienne 5 une et une seule de ces chaines. Dans une premikre partie, nous nous intiressons aux dtcompositions en chaines de longueur non nulle. Etant donnt une suite d’entiers I , , lzr. . . ,I, > 0 de 131

132

A. Bouchet, J.-L. Fouquet

somme tgale au nombre d’arctes de G, nous cherchons s’il existe une dtcomposition en chaines ayant ces longueurs donntes. I1 est clair que le nombre i ( G )des sommets de degrt impair doit &tremajort par 2s pour que la dtcomposition existe. Nous dirons que G est universellement de‘composable en chaines si la condition 2s 3 i ( G )suffit i assurer I’existence de la dtcomposition. Ainsi un graphe eultrien est un cas particulier de graphe universellement dtcomposable en chaines. Nous nous inttressons ici au cas des graphes cubiques. Rappelons qu’il a CtC prouvt que tout graphe complet est universellement dtcomposable en chaines [I]. Dans une seconde partie, nous considtrons des dtcompositions en chaines de longueur tventuellement nulle. Soit C = {Cl,C2,.. . , Cs}une telle dCcomposition. Nous exigeons que chaque sommet de G soit extrtmitt de deux chaines de G exactement en convenant de compter deux fois chaque sommet x pour lequel il existe une chaine dont les extrimitis sont confondues en x. Nous dirons donc que C est une factorisation en chaines. II est clair que les degrts de G doivent &tre pairs pour admettre une factorisation en chaines; on retrouve ainsi une analogie avec les problitmes eultriens. On introduit comme plus haut la notion de graphe universellement factorisable en chaines, et on considitre le cas des graphes 4-rCguliers. II a ttt prouvt dans [3] que tout graphe complet d’ordre impair est universellement factorisable en chaines. Dans une troisiitme partie nous nous inttressons h deux autres types particuliers de dtcompositions en chaines: nous exigeons que toutes les chaines de la dtcomposition soient d’tgales longueurs (Pk -de‘composition). Une preniittre dtcomposition permet ainsi de caracttriser les graphes cubiques qui admettent un couplage parfait, ceci nous a suggCrt un processus simple de construction de cette classe particulikre de graphes cubiques. La deuxibme dtcomposition (Pk-de‘composition ordonne‘e) fait intervenir a la fois I’idte prictdente et le coloriage en k couleurs des arktes d’un graphe d’indice chromatique k, nous caracttrisons ainsi, en termes de Pk-dtcomposition ordonnte, les graphes k -rtguliers bipartis.

2. Graphes universellement dkcomposables en chaines Soit G = ( X , E ) un graphe ayant m ar6tes et i ( G )sommets de degrt impair; soient I,, I*, . . . , 1, q entiers strictement positifs. On dit que G est universellement dkomposable, si une condition ntcessaire et suffisante pour trouver une dtcomposition en chaines de longueurs 11, L , . . . , I q est que

Trois types de dicornpositions d’un graphe en chafnes

133

On dit que G est (k)-universellement dkcomposable si pour tout i (1 s i s q ) 1, est diffkrent de k. Theoreme 2.1. Soit G = ( X , E ) un graphe cubique, les conditions suivantes sont kquivalentes : (i) G est (2)-universellement dkcomposable ; (ii) G posstde une chaine hamiltonienne.

-

Preuve. (i) 3 (ii). G Ctant cubique, on a: IX I = 2s et J E1 = 3s. Soit 3s

=

1+ 1 + 1+ * .

. + 1 + ( 2 s + 1)

s-1

G ttant (2)-universellement dCcomposable, il existe donc une partition de ses aretes en s - 1 chaines de longueur Cgale B 1, et une chaine de longueur 2s + 1. Retirons les deux arttes extrtmes de cette chaine, on obtient clairement une chaine hamiltonienne de G. (ii) 3 (i). Examinons tout d’abord une partition de 3s en s parties distinctes de 2: 3 s = 1 , + 1 z + ~ ~ ~ + 1 (Vi s l ~ i ~l , >sO ) ,.

Supposons de plus que les 1, sont ranges par ordre dicroissant:

1, 3 lr 5 I ,

3

* * =

3

r, 5 I,

5

f,

.

3

1,

(oir t est le plus grand indice tel que I, est supCrieur ou Cgal a 3). Pour tout i (1 s i s t ) soit Ai = 1, - 2, on a clairement:

c (A,+ 1) c (Ii 1

=

i=l

i=l

- 1) =

9(li

- 1) = 2s.

i=l

ConsidCrons une chaine hamiltonienne de G, et t chaines ClCmentaires P , , Pz,. . . , P, placCes sur cette chaine (d’une manikre arbitraire) de telle sorte que chaque sommet de G n’appartienne qu’i une et une seule de ces chaines (la relation (1) nous assure que ceci est toujours possible). Examinons alors le graphe partiel G‘ de G engendrC par les aretes de G qui n’appartiennent i aucune des chaines P,. Dans ce graphe, tous les sommets ont un degrt au plus deux, c’est donc une union de chaines et cycles CICmentaires; orientons alors chacune de ses composantes d’une manikre quelconque. Chaque sommet x de degrC 2 dans G’ sera origine, par rapport B cette orientation, d’une arcte que nous noterons e(x). On construit alors une dkcomposition en chaines C,, C2,.. . ,C, de longueurs I , , L,.. . , l , en procCdant de la faGon suivante:

A. Bouchet, I.-L. Fouquet

134

(a) Pour chaque i (1 S i =z t ) considtrons les deux extrtmitts x, et y, de la chaine P,, ces extrimitts sont de degrt 2 dans G’, nous obtenons C, en prolongeant Pi par les ar6tes e(x,) et e(y#). (b) I1 reste alors s - t arctes qui ne sont pas prises dans G‘, elles constituent les s - t chaines de longueur 1. Considirons enfin une partition de 3s en q parties distinctes de 2 (q > s): 3s = I ,

+ I , + * - . + lq

(q > s).

Regroupons les termes en s paquets de sommes aI, a:, . . . , astoutes distinctes de 2 (un tel regroupement est toujours possible). On construit alors une dtcomposition en chaines C,, C2,.. . , C,rtalisant la partition al,u2,.. . , a, et on casse certaines chaines pour retrouver la partition initiale I , , I*, . . . , I q . 0 On ne peut espirer amtliorer ce thtorkme 2.1, en remplaGant la condition (i) par la condition (i’) suivante: (i‘) G est universellement dtcomposable. La fin de ce paragraph: est consacrte A la construction d’un contre-exemple 6 ce problkme. Soit en effet un graphe cubique G d’ordre 6q (d’ou IE 1 = 9 q ) , soit la partiton de ( E 1 suivante:

-

9q=2+2+2+..*+2+39+2

(433).

3q - 1

Si G est universellement dtcomposable, il admet une partition de ses arCtes en 3 q - 1 chaines de longueur 2 et une chaine de longueur 3q + 2. A chaque extrkmitt de cette chaine otons une chaine de longueur 2. G posskde ainsi une chaine ClCmentaire P de longueur 3q - 2 telle que G - P n’ait aucune composante ayant un nombre impair d’arites. O n a

Or, il est possible vtrifier que dans le graphe de la figure suivante (Fig. l), chaque chaine P telle que G - P n’ait aucune composante impaire vtrifie:

(b)

I(p)
(oil p est le nombre de ‘barreaux’ de chacune des ‘Cchelles’ H I , ...,Hk). (a) et (b) seront des relations clairement contradictoires pour p et 4 suffisamment grands. G ne peut donc 6tre universellement dtcomposable malgrC la prtsence d’un cycle hamiltonien (A fortiori d’une chaine hamiltonienne).

Trois types de dkompositions d ‘un graphe en chaines

135

Fig. 1.

cp=,

On a en effet I ( P ) = I P n Hi 1 - 1. La dhonstration de la propriCtC (b) reposant alors sur le fait que pour chacun des sous-graphes H,, IP n H, est major6e par 4 p + 4.

I

3. Graphes universellement factorisabies en chaines

Soit C = {C,, C2,.. . , C,} une dtcomposition en chaines du graphe G = (X, E). On dira que C est une factorisation en chaines si chaque sommet de G est exactement deux fois extrCmitC par rapport aux chaines de C. PrCcisons qu’un sommet x sera comptC deux fois comme extrCmitC d6s qu’il existera une chaine C,E C dont les extrCmitCs sont confondues et Cgales B x. En particulier nous admettons la presence de chaines de longueur nulle dans C, et la situation prkcidente se produira si C,est une chaine de longueur nulle et rCduite au sommet x. Une faGon Cquivalente de dCfinir C comme factorisation en chaines consiste B dire que chaque chaine de C peut @treorientCe de manikre B trouver chaque sommet de G une fois, et une seule, comme extrimit6 initiale et une fois et une seule, comme extrCmit6 finale.


136

Proposition 3.1. U n graphe admet une factorisation en chaines si, et seulernent si, tous ses degre‘s sont pairs. Preuve. La condition nicessaire est ividente. Riciproquement, chaque composante connexe admet une chaine eulirienne B extrCmitCs confondues. Choisissons donc une extrimit6 double sur chacune de ces chaines eulkriennes; considirons ensuite les sommets manquants comme extrCmitCs doubles de chaines de longueur nulle. 0 Remarque. I1 est Cgalement trks facile de montrer qu’un graphe admet une factorisation en chaines de longueur s > 0 si, et seulement si, tous les degrCs sont pairs et non nuls. Proposition 3.2. Le nombre s des chaines figurant duns une factorisation en chaines d’un graphe est e‘gal au nombre des sommets de ce graphe. Preuve. Cette proposition est une conskquence immtdiate de la dCfinition. I3 Definition. Rappelons qu’un partage d’un entier rn 2 0 est une expression de la forme rn = 1, + 1, + * 1, oh chaque 1, est un entier 3 0. Le graphe G = (X, E ) sera dit universellement f actorisable en chaines (resp. ( k )-universellement factorisable en chaines) si tous ses degrts sont pairs et pour tout partage 1 E 1 = 1, 12+ * . + 1, (resp. tel que I,# k pour chaque i = 1,2,. . . ,s) avec s = ( X 1, il existe une factorisation de G en chaines de longueurs I , , I?, . . . , I,.

+

+

Proposition 3.3. U n graphe rkgulier de degre‘ 4 est (0)-universellementfactorisa ble en chaines si il admet un cycle harniltonien. Preuve. Soit G = ( X , E ) le graphe consider6 avec IX 1 = s, 1 E 1 = 2s et un cycle hamiltonien C. Considirons une suite d’entiers 1, > l2 > . * * > 1, > 0, de somme Cgale A 2s. It s’agit de trouver une factorisation en chaines avec ces longueurs donnCes. Pour chaque i = 1,2,. ..,s, posons A, = 1, - 1. Soit t l’indice maximum tel que A, > 0. Du fait que la somme des 1, (1 S i ZS s ) est tgale B 2s, on diduit que la somme des A,. (1 ZS i S t ) est Cgale A s. DCcomposons donc le cycle hamiltonien C en t chaines Cltmentaires Pi,Pz,. . . , PI deux A deux disjointes. Le nombre des coefficients nuls A,+i, .. , A , est Cgal au nombre des sommets de C qui ne sont extrimitis d’aucune de ces t chaines. Associons donc B chaque valeur nulle A, ( t < i S s ) une chaine Pi de longueur nulle et rCduite B I’un des sommets qui n’est extrCmitt d’aucune des chaines Pi,P 2 , .. . ,P,. L’ensemble P = {P,, P z , , . . ,Pr} apparait ainsi comme une factorisation en chaines du cycle C avec les longueurs A , , Az, . . .,A,.

Trois types de dicompositions d’un graphe en chaines

137

Soit C’ le graphe partiel complCmentaire de C dans G. Ce graphe partiel C’ est un 2-facteur qui admet manifestement une factorisation en chaines P ’ = { P i , PI,. . . ,Pt} oc chaque PI est de longueur Cgale a 1. Orientons les chaines de P (resp. de P ’ ) de telle facon que chaque sommet de G apparaisse une fois, et une seule, comme origine d’une chaine de P (resp. de P’) et une fois et une seule, comme extrCmitC finale d’une chaine de P (resp. de P’). ConcatCnons chaque chaine orientCe Pi E P’ a la chaine orientCe Pi E P telle que I’extrCmitC initiale de P:est Cgale 6 I’extrCmitC finale de P,. On obtient ainsi la factorisation en chaines dCsirCes. 0 Remarque. Le r6le exceptionnel jouC par la longueur 0 dans les factorisations en chaines des graphes 4-rCguliers est analogue 6 celui qui est jouC par la longueur 2 dans les dCcompositions en chaines des graphes cubiques. Notons par ailleurs que la propriCtC d’admettre un cycle hamiltonien est suffisante mais non nicessaire pour qu’un graphe 4-rigulier soit (0)-universellement factorisable en chaines. En fait, on peut montrer avec un raisonnement analogue i celui qui a Ctit utilisC pour les graphes cubiques que tout graphe 4-rigulier qui est (0)-universellement factorisable en chaines admet nicessairement une chaine hamiltonienne. Nous avons Ctabli que cela suffit lorsque la suite des longueurs prend au moins 3 valeurs ou 2 valeurs distinctes de 1. Nous conjecturons que cette condition est suffisante dans tous les cas. Proposition 3.4. Tout graphe re‘gulier de degre‘ 2p est factorisable en chaines de longueur p . Preuve. Montrons la propriitk par ricurrence sur p en notant qu’elle est triviale pour p = 1. La supposant vraie pour p, nous considCrons un graphe G rCgulier de degrC 2p + 2. Extrayons un 2-facteur F du graphe G, et orientons chacun de ses cycles. RCalisons une factorisation en chaines C,, Cz, . . . , C, de longueur p dans le graphe partiel G - F, et orientons ces chaines de telle fagon que leurs extrtmitks finales soient deux i deux distinctes. I1 suffit alors de prolonger chaque chaine de G - F par un arc de F pour obtenir la factorisation voulue. 0

4. Graphes Pk-decomposables en chaines

(a) Soit G = ( X , E ) un graphe, on dit que G est Pk-de‘composable s’il est possible de partitionner ses arites en chaines de longueur k . Le thCor&me ci-dessous est annonce dans [2].


138

Theoreme 4.1. Soit G = ( X , E ) un graphe cubique, les conditions suivantes sont kquivalentes : (i) G est P3-dicomposable ; (ii) G admet un couplage parfait. Preuve. Si le graphe cubique est P3-dCcomposable, on vCrifie aisCment que les arites centrales de chacune des chaines de la dCcomposition constituent un couplage parfait. RCciproquement, soit L = { e l ,ez, . . . ,e , } un couplage parfait de G. G - L est un 2-facteur de G, orientons chacun des cycles du 2-facteur, B chaque sommet x on peut associer ainsi d’une manikre unique une arete e ( x ) (l’arete qui correspond 2 I’arc sortant de x). On construit alors la dCcomposition voulue en adjoignant 2 chaque ar&tee, de L, les ar&tese(x,) et e ( y l )(ob e, = [x,, y,]). 0 Le thCorkme ci-dessus suggkre alors une construction de la classe 2‘ des graphes cubiques qui admettent un couplage parfait. Soit en effet G €9, considtrons une P,-dCcomposition de G, cette P,-dCcomposition induit deux types d’aretes: les arktes centrales (qui constituent un couplage parfait); les aretes extrkmes (qui constituent un 2-facteur). Soient e et e ‘ deux aretes extrkmes de G :

e = [x, y ] et e’ = [z, t ] . Supprimons ces deux aretes, ajoutons deux nouveaux sommets a et b relies respectivement B x, y et z, t ( a et b Ctant de plus, eux aussi, relies). Le graphe ainsi obtenu est un nouveau graphe de 9. It est alors facile de montrer que, par cette procedure on engendre exactement 2 en partant des boucles (une boucle Ctant considCrCe comme arste extrkme).

Exemple (See Fig. 2).

G r a p h e rCduit

a une boucle s a n s somrnet

Fig. 2.

Trois types de dicornpositions d’un graphe en chafnes

139

G r a p h e reduit 5 deux boucles sans sommet

Fig. 2 (continued).

Parmi les 3 graphes ci-dessous (Fig. 3) (ou le thCorkme de Petersen sur l’existence d’un couplage parfait ne peut s’appliquer) seul le second graphe admet un tel couplage.

8

Gz€Y

Fig. 3.


140

(b) Soit G = ( X , E ) un graphe d’indice chromatique k, on dit que G admet une P, -dicomposition ordonnke, s’il existe une coloration 4 des aretes de G en k couleurs, une permutation u de {1,2,.. . , k } , une A-dicomposition de G, et une orientation des chaines de la Pk-dCcomposition telle qu’en parcourant chacune de ces chaines dans le sens donni par I’orientation on rencontre les arktes de couleurs cr(l), u(2),. . . , u ( k ) dans cet ordre.

Exemple. P3-dkcomposition ordonnte de K3., (Fig. 4). 3

3

1

4

Fig. 4.

Theoreme 4.2. Soit G = ( X , E ) un graphe k-rigulier, les propositions suivantes sont kquivalentes : (i) G biparti; (ii) G admet une P k -dicomposition ordonnke. Preuve. Nous dirons qu’une Pk-dtcomposition ordonnte d’un graphe G biparti a la propriiti (E) si tous les sommets terminaux (respectivement initiaux) appartiennent B la meme classe de la bipartition (X= A U B,pour k impair les sommets initiaux et terminaux appartiennent B A, pour k pair les sommets initiaux appartiennent B A et les sommets terminaux a B). (i) 3 (ii). Montrons par induction sur k que G posskde une Pk-dCcomposition ordonnte vtrifiant (E). G ttant biparti, il est d’indice chromatique k. k = 1, la propriCtt est triviale puisque G est riduit A un couplage parfait. Supposons la propritti vtrifite B I’ordre k - 1 et montrons la pour k. Soit L un couplage parfait, G - L est k - 1 rigulier, il admet donc une P k - 1 dCcomposition ordonnte vtrifiant (E). Ajoutons alors les ar&tesde L munies de la couleur k, on obtient clairement une Pk-dkcomposition ordonnee de G virifiant (E). (La permutation u utilisie est la permutation a’ de la Pk-IdCcomposition ordonnie de G - L, prolongte B (1,. . . , k } . ) II n’est pas difficile de dimontrer, en fait, que si G est biparti (k-rtgulier), toute A-dtcomposition ordonnee de G virifie (E). C’est cette propriiti que nous allons utiliser maintenant.

Trois types de dkompositions d ’ u n graphe en chaines

141

(ii) j (i). Par induction sur k, les cas k = 1, k = 2 Ctant triviaux, considtrons une Pk-dCcomposition ordonnCe de G (supposons en fait que u est 1’identitCsur (1,. .., k ) ) . Soit L = { e @ ( e )= k } . G - L est k - 1 rCgulier, la Pk-dkcomposition ordonnCe de G induit trivialement une Pk-,-dCcomposition ordonnCe de G - L. G - L est donc biparti et la p k -,-dCcomposition ordonnCe obtenue vCrifie (E). Supposons qu’en ajoutant les arstes du couplage L on obtienne un graphe G non biparti, c’est-&dire: V ( G - L ) = A U B (G - L biparti); soit [a, b ] E L ( @ [ a ,b ] = k ) telle que a et b appartiennent A A. Soit alors P , la chaine de la A-dCcomposition ordonnCe contenant [ a ,61, P2 celle qui contient les arites de couleurs k - 1 et k - 2 incidentes A a (on a en effet, soit I’arite de couleur k - 1 incidente B a qui n’appartient pas ;i P , , soit celle incidente i 6, on suppose ici, sans restriction de gCnkralitC quec’est a ) . Soit enfin c I’autre extrimit6 de I’arite de couleur k - 1 de P I . Les sommets b et c sont sommets terminaux de la Pk-,-dCcomposition ordonnie de G - L, ils appartiennent donc tous les deux il A ou B: impossible. G est donc lui aussi biparti. 0

I

Bi bliographie [ l ] A. Bouchet and R. Lopez-Bracho, Decomposition of a complete graph into trails of given lengths, Discrete Math., submitted for publication. [2] A. Kotzig, Problem in 10th Southeastern Conf., Boca Raton, 1979, Congressus Numerantium 24, 9B. [3] R. Lopez-Bracho, Decompositions en chahes d’un graphe complet d’ordre impair, in: C. Berge et al., Combinatorial Mathematics, Proc. Internat. Coll. on Graph Theory and Cornbinatorics, Ann. Discrete Math. 17 (North-Holland, Amsterdam, 1983) pp. 427-438. [S] R. Lopez-Bracho, Thtse de 3tme cycle: Etude du nombre achrornatique des &toiles,Universiti de Paris Sud, Centre d’Orsay.



SUR LES COUPLAGES DANS LES GRAPHES LOCALEMENT FINIS* FranGois BRY Universite‘ Pierre et Marie Curie, U.E.R. 48, Equipe de Recherche Combinatoire 4, place Jussieu, 75005 Paris, France We present some results on matchings in locally finite graphs. First, we give an extension to locally finite graphs of a theorem of Edmonds and Gallai giving a decomposition of graphs according to matchings. Then we estimate the number of perfect matchings (I-factors) in locally finite graphs, depending on connexity. These results generalize some theorems of Kotzig, Lovisz and Zaks.

1. Introduction

Edmonds et Gallai ont donne dans [4], [ 5 ] une dCcomposition des graphes finis relative aux couplages maximums. En collaboration avec Las Vergnas, nous avons ktendu cette dkcomposition aux graphes localement finis. Ce thkorkme est CnoncC au Section 3. Kotzig a p r o u d dans [7] que tout graphe fini contenant un et un seul couplage parfait a un isthme appartenant A ce couplage. Un graphe fini 2-arite-connexe contenant un couplage parfait en a donc au moins deux. I1 existe par contre des graphes localement finis (infinis) 2-arCte-connexes ne contenant qu’un seul couplage parfait. Au paragraphe 4 nous en donnons un exemple et nous montrons que tout graphe localementfini contenant un et un seul couplage parfait est au plus 2-connexe. Le thCorkme de Kotzig est le premier rtsultat sur le nombre f ( G ) de couplages parfaits d’un graphe fini G. Auparavent, dans le cas particulier des graphes bipartis finis, Hall [6] avait estimk ce nombre ( f ( G ) an ! , si G = (X, Y ; U ) est un graphe biparti fini tel que = 1 Y 1 et tel que tout sommet de X soit de degrC supkrieur ou Cgal a n ) . Beineke et Plummer ont ensuite prouvk que f ( G ) sn si G est un graphe fini n-connexe contenant un couplage parfait [l]. Zaks a amkliorC ce risultat en montrant dans [lo] que f ( G ) a n !! sous les memes hypothkses. (Si n est impair, cette borne est exacte en

1x1

* L’auteur bCnCficie d’une allocation d e recherche de la D.G.R.S.T. (Contrat no 79317). 143

144

F. Bry

ce sens que le graphe complet K,+, contient exactement n!!couplages parfaits.) Dans le cas particulier des graphes finis non bicritiques (un graphe est bicritique s’il contient un couplage parfait et si le sous-graphe obtenu en supprimant deux sommets quelconques 2ontient encore un couplage parfait) Lovisz [8] a precis6 cette borne infCrieure: f ( G ) > n ! si G est un graphe fini non bicritique n-connexe. Finalement, Mader a donnC dans [9] la borne infirieure exacte du nombre de couplages parfaits d’un graphe fini ne dependant pas de la connexitt mais du degrt minimum du graphe: si G est un graphe fini 2-connexe de degrC minimum n et si G contient au moins un couplage parfait, alors f ( G )5 n !! si n est impair, et f ( G )2 f(S,) si n est pair, oh S, est K,+2moins un couplage parfait. Nous donnons au paragraphe 5 des exemples de graphes localement finis (infinis) n-connexes avec un nombre fini (non nul) de couplages parfaits, pour tout entier n. Des rCsultats analogues B ceux de Zaks et Lovisz sont ensuites tnoncts pour les graphes localement finis. Les preuves seront publiCes ulttrieurement.

2. Terminologie et notations Les graphes considCrts sont des graphes simples [2]. Un couplage d’un graphe G = (X, U) est un couplage parfait si tout sommet de G appartient ii une ar&tedu couplage. Un couplage F d’un graphe G est un coupluge maximum si pour tout couplage H de G tel que U H 3 U F , on a UH = UF. D’apr&s un theorkme de Brualdi ([3] Thtorkme 4), tout graphe localement fini h au moins un couplage maximum. Si F est un couplage maximum d’un graphe G, le dkfuut de F, nott 6 ( F ) ,est le cardinal de I’ensemble des sommets de G insaturks par F. Deux couplages maximums d’un m&megraphe ont mime dtfaut. On appelle difuut d’un graphe G le dkfaut d’un couplage maximum de G,on le note S ( G ) . Un graphe est coupluge-critique s’il ne contient pas de couplage parfait et si le sous-graphe form6 en enlevant un sommet quelconque a au moins un couplage parfait. On vCrifie qu’un graphe couplage-critique est fini et d’ordre impair. Si G = (X,U) est un graphe et si S est une partie de X,on dCsigne par C,,(S) le cardinal de I’ensemble des composantes connexes d’ordre pair du sous-graphe Gs induit par S. De m&me, CJS) dtsigne le cardinal de I’ensemble des composantes connexes couplage-critiques du sous-graphe Gs.

3. DCcompositionrelative aux couplages maximums d’un graphe localement fini

En collaboration avec Las Vergnas, nous avons Ctabli le thtorbme suivant:

Les couplages dans les graphes localement finis

135

Theoreme 3.1. Soit G = ( X , U ) un graphe localement fini, Soient P l’ensernble des sommets de G insature‘s par au moins un couplage maximum de G, Q l’ensemble des sommets uoisins de P ( Q = r, (P)\P) et R = X \ ( P u Q ) . Alors : (1) Les composantes connexes du sous-graphe Gpinduit par P sont couplagecritiques. (2) Soient % ( P ) l’ensemble des composantes connexes de Gp et H = (Q,%( P ) ;E ) le graphe biparti de‘fini par (4, C )E E si et seulement si il existe c E C tel que ( q , c ) E U. H a un couplage qui suture Q et laisse insature‘e une partie de % ( P )de cardinal S(G). Tout couplage de H saturant Q laisse insaturke une partie de % ( P )de cardinal supe‘rieurou e‘gal a S ( G ) .En particulier si S ( G )est fini, Q et P sont finis et IQ I = C,,(P) + S ( G ) . (3) Le sous-graphe GR induit par R a un couplage parfait. (4) Tout couplage muximum de G se de‘compose en un couplage parfait de G R , en un couplage de de‘faut 1 de chaque composante de Gp et en un couplage maximum de H. 4. Graphes localement finis avec un et un seul couplage parfait

Kotzig a prouvB dans [7] que tout graphe fini contenant un et un seul couplage parfait a un isthme appartenant A ce couplage. Cela n’est plus vrai pour les graphes localement finis infinis, comme le montre I’exemple suivant. Exemple 4.1.

Fig. I .

Le graphe de la Fig. 1 est 2-arite-connexe et n’a qu’un seul couplage parfait (reprCsentC par les arites Cpaisses). Un rCsultat semblable au thCor6me de Kotzig est le suivant. ThCorkme 4.2. Tout graphe localement fini contenant un et un seul couplage parfait est au plus 2-connexe. Nous ne sommes pas arrivCs Q caractkriser les graphes localement finis (infinis) ne contenant qu’un seul couplage parfait, bien que nous ayons pu Btablir quelques unes de leurs propriCt6s. Nous faisons la conjecture suivante.

F.Bry

146

Conjecture 4.3. Tout graphe G = ( X , U ) localement fini (infini) 2-connexe contenant un et un seul couplage parfait a un ensemble de skparation S de cardinal 2 tel que C o ( X \ S ) a1.

5. Nombre de couplages parfaits d'un graphe localement fini Si G est un graphe localement fini, F ( G ) dCsignera le nombre de couplages parfaits de G. Si n est un entier F ( n ) dksignera la plus petite valeur de F ( G ) lorsque G parcourt la classe des graphes localement finis n-connexes et contenant au moins un couplage parfait. En ces termes, le ThCorCme 4.2 se traduit par F(3) 3 2. Let exemples qui suivent montrent que F ( n ) est fini pour tout entier n.

Exemples 5.1. Soit n 3 3 un entier. DCfinissons un graphe localement fini infini T.. Soit (X, m E N) une suite d'ensembles deux B deux disjoints, chacun d'entre eux de cardinal n. Posons X,,, = { x Y ' , . . . ,xn"}. L'ensemble des sommets de T, sera U{X, m E N}. L'ensemble U des ar&tesde T, est dkfini par: si rn est impair ou si m = 0,

I

I

{ x ~ , x ~ + U, ~ } E i s i c n,

1 sj s n ;

si m est pair et si m # 0,

{xT,xy''}~U,

l ~ i ~ n .

Fig. 2. Legraphe T,.

On vkrifie aiskment que, pour tout n, le graphe T, est n-connexe et contient exactement n ! couplages parfaits. Si G = (X, U ) est un graphe localement fini n-connexe, posons a ( G )= M a x { k E N I V S C X , k + l s ( S ( < a , C c , ( X \ S ) = S I S l - k } . Si G n'est pas bicritique, on a a ( G )= 0 ou a ( G ) = 1 (dam le cas des graphes finis, on a nkcessairement a (G) = 0).

Les couplages dans les graphes localement finis

147

Theoreme 5.2. Soit G un graphe localement fini n-connexe non bicritique et contenant au moins un couplage parfait. Sia(G)=O, alorsF(G)*n!; s i a ( G ) = l , alorsF(G)a(n-l)!. Theoreme 5.3. Soit G un graphe localementfini n -connexe ( n z=4 ) et bicritique. Si n est pair, alors F ( G )b n !!/2; si n est impair, alors F ( G )3 $n!!

6. Problemes ouverts

( 1 ) Quelle est la borne infCrieure exacte F ( n ) du nombre de couplages parfaits des graphes localement finis infinis n-connexes contenant au moins un couplage parfait? (2) Quelle est la borne infkrieure exacte F B ( n ) du nombre de couplages parfaits des graphes localement finis infinis n -connexes et bicritiques? (3) Peut-on donner un rCsultat semblable ii celui de Mader [9] pour les graphes localement finis infinis?

Bibliographie [I] L.W. Beineke and M.D. Plummer, On the I-factors of a non separable graph, J. Comb. Theory 2 (1967) 285-289. [Z] C. Berge, Graphes et Hypergraphes (Dunod, Paris, 2nde Bd., 1973). [3] R.A. Brualdi, Matching in arbitrary graphs, Proc. Camb. Phil. Soc. 69 (1971) 401-407. 141 J. Edmonds, Paths, trees and Rowers, Canad. J. Math. 17 (1965) 449-467. [ 5 ] T. Gallai, Maximale Systeme unbhangiger Kanten, Math. Kut. Int. Kozl. 9 (1964) 373-39.5. 161 M. Hall, Distinct representatives of subsets, Bull. Amer. Math. Soc. 54 (1948) 922-926. [7] A. Kotzig. 2 te6rie konetnqch grafov s lyneflrnym factorom 11, Mat. Fyz. casopis 9 (1959) 136-159. (81 L. Lovasz, On the structure of factorizable graphs, Acta Math. Acad. Scient. Hung. 23 (1972) 179-1 8.5. [Y] W. Mader, Uber die Anzahl der 1-Faktoren in 2-fach zusammen hangenden Graphen, Math. Nachr. 74 (1976) 217-232. [lo] J. Zaks, On the I-factors of n-connected graphs, J. Comb. Theory 1 1 (1971) 169-180.



A DETERMINISTIC ALGORITHM FOR FACTORIZING POLYNOMIALS OF IF, [XI Paul CAMION C.N. R.S.-I. N.R.I.A., Paris, France The general problem under consideration is actually solved for every semi-simple, commutative algebra A of finite dimension over a finite field IF,. A probabilistic algorithm for decomposing an idempotent u of A into a sum of primitive idempotents was introduced in [l] and [2]. Here the probabilistic algorithm gives place to a deterministic one’which, in particular, allows the factorization of polynomials of IF, [ X I .

1. Introduction After Rabin [6] we call ‘probabilistic’ an algorithm where a finite field IF, is concerned, when each step of the algorithm essentially consists in picking up an element at random in IF, apd then performing a given sequence of operations involving that selected element. If we know that, say, about half the elements in IF, will lead the algorithm to end, then the average number of steps will be close to 2 . However the probability that the algorithm does not end within s steps is close to 2 - s for s small and q large. What we want to do here is to define a small subset P of IF, of which the size is bounded, say, by 2 log2q, and in which the elements will be selected for the algorithm. Thus we will here be sure that the algorithm ends within 210g2q steps. In some applications, as correcting codes, it will be possible to perform simultaneously the required sequence of operations for the Card P elements of P. Here, ending in one step is a certainty. In other applications, the algorithm will be lead by taking elements from P in a given order, arbitrarily fixed, once for all, and the average number of steps will be smaller than the one for the probabilistic algorithm outlined here above.

2. Previous results

2.1. Previous papers on the topic

In Camion [l], we give an algorithm for constructing the primitive 149

150

P. Camion

idempotents of any ideal of A = Fq [XI,. . . ,x , ] / ( t i ( X i ).,. . , tr ( X ) ) in the case where all t i ( X i ) ,i = 1,. . . , r, have simple roots, i.e., when A is a semi-simple ring. For r = 1, that algorithm allows the factorization of any f ( X ) E F, [XI in any overring L ’ [ X ] ,F, s L ’ , for whatever finite field F, q odd. All needed operations are performed in A. That algorithm is investigated more extensively in Camion [2] where the finite fields F, with characteristic 2 are also dealt with. Several examples show in that paper how the algorithm works. Notice that the paper by Cantor and Zassenhaus [7] gives actually the same algorithm although restricted to the purpose of factorizing the polynomials of OF, [ X I . In Camion [3], it is shown how a technique introduced by McEliece can be exploited to decrease the average number of computations.

2.2. The Berlekamp sub-algebra of A Let first A be any commutative algebra over any field K. An idempotent u of A is a non-zero element u such that u z = u. Such an idempotent is called primitive when the principal ideal ( u ) of A is a minimal ideal (or simple ideal). We know that the following property characterizes semi-simple rings R (see Lang [5, 1972, pp. 446-4471): R is the direct sum of its minimal ideals,

R =R1$**.$R,. Each Ri is a simple ring. If ei is its unit element, then 1 = e l + . - .+ e k and Ri = Rei. We have eiei = 0 if i # j . When R is a commutative ring such as A, then Ri = Riei is a field and it contains Kei. We may write A = (ei)@I @ (ek).

(1)

Now consider the direct sum

B =Kei$-..$Kek.

(2)

We call B the Berlekamp sub-algebra of A, since Berlekamp [4]introduced an algorithm for constructing a basis of the k-vector space B which is valid whenever K is a finite field OF, and when A has a finite basis ...,en. This relies on the following. Property 1. The algebra B is made of all elements g in A such that g q = g.

A deterministic algorithm

Let first g be in B. We have g = with p s = q, then gq

=

2

Rei =

Icick

2

151

zlsiSk liei and since A has characteristic p

liei = g.

(3)

Isick

Now if g = & s i s k aiei,ai E Ri and g q = g, then are? = aiei by (1). This shows that aiei,belonging to the finite field Riei with characteristic p , is then fixed under the galois automorphism a -+ a q of Riei and thus belongs to the subfield Fqei of Riei.Hence a4eS = aPei = liei for some li in F, ; g = liei E B.

zlsisk

2.3. The algorithm of Berlekamp Let

be a basis of A over K and

{Ei}lrisn

zIrisn

bi (aii- 8,) = 0 for j = Consider the n-tuples (bl,. . . ,6.) such that 1, ..., n. They form a vector-space of which a basis may be constructed by diagonalizing an n X n matrix. That vector-space is B since it is made of all elements g = b i g i of A verifying

xlsisn

= Isisn

bi

2

lsjsn

&,E,= g.

3. Factorizing polynomials of F, [XI

3.1. Considering a particular semi-simple ring for factorizing polynomials Considering a polynomial f (X) in IF, [XI, it is easy to find out if it has multiple roots, for example [8], and then to reduce the problem to factorizing a polynomial f ( X ) with simple roots. Now the algebra A = F, [X]/ ( q - 3 ) k ,

which is obtained by considering k-subsets in IF *, and in A & U A in place of k-tuples.

(21)

respectively,

5.2.2. Proof of (9) Let C,, i = 0 , 1 , 2 be the cosets of Co in IF:. Here, for every x E IF: then ( x ) denotes i iff x E C,. We will denote by A:, the set of elements x in C,such that x + a is in C,. We have the following. Property 3. For every a in FI ,:

we have that

Card A :i = (q - 4)/3. rt{O.l.Zt

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156

Now, since IF, has characteristic 2, we have that v x €IF,:

(x

+ 1)2= x 2 + 1,

(24)

and thus Card A i., = Card A L.

Card A A, , = Card A

(25)

On the other hand, (13) here entails that

Vi E {0,1,2}: Card A !.,= Card A !,,,-,,

(26)

when computing mod 3 in {0,1,2}. In particular, we have that Card A I.,

= Card

Ah.

(27)

Finally, the mapping x --* x + 1 is an involution over F q , which shows that A i,,,= A b,2. This together with (25) achieves the proof of (22) since VU €IF*,:

uA:.,=Af+,a).l+co).

The rest of the proof for (9) is essentially the same as for (8).

5.3.Proofs of (10) and (11) Let first 4 be odd. For every x in F:, let (x) = i iff x f Ri, i = 0 , l . Let P be a separating set for F,. We have that V{a, b } C IF*, \ ( - P ) , a # b and ab E Ro;

3 x EP: (a + x ) # ( b +x).

Since Card(Fj; \ ( - P ) ) = q - k - 1, with Card P = k, we have that max{Card(Ro\( - P ) ) , Card(R, \ ( - P ) ) )3 (q - k - 1Y2. Let Card(Ri \ ( - P ) ) a ( 4 - k - 1)/2 for i = 0 or 1 and consider the k-tuple ...,xI, where xi runs over a11 distinct values in F. Then, we must have that be distinct for a E R, \ ( - P ) . This all k-tuples ((x, + a ) , . . . ,( x k + a ) )E (0, entails (x,,

( 4 - k - 1)/2 C 2',

4 < 2'+?,

which proves (10). We prove (11) with the same argument, pointing out that Card(G \ P ) 3 (q - k - 1)/3 for i = 0 , l or 2.

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157

References [ I ] P. Camion, Un algorithme de construction des idempotents primitifs d’idtaux d’algirbres sur IF, C.R. Acad. Sci. Paris 291 (20 October 1980). [2] P. Camion, Un algorithme de construction des idempotents primitifs d’idtaux d’algkbres sur IF, , in: Theory and Practice of Combinatorics, Ann. Discrete Math. (1982) to appear. [3] P. Camion, Factorisation des polynomes de Fp[X], Rappt. INRIA, No. 93, Septembre 1981. [4] E.R. Berlekamp, Algebraic Coding Theory (McGraw-Hill, New York, 1968). [S] S. Lang, Algebra (Addison-Wesley, Reading, MA, 1972). [6] M.O. Rabin, Probabilistic algorithms in finite fields, MIT/LCS/TR-213, Janvier 1979. [7] D.G. Cantor and H. Zassenhaus, A new algorithm for factoring polynomials over finite fields, Math. Comp. 36 (1981) 154. [8] D.E.Knuth, The Art of Computer Programming, Vol. 2 (Addison-Wesley, Reading, MA, 1971).



ON THE COMPLEXITIES OF PERIODIC SEQUENCES Agnes H. CHAN, Richard A. GAMES and Edwin L. KEY The MITRE Corporation, Bedford, MA 01 730, U.S.A.

The shortest linear recursion which generates a periodic q-ary sequence is defined to be the complexity of the sequence. A nonlinear feedback shift register sequence with maximum period q" is called a 9-ary de Bruijn sequence of span n. It is shown that, for 9 = w ' where w is a prime, the complexity of a q-ary de Bruijn sequence of span n is bounded by 9" - 1 from above and w'-'9"-' + n from below. For q-ary sequences of arbitrary period p, it is shown that their complexities are bounded by rhe period p from above, and their lower bounds are determined by the prime factors of p.

1. Introduction

Pseudo-random q -ary sequences generated recursively using n -stage feedback shift registers (FSR) have many applications in modern communication systems. Each element of these sequences {si}= (so,sI, . . . ,s i , .. .) is a function f of the previous n elements, that is, s i + n =f(si,

si+l,. . ., s i + n - l ) -

If the function is linear, then the feedback shift register is called a linear feedback shift register (LFSR). Otherwise, it is called a nonlinear feedback shift register (NLFSR). In order to ensure that these q-ary sequences are sufficiently random, it is desirable to have long periods (throughout the paper, we use period to mean least period), balanced statistics, and unpredictability. An n-stage LFSR can generate a sequence with a long period of q " - 1 if the feedback function f is properly chosen. (For a detailed treatment of shift register sequences, Golomb [7] is an excellent reference.) However, such a LFSR sequence has the disadvantage that only 2n consecutive terms of the sequence are needed to determine the feedback function f, and hence the remaining terms of the sequence, by solving n linear equations in n unknowns. In fact, any LFSR sequence is susceptible to such a linear attack. A very powerful algorithm for this purpose has been devised by MaSsey [Ill. Any sequence with period p can be generated by a LFSR, namely the p-stage cyclic shift register. It is natural, therefore, to define the complexity of a periodic 159

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sequence as the smallest number of stages in a LFSR that generates the sequence. Sequences generated by the less theoretically understood n -stage NLFSR can have complexities that approach the period of the sequence, as will be shown in the paper. This fact makes such sequences important in applications since they are not easily susceptible to the linear attack. The output of a FSR can be viewed either as a 9-ary sequence or as a sequence of 9-ary n-tuples which represent the successive states of the shift register. The latter view provides a tie between these generators and the de Bruijn graphs [ 3 ] . The de Bruijn graph G,is a directed graph with a vertex for each binary n-tuple. An edge is drawn from x = (x,,x2,. . .,xn) to y = (y,, y,, . . .,yn) exactly when y = (x2.. . . ,x,. yn). In [1,3] de Bruijn and van Aardenne-Ehrenfest showed that there are (9!)4'-'-" distinct Hamiltonian circuits in G,. It was discovered recently that the result for 9 = 2 was derived by Flye Sainte-Marie in 1894 [5]. These Hamiltonian circuits correspond to shift register sequences with period 9", called de Bruijn sequences. The de Bruijn sequences are of special interest because of their maximum period but also, as will be shown, they cannot be generated by a linear recursion with few stages, that is, they have high complexities. The next section introduces concepts, including the concept of complexity of 9-ary sequences, that are essential to the paper. Section 3 establishes upper and low bounds on the complexities of de Bruijn sequences. Finally, Section 4 addresses the complexities of periodic sequences of arbitrary period p.

2. Concepts and notations

In this section we introduce concepts, definitions and notations that are used throughout the paper, For simplicity, we denote GF(9)" by O ( n ) ,where 9 = w ' and w is a prime. A periodic sequence {s,} = (st), s l , . . . , s i r ...) of period p is completely specified by the elements of a single period, and will be denoted by a q-ary p-tuple (or vector) s E Q(p) where s = (so,

SI,.

. . ,s p - 2 ,

sp-,).

We define

S, (p)= { s : s is a q-ary sequence of period p } . A 9-ary sequence generated by an n-stage FSR is said to have span n. A sequence {s,} = (s,,,sl,.. . ,si,.. . ) satisfies a linear recursion of degree rn if, for all i 2 0 , m

The complexities of periodic sequences

161

The recursion (1) can also be expressed as a difference equation

where E is the shifting operator of finite differences, i.e., Esi The polynomial equation

= s,+~

m

is called the characteristic equation in finite differences, and the polynomial is called the connecting polynomial in shift register theory (because of its relationship to feedback connections). It is clear that any periodic sequence s E S , ( p ) satisfies a linear recursion, namely ( E P- 1)s = 0, i 2 0. The sequence s might also satisfy a linear recursion of degree less than p . Let f(E)s, = 0, i 3 0, be the linear recursion of least degree satisfied by s, then s is said to have complexity c(s) where c (s 1 = deg(f(E )I.

I

From the division algorithm it can be seen that f ( E ) EP- 1. Thus, to determine the complexity c(s), the degree of f ( E ) must be determined, which is equivalent to studying the factorization of EP- 1 in GF(q)[E]. For p = q", Eq"- 1 = ( E - 1)4" in GF(q), and so f ( E ) = ( E - 1)"". In particular, since all q-ary de Bruijn sequences have period q " , their connecting polynomials are all of the form (x - 1)"'. For simplicity, we sometimes call the connecting polynomial associated with the linear recursion simply the linear recursion. The operator D = E - 1 plays an important role in the study of sequences of period q". For a sequence s = (so, sI,s 2 , . . , sp-l)E S, ( p ) ,

D S= ( S ( , - S ~ , S ~ - S ? ,..., S ~ - ~ - S ~ - ~ , S ~ - I - S ~ ) . Consider a sequence s in Q ( q " )with complexity c(s), then ( E - lY"'s, = 0 for all i 2 0, that is, ( E - l)""-'Ds, = 0 for all i 2 0. Thus, applying D reduces the complexity of a sequence of period q" by one. Furthermore, if Dr("-'is applied to s, then a constant nonzero sequence ( a ) of period one is obtained. Consequently, the complexity of a sequence of period q" is c(s) if, and only if, for all i 5 0, ( E - l)c(s)-ls, = a, where a E GF(q)- (0).

3. Bounds on the complexities of de Bruijn sequences

In this section upper and lower bounds on the complexity of a q-ary sequence with period q", n 1, are derived. Next, de Bruijn sequences are considered and

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improved bounds are determined for this case. The upper bound is attained by a de Bruijn sequence obtained from appending the 0 vector to a sequence of maximum period q" - 1 generated by linear recursion. Examples of binary sequences for 3 S n S 6 show that the improved lower bound is best possible, but it remains open as to whether this is the case for all n 2 7 .

Theorem 1. If s E S, ( q " ) , then the complexity of s satisfies

w '-'q"-l+ 1 S c ( s )S 4". Proof. Since s E S, ( q " ) , for all i a 0, ( E 4 "- l ) s i = ( E - l)S"s,= 0

over G F ( q ) ,

but since s f f S , ( w ' - ' q " - ' ) , there is some i 2 0 , such4hat (E"'-'4"' - l)s, = ( E - 1)"'

+

and so, w' -Iqn-l 1 S c ( s )

S

' a # 0 over G F ( q ) ,

4".

The next two theorems characterize q -ary sequences which attain the bounds of Theorem 1. If u = (al,az,. . . ,a N )and b = (bl,b2,. . . ,b M ) , then ( a b ) denotes the vector (at, az,. . ., u N , bl,bz,. . . , b M ) .

I

Theorem 2. L e t s E S, ( 4 " ) .The complexity ofs, c(s) = w ' - ' q " - '+ 1 if, and only if, s = ( r r + a r + 2a r + ( w - 1)a) for some r E Q ( w ' - ' q " - ' ) , and a E GF(q1 - (0).

I

I

I- I

Proof. c(s) = w'-'q"-' + 1 if and only if for i 2 0 ( E - l ) " ' - l q i = a, a E GF(q)(E"' ',"

I

m,

- l)s, = a,

that is,

If we denote s, as r i , i = 0 , 1 , . .., w ' - l q " - l - 1, then s can be written as s = ( r r + a r +2a * . r + ( w - 1)a).

1

I

1. 1

To characterize q-ary sequences s of complexity q", we need to define Add(s) = so+ s I+.

1

+

in G F ( q ) ,

where s = (so, sI,. . For a E G F ( q ) , we define wt. ( s ) as the number of occurrences of u in one period of s.


163

Theorem 3. Let s E S,(q"), then c ( s ) = q" if, and only if, Add(s)#O. Proof.

E4"-1 - E4"-1+ E 4 " - 2 + . . . + E + l ( E - I>'"-' = (E-1)4" - ___ (E-1) E-1 so that c ( s )= q" if, and only if, there exists a nonzero element a in GF(q)such that for all i 3 0 .

Coroliary 4. If s is a q-ary de Bruijn sequence of span n, then c(s) s q" - 1. Proof. Since every n-vector appears exactly once in a q-ary de Bruijn sequence s of span n, each element a in GF(q)appears exactly q " - ' times in one period of s. Therefore, wt, (s) = q"-I for all a E GF(q), and Add(s) = 0. To see that the bound in Corollary 4 is attained, consider the d e Bruijn sequence obtained by appending an additional 0 to a maximum length sequence r E S, (q" - 1) generated by an n stage LFSR, so that the 0 vector is included in the sequence. In particular, suppose r,, = r , = * * = r.-z = 0 and define s E S, ( 4 " ) by si = ri

for i = 0,1,. . . ,q" - 2,

Sqm-I

= 0.

A de Bruijn sequence obtained in this way is called an ML-sequence. To show that the complexity of s in this case is q" - 1, we apply a result of Massey [ l l , Theorem 11.

Theorem 5 (Massey). If some LFSR of length L generates the sequences sII,sI,. . . ,sh.-],but not the sequence of so,sI,.. . , s , w - l , ~ N , then any LFSR that generates the latter sequence has length L' satisfying L ' a N + 1 - L. Corollary 6. Every ML-sequence s has maximum complexity c (s) = q" - 1. Proof. Let r E S,(q" - 1) be generated by an n-stage LFSR, and assume that TI, = r l = . = rn-2= 0, and s, = ri for i = 0,1,. . . ,q" - 2. In addition, s,.-~ = 0. The same LFSR generates 1

so, SI,

. .. ,S q " + n - 3

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164

but does not generate Sf,,

s I ,. . . ,sq n + n

.3,

s, "+ "

2

~

because the latter ends with the n consecutive zeros: = S," = . . . = Sq"+"-Z = 0.

So, Massey's Theorem asserts that c ( s ) = q" - 1 by Corollary 4.

C ( S ) ~(9"

+ n - 2)+ 1 - n = q" - 1, and thus

The lower bound of Theorem 1 can be improved in the case of de Bruijn sequences. First, let us consider the function A

=

af,E"+a l E ' t *

* *

+ a,-lE"-',

The action of A on a q-ary sequence 3 : GF(q)" + GF(q) by 3

:(&,&+I

s

a, E GF(q).

in S, (4') induces a linear transformation

) . . . )s,+"-l)--+afls, + a l s , + l + . . . + a " ~ l s , + " L l .

Theorem 7 . Let s be a q-my de Bruijn sequence of span n. For every nonzero linear transformation A, wr, (As)= q " - ' , a E GF(q). Proof. Since A : GF(q)"-+GF(q) is a linear tranformation, Im A s GF(q)"/Ker 3 where Im A = range of 3 and Ker A = kernel of A. For every nonzero linear transformation A, Im A = GF(q); hence (KerA 1 = q" I . I t remains to show that wt, (As) = I Ker ill. Let us consider the de Bruijn sequence s as a sequence of n-vectors, s = ( s o , s I ,.. . ,s q n L l ) where s, = (s,,s,,,, . . . ,s,,, ]). Since every n -vector occurs exactly once in s, (4:0 6 i S q" - 1) = GF(q)" and implies that Im A = {As,: 0 s i s q" - 1). Thereso, As = (As,,,A s l , . . . , fore,

w r f l ( A s ) = I K e r 3 ( = q nI .

For a E GF(q)-{0), let u denote a vector in GF(q") with Au = a, then wt,(As)=

10

+KerA 1 = q " - I .

Corollary 8. Lets be a q-ary de Bruijn sequence of span n, then for a E GF(q), Wt"

( s ) = wt,

(Ds)= .

*

= wt,

( D " - ' s )= q"?

Proof. For each k, 0 c k < n - 1, Let A = D k = ( E - l ) k .Then the result follows immediately from Theorem 7.


165

Remark. The converse of the corollary is not true. For example, let n = 4 and 9 =2 s = (1,0,0,0,1,0,0,1,1,1,1,1,0,0,1,0)

has the property that w t ( s )= w t ( D s ) = w t ( D ' s ) = w t ( D 3 s )= 2', but it is not a de Bruijn sequence.

Theorem 9. Let s be a de Bruijn sequence of span n 2 2, then the complexity of s satisfies

+

c ( s ) a w r - ' q n - ' n,

except when n = 2 and 9 = 2 .

Proof. Consider D"-'s, recalling that c ( D n - ' s )= c(s)- (n - 1 ) . We show that c ( D " - ' s ) awr-'9"-'+ 1, so that c ( s ) a w r - ' q n - + ' n. Suppose to the contrary that c ( D " - ' s ) <w ' - ' q " - ' + l . By Theorem 1 , c ( s ) a ~ ' - l q " - 1~, +so since c ( D ' s ) = c ( s ) - i , thereexists k , O s k s n - 2 , s u c h that c ( D k s ) =w ' - ' q " - ' + l . By Theorem 2, one period of D k s has the form, for p = 9" and a E GF(9) - {0}, D k s = ( r l , r ? , . . . , r p / w ,rl + a , . . ., rp/w+ a , . . . , r l + ( w - l ) a ,. . . , r p / w + ( w - l j a ) , so that D'+'S

1

.. ,

= (rl - r2,r2 - r 3 , .. . ,rpIw - rl - a rl - r 2 , .

rpIw - rl - a

1 I rl - r 2 , .. . , rpIw+ ( w - 1)a *

TI).

But ( w - 1)a = - a over GF(q), and so

D k + l s = (rl - r2,r2 - r,, . . . , rpIw- rl - a

1. . I r , - r2,

r2 - r3, . . . , rpIw - r l - a ) = ( r Ir I . . . l r ) where r = (rl - r2,r2 - r , , . . ., rpIw - rl - a ) . Now, 1 s k lary 8 implies that for a EGF(q),

+1 6 n -1,

and Corol-

wr, ( D k + ' s=) 9 " - ' ,

and so wt, ( r ) = q " - ' / w = ~ ' - ' 9 " hence -~;

For n > 2 or t > 1 , ( 2 ) has value 0 over GF(q). For n = 2, z a E G F ( q ) a = 0 if 9 # 2.

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166

Thus, Add(r)=O except when n = 2 and q = 2 . But Add(r) = (rl - rz) + (rz - rj) + . *

+ (rPiw- rl - a )

= -a,

contradicting that Add(r) = 0. Thus, c ( D " - ' s )3 w r - l q n - ' + 1 and the theorem is proved. We end this section with examples of binary de Bruijn sequences for 4 G n =z 6 which attain the lower bound of Theorem 9. Example. n = 4, c ( s ) = 12, s = (0000111101001011).

Example. n = 5, c(s) = 21, s = (000001001111101 0001101011100101).

Example. n = 6, c ( s ) = 38, s

= (000000111010001

01 11000100101011

01010011111100110010000101111011).

4. Bounds on the complexities of sequences of period p

From the previous section we observe that de Bruijn sequences have high complexities; however, the synthesis problem of de Bruijn sequences remains a difficult one. Most of the q-ary sequences generated by n-stage NLFSR's have periods p less than q " ; thus it is important that we consider the complexities of q-ary sequences of arbitrary period p. Clearly the complexity of any q-ary sequence of period p is bounded by the period p from above. In particular, the sequence with p - 1 0's and exactly one 6 where 6 is a nonzero element in GF(q),has complexity p. Thus, c(s) s p for all q-ary sequences of period p. We are left to consider the lower bounds of q-ary sequences. Let s be a q-ary sequence of period p. The complexity c(s) is given by the degree of the least linear recursion f ( x ) satisfied by s, which divides x p - 1. Thus the study of the bounds on c ( s ) reduces to the study of the factorization of x p - 1. For a polynomial f ( x ) over GF(q), the period of f(x) is defined to be the smallest p such that f ( x ) I x p - 1. Obviously, the period of any sequence s generated by f ( x ) divides p and is therefore at most p .


I67

We first show that we only have to consider the polynomial x p - 1 with p relatively prime to q. Let s be a q-ary sequence of period p , and p = w 6 eq'm, where m is relatively prime to q (that is, the greatest common divisor of q and n, denoted by (4, m ) , is 1). Let f ( x ) be the least linear recursion that generates s.

Lemma 10. Iff( x ) = g, (x)~',with g, (x) as irreducible polynomial over G F ( q ) . Then, (1) g, (x) x m - 1 for all i B 0 and g, ( x ) has period m, (2) d, 3 w '-'q'-'+ 1 for all i 3 0.

n

1

I

Proof. Since s is of period p , f ( x ) x p - 1. Now ( x p - 1) = ( x m - 1)w6q' over G F ( q ) , this implies that every irreducible factor g, ( x ) of f ( x ) divides x"' - 1. Clearly, g, (x) has period p' m ; indeed, gt ( x ) x p - 1. Since the multiplicity of any root of g, ( x ) can be at most w6q', d, s w 6 q ' ;thus g, ( x ) ~ ' ( x p - l)wsq', and f ( x ) I x p wsq' - 1. But p = w6q'm is the period of s, therefore, p ' = m. To show (2), suppose d, 6 w'-'q'-' for all i, then f ( x ) l (x"' - l)w' Since f(E) s, = 0 for all i B 0, ( E m- 1 ) w t ' q 1 - I S , = ( ~ m w ''ql - l)s, = 0 and s has period dividing rnw'-'q' I ; this contradicts that s has period p = mw'-'q'.

I

n

n

I

n

1

I.

1

Theorem 11. Let s be a q-ary sequence of period p with p = mwsq', where ( m ,q ) = 1. Then

c ( s ) z = ( w ' - ' q ' - ' +l)minC,, where min C, = minimum complexity of q-ary sequences of period m. Proof. Let f ( x ) be the least linear recursion that generates s and f ( x ) = rI g , ( ~ ) ~ Then a. c(s ) = deg f ( x ) =

= (w'-Iq'-l

i

d, .deggi ( x )

+ 1)deg

gi ( x ) .

(3)

From the previous lemma, we observe that n g i ( x ) generates a sequence r of period m ; therefore, deg

fl gi ( x ) = c ( r )

B min

C,.

Combining (3) and (4), we obtain

c ( s ) a ( w ' - ' q ' - ' +l)minC,,,.

(4)

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Henceforth, we assume that the period of the q-ary sequence s is p , with ( p , q ) = 1. Since the factorization of x p - 1 determines the complexity of c ( s ) ,we shall quote some of the results given in [2, Chapter 61.

n

Theorem 12 (Berlekamp, Theorem 6.21). If f ( x ) = g, ( x ) where g, ( x ) are irreducible polynomials of period p , over GF(q), then the period o f f ( x ) is the least common multiple of the p , .

n

n

Corollary 13. Let f ( x )= g, ( x ) and f ( x ) has period p. If p = w, where w, are distinct primes, then, for each j , there exists a g, ( x ) with period p , such that w, p , . Proof. Since the period p of f ( x ) is the 1.c.m. of p , and p = w, Ip, for some i.

I

n w, , for each j ,

Lemma 14. If a is a root of an irreducible polynomial g, (x) with period pi over GF(q), then (1) pt = ord(a, q), where ord(a, q ) = order of a in the extension GF(q)[a], (2) deg gi ( x ) = ord(q, p i ) . Proof. Since g, ( x )is an irreducible polynomial, all roots of g, ( x ) have the same order, ord(a, 9). Let rn = ord(a, q), then g, (x) x m - 1, and p, s m. I f p , < m, then g, ( x ) xpi - 1 implies that a is a root of x p -~ 1 contradicting ord(a, q ) = rn. To show (2), we let deg g, ( x )= d and observe that g, ( x ) = ( x - a4’), this implies that a q d= a or q d = 1 mod p , . Hence, deg g, ( x )= ord(q,p,).

I

I

nfl:

Theorem 15 (Berlekamp [2, Theorem 6.511). If n = ord(q, n ) = l.c.m.{ord(l, n:)}. Theorem 16. Let s E S,, @) with p =

n n:, n, distinctprimes, then

n,,, u,, u, distinct primes. Then

where minimum is taken over all partitions rr of I, and the sum is taken over all subsets J in the partition rr. Proof. Let f ( x ) =

n:=,)g, ( x ) with each g, ( x ) having period p,. Then, ,c deg g, I

c (s ) = deg f (x) =

=

=I1

(x)

2 ord(q,p,) 21.c.m. (ord(q, ui): n ui =

j =O

j -0

i€J,

= pi

(5)

Since the period of f ( x ) is the 1.c.m. of the p,’s and is p , each ui must appear as a factor of some pi and only uI’s can appear as factors of p, ; thus, each J, is a


u

subset of I and J, = I. If for a distinct pair j and j ‘ such that we can consider 4 and .lj*- 4. Since 4. - .lj C .lj.,

169

4 fl 4.# 0,then

l.c.m.{ord(q, ui): i E 4,- 4 ) C l.c.m.{ord(q, ui): i E 4); thus the expression in (5) is at least

where 4. is in a partition 7~ of I. Therefore, the lower bound is given by the minimum taken over all partitions of I. Theorem 17. Let s E S, @) with period p = u d , u is a prime. Then,

c(s)>ord(q,p). Proof. Let f ( x ) = Then,

n:=,g, (x) with each g, (x) having period p, and p, = udi,d, s d.

c ( s ) = d e g f ( x ) = i d e g g , ( r ) = Aord(q,p,)=f:ord(q,u’J). j=O j =O

j =O

Since the period of f ( x ) is l.c.m.{udi) and is p, l.c.m.{ud,} = umaxjdr’ = ud. Hence, there exists a factor gi(x) such that period of gj(x) = u d , so deggj(x) = ord(q, u d ) , and c(s) 2 ord(q, p ) . Theorem 18. Let s E S, @ ) with p = C(S)Z

min

W E l ( f )

[

,En

n up‘, ui distinct primes. Then

I.c.m.{ord(q,u?): i EJ}]

where minimum is taken over all partitions subsets J in the partition 7 ~ . Proof. Let f(x) = 0 < e, s d , . Then

7~

of I and the sum is taken over all

n:=,g, (x) with each g, (x) having period p,, and p, =

ord(q, up): j =O

n

i€J,

u? = pj) .

ur,

A.H. Chan et al.

I70

Since the period of f(x), p , is the 1.c.m. of the p,’s, each u, must appear as a factor of some p,. In fact, using the same argument as in the proof of Theorem 17, u $ must appear as a factor of some p,. For each j , 0 S j S 1. Let J:={iEJi: e i = d , } ;

then, the expression in (6) is at least

2 l.c.m.{ord(q,

u$): i € J:}.

j-0

u:=,,J;

Noting that = I, we can apply the same argument as in the proof of Theorem 16, and obtain the result.

Acknowledgement

The authors would like to thank Dr. J. Rabinowitz for many of his suggestions and comments during the preparation of this manuscript.

References [ I ] T. van Aardenne-Ehrenfest and N.G. de Bruijn, Circuits and trees in oriented linear graphs, Simon Stevin 28 (1951) 203-217. 121 E.R. Berlekamp, Algebraic Coding Theory (McGraw-Hill, New York, 1968). [3] N.G. de Bruijn, A combinatorial problem, Proc. Kon. Ned. Akad. Wetensch. 49 (1946) 758-764. [4] A.H. Chan, R.A. Games and E.L. Key, On the complexities of de Bruijn sequences, J. Comb. Theory, Ser. A, to appear. [S] C. Flye Sainte-Marie, Solution to question nr. 48,I’Intermediaire des Mathematiciens 1 (18Y4) 107-110. 161 R.A. Games and A.H. Chan, A fast algorithm for determining the complexity of a binary sequence with period 2”. IEEE Trans. Inform. Theory, 1980, submitted. [7] S. W. Golomb, Shift Register Sequences (Holden-Day, San Francisco, 1967). [X] 1.J. Good, Normal recurring decimals, J. London Math. Soc. 21 (1947) 167-169. [9] E.L. Key, An analysis of the structure and complexity of nonlinear binary sequence generators, IEEE Trans. Inform. Theory IT-22 (1976) 732-736. [lo] A. Lempel, On a homomorphism of the de Bruijn graph and its applications to the design of feedback shift registers, IEEE Trans. Comput. C-19 (1970) 12041209. [ I 11 J.L. Massey, Shift registers synthesis and BCH decoding, IEEE Trans. Inform. Theory IT-15 (1969) 122-127.


THE EXTENDED REED-SOLOMON CODES CONSIDERED AS IDEALS OR A MODULAR ALGEBRA Pascale CHARPIN Dipartement de Mathimatiques, U.E.R. Sciences, 123 rue Albert Thomas, 87060 Limoges, Cedex, France

The extended Reed-Solomon codes are ideals of a modular algebra A. Some properties of A are described which permit the lower bound for each principal ideal dimension of A to be defined. The results are used to characterise among Reed-Solomon codes those which are A principal ideals.

1. Definitions and previous properties

p is a prime, rn and r are integers different from zero, and K and G are, respectively, the Galois fields GF@') and GF(p "). A is the modular algebra KG. It is the polynomial space:

with the usual operations of polynomial multiplication and addition. The radical of a ring is the intersection of all its principal ideals. As A has characteristic p , an element of A is either invertible, or nilpotent, since

An ideal of A, different from A, has only nilpotent elements. Hence A has only one maximal ideal which is also its radical: P = { x E A Ixp=O}.

Pi, j 5 1, is the j-power of the radical of A. Piis the subspace of A generated by the set:

Let M = rn ( p - 1). I,, 0 S j

S

M, is the subset of N":

171

P.Charpin

172

Theorem 1. Let {el,.. . ,e m )be a basis of the F,-vector space G and let for each j, O S j S M , thesubsetofA:

1

~ j = k =(l f i ( ~ e l ~( i -l , l. . ~. , i , , , ) ~.~ Then Bo is a basis of the K-vector space A and B', 1 C j

Proof. Let j, 0 s j x=

C

6 M, is a

basis of PI.

M,and let x be a linear combination of vectors of B':

Akvk,

AkEK*,vk€B'.

kaRClj

Let i E R, i

= ( i , ,. .

. , L ) , so that

Vk, k E R , k = (k,,...,km),

Then

This means that x is different from zero if R is not empty. The vectors of B' are linearly independent. Ba has p" elements, since

pol=

I[O,p - 11" I = p " .

Then Bo is a basis of A. The space generated by B' is a hyperplane of A which is in P. Then B ' is a basis of P. So is becomes, from the definition of P', j > 1, that each element of P' can be written as a linear combination of elements of B'. So B' is a basis of Pi.

Remark. The index of nilpotency of P is M

+ 1 since P'+'

= (0).

Theorem 2. Let i E [l, M I ; s and t are, respectively, the quotient and the remainder of the division of j by p - 1. Then P' is the ideal of A generated by the subset of A.

I

S J = { ( x s-~ 1)p-l. . - I)"-'(xs=+~ - 1)' g l , .. . ,g,,' are linearly independent in G } . ( x ~ Z

Proof. The ideal generated by 9' is in P' and not in PI+' since each element of 3' is the product of j = s ( p - I)+ t factors (X't - 1). Let cr be an automorphism of the Fp-vector space G and let the A automorphism be defined so that

173

The extended Reed-Solomon codes

It is clear that &(%') = 9'. Poli shows in [I] that if an ideal of A is invariant under the group {& (T E GL(F,, m ) } , it is one of the powers of the radical of A. Hence, the ideal generated by %' i s Pi.

1

Corollary 1. Let j , j E [ l , M ] ; j = s ( p - l ) + t , there is a y, y E %"-j such that; Y X = A (X'l-

1y-I-* (Xc- ly-',

rE[O,p-l[.

If x E P ' \ P ~ + ~ ,

A E K*.

Proof. Let x E P' \P'+'. There is a 2, z E P'-', such that z x # 0. Since % M - J generate P"-j, then 3y, y E 9Y-1, y x Z 0 . Since yx E P", we have that yx = A(X'11Y-I- * ( X ' m - 1 Y - I with A E K*. The powers of the radical P of A are the Reed and Muller codes when p' = 2 and the generalized Reed and Muller codes when p' > 2 [2] and [3]. Theorem 2 is the generalization of a well-known property of Reed and Muller codes [4, p, 3851. 2. The dimensions of the principal ideals in A

If x is an element of A, the principal ideal of A generated by x is denoted (x); dim@) denotes the dimension of the K-vector space (x). Property 1. Let x, n E PI \Pi+', 1 c j s M. Then each y, y E (P' \Pj+') n (x) is such that ( y ) = (x). Proof. Let y , y E (x) and y E P j \ P i + ' . Then y = ax with a E A \ P : so a is invertible; that proves the property (x) = ( y ) . Theorem 3. Let j , 1 S j 6 M and j ' = M - j ; s ' and t' are, respectively, the quotient and the remainder of the division of j ' by p - 1. Then, Vx, x E Pj

\PI,

dim@) 2 p"'(t' + 1).

I f x isan elementof$? (Theorem 2), then dim(x) = p " ( t ' + I). Proof. Let x E P'\P'+'. From Corollary 1: 3y, y E gM-I,y x f 0. So y = (XSt - 1y-1 . . . (XB,.- l)(X..+l- 1)''

where (gl,. . . ,g,.,') are linearly independent in G.

(1) (2)

P. Charpin

174

We note by I the subset of W'+I :I = [O,p - 1]"X [0, t'] and Vi, i E I , i = ( i l , . . . , is.+l),u i = ( X ~-I1)iI . . . (X~,,+I -~Y.+I,

I

Q = { u i x i E I}. The cardinal of % is p"'(t'+ 1).Let z be a K-linear combination of elements of Q:

and let i, i E R, i = ( i l , . . . ,i,,+,) so that for each k, k E R, k = ( k l , .. . ,k,.+l),

( k l , .. . ,k,.) = ( i l , . . . ,is,)

+ is,+l< k,z+l,

Then (X81 -

l)I-l-i~ . . . ( X G - 1 ~ - 1 - i , ~ ( X 8 , 1Y'-* ~ + ~, . +- I-Z = y x # 0.

So z # 0, if R is not empty. 4'2 is a system of p s ' ( t ' + 1) linearly independent vectors of the K-vector space (x). ( 1 ) is proved. We now suppose that x = (Xgs.+I- 1)P-I-''(Xgs,+2 - I)'-' * (X*- - l)I-', where (gl,...,g,,,) is a basis of G. Bo is expressed from ( g l , ... ,g m )(Theorem 1). So, if u is in B', either ux E % or ox =O. Then Q is a basis of ( x ) . (2) is proved.

-

3. The extended Reed-Solomon codes considered as ideals of A

Notations. n = p m - 1, S = [0, n]. V k , k E S, the weight of k is w ( k ) : In-l

w ( k ) = C ki,

m-I

kiE[O,p-11,

i=O

C kip'zk.

i =D

Vj, l S j S M , S , = ( k E S I w ( k ) < j } . V x , x E A , x = C p e G x g X gand , V k , k E S, x ( k ) = CgEGx8gk. x ( k ) is calculated in an overfield of K and G. Property 2. V j , l s j s M , P ' = { x E A IVk, k E S j , x ( k ) = O } . Proof. For the proof, cf. [3].

"he extended Reed-Solomon codes

175

Henceforth K = G.The Reed-Solomon code, here denoted by cd, of length n, with minimum distance d over G, is the cyclic code with generator

where a is a primitive element of G. The extended Reed-Solomon code, here denoted by e d , is invariant under the affine permutation group on GF(pm).(Theorem of Kasami [5].) It is therefore an ideal of A, expressed as e d

= {x E A

The dimension of

Ix ( k ) = 0 for k = 0 , 1 , . ..,d - 1).

(3)

6 is dim 6 = n - d + 1 [ 5 ] .

Theorem 4. The extended Reed -Solomon code in the set:

edis a principal ideal of A if d is

j E [ l , p - 11, k E [0, rn - 11 D = [ d , = j p * +i = k + l (p-1)p'I I = j + ( p - 1)J[k + 1,rn - 111 m-1

(If k word

then dr = j p m - ' . ) If d = d r , dr E D,then extended.

= rn - 1, gd

I

ed= (id), where

gd

is the

Proof. (1) First we suppose that d ED. Then 3I,

I

=j

+ ( p - 1)J[k + 1, TTI - 11 I,

d = dr.

We have

= P * ( P -1).

But d is such that for all i, i E S and w ( i ) < 1, then i < d. Therefore it follows from (3) and from Propqrty 2 that e d C P'. idis such that g d ( ( I ) # 0 by the definition of the generator g d . Then g d d P ' + ' . We have shown:

Property 3. If d = d,, dr E D , then gd E P' \PI+' and then

ed C PI, edz PI+'.

Then, appealing to Theorem 3, we have dim(gd)3p'(t + 1) where s = m - ( [ k + l , r n - l ] ( - l = k and r = p - l - j . We have ( i d )

Therefore

c d

c e d and dim(gd) 3 dim e d . = (id).

P. Charpin

176

(2) We suppose now that d E D. Let I be the first index such that d < d l .Since c it follows from Property 3 that ede PI+'. If I = I, e d c P and e d z P*. If I > I, t d c SO, it follows from Property 3 that t d c PI-'. But c d l . , = If there is one x, x E e d n (P'-'\P'), then, by Property I, = (x> with (x) c e d . SO e d = e d l - , . This equation is impossible because d > dl-l.SO c d c PI. In all the cases, the definition of the generator gives g d !Z PI+'.We have proved edl

ed,

Property 4.

Property 4. Let d be such that d 6Z D. If 1 is the first index such that d < dl, then & c PI, c d $ ? PI+' and i d f PI \PI+'. Then, if c d is a principal ideal of A, it follows from Property 4 and Property 1 that c d = ( g d ) = ( i d l ) . This equation is impossible because d < d l . Theorem 4 is thus proved.

References [l] A. Poli, Codes stables sous le groupe des automorphismes isomktriques de A = F, [ X , ,. . .,X , ] / ( X : - 1 , . . . ,XP,- I), C.R. Acad. Sci. Paris (1980). [2] S.D. Berman, Kibernetika 3(1) (1967) 31-39. [3] P. Charpin, Puissances du radical d'une algkbre modulaire et codes cycliques, Revue CETHEDEC (1981). [4] F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes (North-Holland, Amsterdam, 1977). [5] J.H. Van Lint, Coding Theory (Springer, New York, 1971).


ODD ROOTED ORIENTATIONS AND UPPER-EMBEDDABLE GRAPHS 0. CHEVALIER, F. JAEGER, C. PAYAN and N.H. XUONG I.M.A.G., BP 53X, 38041 Grenoble, Ckdex, France In this paper, we give new characterisations of upper-embeddable graphs. Our study leads to the following result: Let GI = (V,, E,) and G, = (V,, E,) be two edge-disjoint graphs. If GI and G, are upper-embeddable with even Betti number then G = (V, U V,, E , U E2U E,) (where E, is any collection of edges with one end in V , and the other in V,) is upperembeddable.

1. Introduction A connected graph G = (V, E ) is said to be upper-embeddable if there exists an orientable surface in which it has a 2-cellular embedding with one or two faces according as its Betti number ( = IE I - I Vl + 1) is even or odd, respectively. It is proved in [33 that G is upper-embeddable iff it contains a spanning tree T such that for at most one component C of (V, E - E ( T ) ) ,lE(C)l is odd. (For various details concerning the concept of upper-embeddability and related subjects, the reader is referred to [2].) In this paper, using the concept of orientation, we state other characterizations of upper-embeddable graphs. These propositions are then used to provide new classes of upper-embeddable graphs, extending thus the result proved in [l].

2. Definitions and notations

(1) A11 graphs are finite and undirected. Loops and multiple edges are allowed. Given any graph G, we can obtain a digraph from G by specifying, for each edge, an order on its ends. Such a digraph is called an orientation of G. The will be used to designate an orientation of G. notation (2) For any digraph, the indegree d - ( x ) of a vertex x is the number of arcs with head x. A vertex x is called a root if every vertex is reachable from x. 177

0. Chevalier ef al.

178

(3) An orientation of G is said to be an odd rooted one if it admits a root r and if, for every x # r, d - ( x ) is odd.

3. Two characterizations of upper-embeddable graphs Let G be a connected graph whose Betti number is odd. As was shown in 131, the following properties are immediate consequences of the definition of upper-embeddable graphs: G is upper-embeddable iff there is an edge e such that G - e is upperembeddable. If the graph G + e obtained from G by adding a new edge e is upperembeddable then G is upper-embeddable. These remarks enable us, in the sequel, to consider only graphs with even Betti number.

Lemma. Let G = (V,E ) be a connected graph. Let S be any subset of V. Then there is an orientation of G such that Vx (x E S

e d - ( x ) is o d d )

i f I S l - I E l (mod 2). Hints. One can notice that, in a digraph, by reversing the orientation of each arc of a given (undirected) path fiom x to y , one changes the parity of the indegree of x and y. It is then easy to verify that an orientation of G which minimizes the quantity I{x E S d - ( x ) is even}l agrees.

I

Theorem 1. Let G be a connected graph with even Betti number. Then G is upper-embeddable if it admits an odd rooted orientation with arbitrarily specified root. Proof. (a) Necessary condition. Suppose that G = (V, E ) is upper-embeddable. It follows that G contains a spanning tree T such that, for every component C, of ( V , E - E ( T ) ) , IE(C,)l is even. Then, by the preceding lemma, each C, admits an orientation such that, for every x E V(C,), & , ( x ) is even (take S = 0 in the lemma). Let be the orientation of T rooted at some vertex r arbitrarily specified. For every x # r, we have & ( x ) = 1. Clearly, and the c’s constitute an odd rooted orientation of G with root r. (b) Sufficient condition. Let G be any odd rooted orientation of G with root r. It is known that G contains a rooted spanning tree f with root r. For every x # r,

ci

Odd rooted orientations and upper-embeddablegraphs

179

we have b r ( x ) = 1. Therefore, every vertex distinct from r has even indegree in (V, E((?)- E ( F ) ) .Hence every component of ( V ,E(G)-E ( T ) )not containing r possesses an even number of edges. It follows that G is upper-embeddable.

Remark. The proof of (b) is independent of the parity of the Betti number. Hence every graph with an odd rooted orientation is upper-embeddable.

Theorem 2. Let G = ( V ,E ) be a connected graph with even Betti number. Then G is upper-embeddable iff, for every S V such that IS I = I E I (mod 2), it admits an orientation satisfying the following conditions : (i) Vx ( x E S d - ( x ) is odd), (ii) Vx E S, there is a vertex in V - S from which x is reachable. Proof. First, observe that I E I = I V 1 - 1 (mod 2) because 1 E I - I V I + 1 = 0 (mod 2). It follows that, for every S C V such that IS I = IE I (mod 2) we have I S l = l V l - l (mod 2) and hence, V - S Z 0 . (a) Necessary condition. If G is upper-embeddable then it contains a spanning tree T such that, for every component Ci of ( V , E - E ( T ) ) , lE(Ci)l is even. Let S be any subset of V such that IS 1 = 1 E 1 (mod 2). If follows from I E l = ( V I - l (mod2) and I E ( T ) ( = l V J - l that I S l = l E ( T ) I (mod2). Therefore, by the lemma, T admits an orientation in which

x E S W d r ( x ) is odd. Furthermore, because T is acyclic, any vertex of odd indegree is reachable from some vertex of even indegree (i.e. in V - S ) . It remains now to show that there is an orientation of (V, E - E ( T ) )such that every vertex has an even indegree. This is clearly the case, because every component of (V, E - E ( T ) ) has an even number of edges (take S = 0 in the lemma). (b) Sufficient condition. We shall prove that G admits an odd rooted orientation and the result follows from Theorem l(b). Indeed, choose any subset S of V such that IS 1 = I VI - 1 . Therefore, IS 1 = I E I (mod 2). Let r be the vertex not contained in S. By the hypothesis, G has an orientation such that

x ES

e

d - ( s ) is odd

and every x E S is reachable from r. In other words, G admits an odd rooted orientation with root r.

180

0.Chevalier et al.

4. Application

In [2] we proved the following result: every graph obtained by connecting (with any number of edges) two uerrex-disjoint upper-embeddable graphs with even Betti number is upper-embeddable. Using Theorem 1 and Theorem 2, we are now able to state a more general theorem.

Theorem 3. Let G I = ( V,, E l ) and G2= ( V,, E,) be two edge-disjoint graphs. If GI and G2m e upper-embeddable with euen Betti number ?hen G = (V, U Vz, El U E , U E,) (where E , is any collection of edges with one end in V, and the other in V2) is upper-embeddable. Proof. By Theorem l(b) it is sufficient to prove that G or G + e (where e is a new edge) admits an odd rooted orientation. Since G I is upper-embeddable, it follows from Theorem 1 that G I admits an odd rooted orientation with root x , . Let I?, be the set of arcs of this orientation. Denote by J?, the set of arcs obtained from E , by directing each edge of E , from V, to V2. Let D be the digraph (V, U V,, g, U I?J. V2 is partitioned into two disjoint subsets Vi and V; such that

x

E V:

x E Vg

e d & ) is odd, e d ; ( x ) is even.

Note that every vertex in V: is reachable from x , . Two cases are considered: Case 1. I V i l = lE21 (mod 2). From 1 E21= 1 V2(- 1 (mod 2) (recall that I E2( - 1 V21+ 1 is even) it follows that Vi # 0. Since G2is upper-embeddable, by Theorem 2, there is an orientation GZ of G2 such that

x E V;

e ( d & , ( x )is odd and every vertex of

V; is reachable from some vertex of V:).

Then every vertex of V; is reachable from x I . Let us designate by 2, the set of arcs of G2. It is clear that (V, U V2, glU 8, U 8,)is an odd rooted orientation with root x i of G. Case 2. VS = / E 21 - 1 (mod 2). Let x 2 be any vertex in V,. Consider a new arc e' having head x2 and tail x , . Let D' be the digraph D + L Set

w = v; a{x,}, where A is the symmetric difference. We have d D ' ( x 2 ) = dD(x2)+ 1. Therefore x E W e d ; . ( x ) is even.

Odd rooted orientations and upper-embeddable graphs

181

Since I W I = 1 V ; l + 1 = lE21 (mod 2) we can repeat the argument used in the first case. In other words, G + e is upper-embeddable with even Betti number. Hence G is upper-embeddable with odd Betti number. This completes the proof.

References [ l ] F. Jaeger, C. Payan and N.H. Xuong, A class of upper-embeddable grdphs, J . Graph Theory 3 (1979) 387-391. [2] R.D. Ringeisen, Survey of results on the maximum genus of a graph, J. Graph Theory 3 (1979) 1-13. [3] N.H. Xuong, How to determine the maximum genus of a graph, J. Comb. Theory, Ser. B 26 (1979) 217-225.



NOTES ON THE ERDOS-STONE THEOREM V. CHVATAL and E. SZEMEREDI School of Cornpurer Science, McGill University, Montreal, Canada

One of the main subjects of extremal graph theory is the largest number ex(F, n ) of edges in a graph on n vertices containing no subgraph isomorphic to a forbidden graph F. The fundamental theorem of Erdos and Stone [6] concerns the case of complete ( d + 1)-partite graphs Kd+l(t)with t vertices in each part: 1 lim ex(Kd+,(t),n) '1-d.

As pointed out by Erdos and Simonovits [ 5 ] , the Erdos-Stone theorem determines the asymptotic behaviour of ex(F, n ) in terms of the chromatic number x ( F ) of E More precisely, if x ( F ) = d + 1 and if F has at most t vertices then F is a subgraph of K d + l ( f )and , so ex(F, n ) G ex(Kd+t(t),n). On the other hand, ex(F, n ) is at least the largest number of edges in a d-partite graph with n vertices, which is asymptotically equal to n 2 ( d - 1)/2d. Thus

for all graphs F. With t(c, d, n ) standing for the largest t such that every graph with n vertices and at least

$n2(1 - l/d)+ cn2 edges contains a

Kd+l(f),

f(c, d , n ) = lim "-=

the Erdos-Stone theorem states that

+ m.

How fast does t(c, d , n ) grow? The case of d = 1 has been studied by Koviri, S6s and Turin [7], who proved that 183

V.Chvriral, E. Szemeridi

184

lim L n-=

@ J 2 l a L

logn

log(l/c)

*

More recently, Bollobis, Erdos and Simonovits [2] proved that

d, n> a log n d log(l/c)

t(C,

for some positive constant a and conjectured that this bound can be improved into

t(c, d, n ) a- b log n log(l/c) for some positive constant b. In [4] we have settled an asymptotic version of this conjecture by proving that lim n-=

wa log n

1 50010g(l/c)

’

(In a sense, this result is best possible: as shown by Bollob6s and Erdos [l], our constant 0.002 cannot be increased beyond 5.) The purpose of these notes is to outline the proof in a less rigorous but more intuitive way than that given in [4]. Our argument hinges on a theorem asserting that every sufficiently large graph can be partitioned into a small number of classes in such a way that the partition exhibits strong regularity properties. To state the theorem in more precise terms, we need a few definitions. When A and B are nonempty disjoint sets of vertices in some graph then we denote by d ( A ,B) the density of edges between A and B : this is the number of edges with one endpoint in A and the other endpoint in B divided by 1 A I 1 B I. The pair ( A ,B) is called E -regular if X C A , 1 X I 3 e I A I , Y C B and I Y / a & / Bimply ( that l d ( X , Y ) - d ( A , B ) I < E ; otherwise the pair is &-irregular. A partition of a set V into classes C,,C,, . . . ,Ckis called &-regularif IC,lS E 1 VI, ICi I = IC, I whenever 1 S i < j S k and if at most ek’ of the pairs (Ci,C j ) with 1 S i < j G k are &-irregular. The Regular Partition Theorem, proved in [lo], asserts that for every positive E and for every positive integer m there are positive integers M and N with the following property: If the set V of vertices of a graph has size at least N then there is an &-regular partition of V into k + 1 classes such that m S k S M. This theorem has been applied to several other extremal problems in combinatorics and elementary number theory (for instance, see [3], [8] or (91);in our proof, it is used to establish the following.

-

Lemma. For every choice of positive numbers c, d and E such that d is an integer and E < c/10 there are positive numbers N and S with the following property. If

Notes on the Erdos-Stone theorem

185

n 3 N then every graph with n vertices and at least

edges contains pairwise disjoint sets C1,C,,. . ,Cd+,of vertices such that IC, I 2 Sn for all i, d (C,,C,) 3 d

-1

+ cd

for all j ,

i#j

and such that every pair (Ci,C,) is &-regular. We use this lemma with an E which is extremely small with respect to c and d. The rest of the proof consists of finding a subset Mi of each Ci such that

and such that every two vertices belonging to distinct sets Mi are adjacent. For simplicity, let us first assume that

d(C;,Cj)a0.4+~ whenever 1 c i < j C d + 1 . In this case, the desired sets MI,M 2 , ... ,Md+lcan be found by an iterative procedure whose rudimentary version is described below. This rudimentary version does not quite produce the desired sets Mi. Instead, it delivers certain sets K1,K 2 , ... ,K d + ,whose common size k is the largest integer such that k(ed)‘ C E IC,I for all t. (Note that k = l o g n / ( l +logd).) It will be convenient to write N(X,Y) for the largest integer such that each vertex in X has at least N(X,Y) neighbours in Y. Step 0 . [Initialize.] Set t = 1 and Si = C, for all j . Step 1. [Construct K,.] Let S consist of all those vertices in S, which have at least

(d(Cr,C,)-E)ISj 1 neighbours in each Sj with t < j =sd Ki (1 d i < t ) such that IK, I = k,

+ 1. Find a subset K, of S and subsets NI of

1 N f l = N ( S , Ki), and such that each vertex in K, is adjacent to all the vertices in each Nf. From each Sj Step 2. [Remove deficient vertices from S,,,, S,+2,.. . , with t < j s d + 1, remove all the vertices having fewer than

neighbours in K , .

V. Chvdtal, E. Szemerkdi

186

Step 3. [Done ?] If t = d + 1 then stop; otherwise replace t by t + 1 and return to Step 1. Before proving that Step 1 can be executed, let us observe that the sets Sj shrink only moderately with each execution of Step 2. More precisely, we claim that ISj 12 c f IC, 1 after t iterations. This claim is easy to justify by induction on t. Since the number of edges between K, and any Sj with f 4 j S d + 1 is at least (d(C1, C,) - & ) 1 Kt

I1 1 *

sj

9

the number of deficient vertices in Sj is less than (1 - c)l Sj I. In particular, when the execution of Step 1 begins, we have

1 S, I a cl-l I C,13 (d + 1 ) 1~Cl 1. Since each of the pairs (C,, C,) is &-regular,at most d~ 1 Cl I vertices belong to S, - S, and so I S I 3 E IC, 1. Let us label each o E S by a sequence of sets N l ( v ) ,N 2 ( o ) ., . . ,Nl-l(o) such that tr is adjacent to all the vertices in Ni(0) and N, (0) C K i , IN,( o ) l = N ( S , , K , ) whenever 1 6 i < t. If L denotes the number of all conceivable labels, then

and so some k vertices in S must have the same label. When the rudimentary procedure has terminated, one may attempt to produce the desired sets M, by letting each of them consist of the intersection of all the sets N : with i < t 6 d + 1. Now any two vertices belonging to distinct sets M, are adjacent. The only problem is that these sets may be too small. In the worst case, we may have d+l

IM,I=)K,)-

2

d+l

2

)K,-N:J=)KI-

(lK/-N(S,K)),

l=l+l

1=1+1

and

IKII-

d+l

d+l

=2

1=2

2 ( ~ ~ ~ ~ - N ( S l , Kkl(1) )-=

I

(1 - d(Cl, C,)+c ) ) = 0.

Nevertheless, M , may get disastrously small only if the sets N : overlap in a very special way. We claim that, in case IMi I < 0.02k, the relative positions of these sets must be far from random in the sense that d+l

2

j=f+l

d+l

IN1nK*IzO.O2k+(K*)

2

j=r+l

(d(Cj,C,)-c)

for some t and for some subset K * of K, such that 0.2k

S

( K * (s 0.8k.

Notes on the Erdos-Stone theorem

187

To justify this claim, write M!;for the intersection of the sets N f with i < s S j , let t stand for the smallest subscript with lMfl s 0.8k and set K * = Ki - M i . Now

d+l

=IK*l

2

j=r+I

(d(C,,C)-c)

d+l

C

+IMfJ

(d(C,,Ci)-c -l)+(lM:l-lMi().

,=(+I

Since

M f= Mf-' n Ni = MI-' - ( K i - Nn, and

IMi-' 1 3 0.8k,

1 Ki - Nil s k (1 - d (C,, Ci) + c ) s 0.6k,

the size of K * is within the prescribed bounds. Furthermore, since

0.2k S I K - K * J S

2 ( K i- N : : l s k t: (l-d(C,,Ci)+c), j=i+l

j=i+l

and

x

d+l

(1 -d(C,, C , ) + c ) S

C (1 - d(C,, Ci)+ c ) S 1,

I#i

l=i+l

we have d+l

2

j=r+l

(1 - d(C,, Ca)+ c ) s 0.8.

Hence

and the claim is verified.

To be able to draw an even stronger conclusion from 1 Mi I < 0.02k, we modify the way of updating the sets Sj in Step 2. Having deleted all the deficient vertices as above, we obtain a subset S y of Si. Rather than setting S, = S : as above, we choose in each S : subsets Tj = Tb (1 S i < t) of size LIST 1/2d] such that, for each choice of u € T i and w ES:-Tj, the number of neighbours of u in

V.Chvcital, E.Szemeridi

188

K, - M, is at least the number of neighbours of w in K, - M,. Then we replace S, by the set obtained from S : by deleting the t - 1 sets T,. The point of this modification is that each vertex in the new S, has at most N(TJ, K , - M,) neighbours in K, - M,, and so I N : - M, 1 s N(T,,,K, - M,). Thus we may conclude that, in case I M, I < 0.02k, there are large subsets such that

TI of S,

d+l

j=f+l

N ( T j , K * ) 3 0 + 0 2 k + I K *C ( (d(C,,C,)-c) ;=,+I

for some t and for some subset K * of Ki such that 0.2k c IK* I s 0.8k. Now a way of changing the disaster into an advantage suggests itself. We may Mlf2,M i + 3 , .. . for all i and, as soon monitor the sizes of the intersections M:+’, as one of them drops below 0.2k, locate the appropriate sets K * and ‘I;Then . we backtrack to the ith iteration, replace Ki by K * and replace each previously constructed Si by Ti whenever t < j d d + 1 and N ( K j , K*)3 ( K * I(d(C,, Ci)- c ) . After these adjustments, the size of M, is guaranteed to stay above 0.02k. If J denotes the set of subscripts for which Sj has been adjusted then

C N(S;,K*)SO.O2k+IK*I C ( d ( C , , C i ) - c ) , j€J

fEJ

and so d+l

IMi(*IK*J,-

C

f=i+l

d+l

I K * - N f J = I K * I +2 ( I N f T 1 K * ( - I K * ) ) t=i+l

d+l

3

IK * I (1 + C

(d(C,,

r=i+l

c ) - c - I)) + 0.02k 3 0.02k.

The backtracking can be avoided altogether if, rather than waiting for the size of Mi to drop below 0.02k, we ask at once whether such a drop is possible. An implementation of this idea involves a careful choice of numbers S1, ti2,,. . ,6, such that &

@61e6z cd. From each of the two sets C, ( d s j S d + l), we delete all the vertices which are deficient in the sense of having fewer than (d (C, ,Ct - E ) I CtI

neighbours in at least one of the sets C, (I S i < d). Since all the pairs (C,, C,) are &-regular,there are no more than ( d - 1 ) I ~ C, 1 deficient vertices in each of the two sets C, and the resulting subsets S, of C, have d ( S d , s d + l ) s cd - E . Hence the theorem of Kovari, S6s and Turin, or its improvement due to Zn5m [ 111, guarantees the existence of sets Kd, K d + , such that K, C S , , I K , I s logn/lOOlog(l/c) for both j and such that every two vertices belonging to different sets K, are adjacent. To extend this bipartite graph into the desired ( d + 1)-partite graph, we first delete from each C, with 1 < i < d all those vertices which are deficient in the sense of having fewer than

(2d(C,,C , ) - 2 c

- l)lK/

1

neighbours in at least one of the two sets K,. (An easy computation shows that there are no more than (1 - 2c)l C, I deficient vertices in each C,.) Then, in each of the resulting sets Cf , we label each vertex u by two sets Nd( u ) , N d + l ( u ) such that

N/(u)cK/,

) N / ( u ) l = N ( C T , K J ) 5

and such that u is adjacent to all the vertices in N , ( u ) for both j . Now it will suffice to find, in each C: with 1 s i < d, a subset M,such that (i) 1 M,13 log n/500log(l/c), (ii) every two vertices belonging to distinct sets M, are adjacent, (iii) every two vertices belonging to the same set M,have the same label.

V.Chvrital,E. Szemeridi

190

Indeed, if Nj stands for the common value of 4 . ( u ) with u E Mi and if M, with d sj zs d + 1 stands for the intersection of all the sets Nj then

and

1-

d-l

(1 -(2d(G, c,)-2c - l))B0.2. i=l

Sets M1,Mz, . . .,MdVlsatisfying the requirements (i) and (ii) could be found by induction; the requirement (iii) leads us to an inductive proof of a more complicated statement involving graphs whose vertices are labeled. The arguments described here may be modified to apply in the more general context; the technical details can be found in [4].

References [I] B. Bollobhs and P. Erdos, On the structure of edge graphs, Bull. London Math. SOC.5 (1973) 3 17-32 1. [2] B. Bollobds, P. ErdBs and M. Simonovits, On the structure of edge graphs 11, J. London Math. SOC.(2) 12 (1976) 219-224. [3] 8. Bollobhs, P. Erdos, M. Sirnonovits and E. Szernertdi, Extremal graphs without large forbidden subgraphs, Ann. Discrete Math. 3 (1978) 29-41. [4] V. Chvdtal and E. SzemerCdi, On the Erdos-Stone theorem, J. London Math. SOC.23 (2) (1981) 207-214. [5] P. Erdos and M. Simonovits, A limit theorem in graph theory, Studia Math. Sci. Hungar. 1 (1966) 51-57. [6] P. Erdos and A. H. Stone, On the structure of linear graphs, Bull. Amer. Math. SOC.52 (1946) 1089-1091. [7] T. Kovdri, V.T. S6s and P. Turdn, On a problem of Zarankiewicz, Colloq. Math. 2 (1954)50-57. [8] I.Z. Ruzsa and E. Szemertdi, Triple systems with no six points carrying three triangles, Colloq. Math. SOC.Jdnos Bolyai 18, Combinatorin, Keszthely (1976) 939-945. [9] E. SzemerCdi, On sets of integers containing no k etements in arithmetic progression, Acta Arith. 27 (1975) 199-245. [lo] E. Szemerkdi, Regular partitions of graphs, Problembs en Combinatoire et ThCorie des Graphes (C.N.R.S., Park, 1978) 399-401. [ l l ] 3. Zndrn, Two improvements of a result concerning a problem of K. Zarankiewicz, Colloq. Math. 13 (1965) 255-258.


SOUS LES PAVES..

.*

M. COCHAND E. P.E.L., 61 an Cour, 1007 Lausanne, Switzerland

P. DUCHET E.R. “Combinatoire”, CMS 54 bd. Raspail, 75270, Paris, Cedex 06 France Various combinatorial properties of Euclidean boxes are investigated. A box is defined as a product of intervals of the real line. Every intersection of boxes is a box, so boxes define a convexity structure whose usual parameters, Radon, CarathCodory, Helly and exchange numbers, have been evaluated by various authors. Related to the Erdos-Szekeres theorem on monotonic sequences, we prove the following: Among 22”-’ I points in R“,there exist three of them x , y, z such that x is ‘between’ y and z : every box containing y and z must contain x .

+

1. Introduction a la convexit6 abstraite

Ce travail n’est que le moment d’une recherche (les seuls rCsultats prCsentant une difficult6 sont les ThCorhmes 3.5 e t 3.6). I1 se veut une invitation faite aux combinatoriciens au domaine de la convexit6 abstraite. Etudier les Convexire‘ abstraites suivant une idCe dont l’origine remonte h Motzkin [ 141 ou les espuces h conuerite‘ suivant la terminologie moderne [ 11, 12, 13, 18, 31, c’est simplement Ctudier une famille de parties stables par intersection, ou, ce qui revient au mime, Ctudier un espace B fermeture, c’est-&dire un ensemble muni de la structure dCfinie par une application idempotente dans I’ensemble de ses parties. PrCcisCment: Definition 1.1. Un espace Ci convexire‘ est un ensemble S muni d’une famille % C 2’ de parties de S vCrifiant: OECe

(1)

et SECe.

Toute intersection d’ensembles de Ce appartient h Ce.

(2)

% sera la convexire‘ considCrCe, les membres de Ce Ctant les ensembles Ce-convexes ou simplement convexes ( )% dksignera l’enveloppe conoexe :

(A)* = QC,A c C E %).

(3)

* Cf. “Sous les pavCs, la plage”. Poetry in the Street, Quartier Latin, May 1968, d’aprks Confucius. 191

M . Cochand, P. Duchet

192

On Ccrira (x1,x2,. . . ) pour ({x,, x2,. . .}). Presque toujours, une convexit6 % peut se dCfinir 5 partir d'une fonctioninterualle, c'est-&dire une application I : S x S + S satisfaisant aux axiomes:

Si z E Z(x, y), alors I ( x , I ) C I ( x , y).

(6)

Un tel point de vue a CtC notamment abordC par Calder [3] et Mulder [15]. Pour I : S x S + S donnCe, on dCfinit une convexit6 El par:

C E %, ssi I ( x , y ) C C pour tout x, y E C.

(7)

Pour %' C 2' donnCe, on dCfinit 1%simplement par:

I ( x , y ) = (x, Y )%. On a alors I%, = I pour toute fonction-intervalle I. Par contre, en gCnCral %+

%Iw.

Definition 1.2. Une conuexite' d 'interualle est une convexit6 qui satisfait: %,* = %.

La relation " z E (x, y)" se lira z est entre x et y.

Definition 1.3. Un ensemble A C S est %-independant si on a: a $Z (A \ a ) pour tout a E A.

(11)

La cardinalit6 maximum d'un indkpendant sera appelCe, bien que l'on n'ait pas de structure matroidale, la dimension de la convexit6 %. Un ensemble non independant sera dit dipendant. Nombreux auteurs ont CtudiC les analogues des thtorkmes de Radon, Helly, CarathCodory en convexit6 abstraite [6,7, 17, 18,201. Rappelons les definitions; conv dCsigne ici la convexit6 habituelle dans W" (une partie de R" est dans conv si elle contient tout segment affine dont elle contient les extrkmitks) dite convexit6 ordinaire.

Theoreme 1.4 (Radon [ 161). Si une partie A de W" contient au moins n + 2 points, il existe une partition de A en deux ensembles dont les enveloppes convexes ordinaires ont un point commun.

Sous les p a v h . . .

193

ThCor6me 1.5 (Helly [lo]). Soient C,, . . . ,C, une famille d’ensembles convexes (ordinaires) duns W“.Si n + 1 quelconques des Ciont un point commun, alors tous les Ciont un point commun. ThCoreme 1.6 (Carathtodory [4]). Si un point p de W” est duns l’enveloppe convexe (ordinaire) d’une partie A, il existe au plus n + 1 points de A dont l’enveloppe convexe (ordinaire) contient p.

D’oii les dtfinitions (% &ant une convexitt abstraite sur S):

Nombre de Radon r ( % ) = plus petit entier r tel que toute partie A de cardinal3 r admette une partition B, C telle que ( B ) n (12)

(C)# 0. Une telle partition est appelte partition de Radon de A.

Nombre de Helly h ( % ) = plus petit entier h tel que toute famille finie d’ensembles %-convexes dont l’intersection est vide com- (13) porte au plus h ensembles d’intersection vide. Nombre de Carathkodory c(%) = plus petit entier tel que pour toute partie A de S, on ait: (A)=

U ( ( F ) , I F I Q c et F C A ) . F

Le Nombre d’kchange e ( % )introduit par Sierksma [18] est le plus petit entier e tel que pour tout ensemble A de cardinal fini 3 e et tout point p de S on ait: (A)C

U (pU(A\a)). OEA

1 J

et [

1 dtsigneront les parties entibres par

dtfaut et par ex&.

2. ConvexitC par pavCs

Nous dtsignerons par i : R“ --* W le ieme application coordonnte. Un pave‘ P de W” est le produit de n intervalles de R:

avec ai,bi E W U { + m,

- m},

et f dtsigne [ ou

1.

M.Cochand, P. Duchet

194

Une intersection quelconque de pavCs est un pavC. Les pavts sont donc les La notation ensembles convexes d'une convexit6 abstraite sur W" notCe 9,. ( )8m sera abrCgCe en ( ). et m&me( ). Un point z est entre x et y si et seulement si, pour tout i = 1,.. ., n le ibme coordonnCe de z est entre les icme coordonnCes de x et y .

Proposition 2.1. 9,, est une conuexitt d 'intervalle. Proposition 2.2 (Domain-finiteness,cf. [9,19]). P,, est de type fini; autrement dit, pour tout A C W",

Proposition 2.3 (Join-Hull commutativity cf. [ll, 181). 9,, est 'pyramidale'; autrement dit pour tout A C W" et p E W"

Ces propriCtts sont faciles E i vCrifier comme consequences de la remarque suivante:

Lemme 2.4. Pour A C W", on a

n n

( A )= n { P ; P paut qui contient A } =

(i(A))

i=l

(18)

et si A est fermt:

On doit Cgalement remarquer que 9,est une conuexite' produit au sens de Eckhoff [6,7] et Sierksma [HI. Eckhoff a obtenu le nombre de Radon (thCorkme 3.7). Nous donnons en Section 4 une adaptation fraqaise de sa preuve. Du travail de Sierksma on peut dCduire les propositions 2.3 et 2.4 ainsi que les valeurs exactes des nombres de Helly, CarathCodory et du nombre d'Cchange. Nous donnons en Section 4 des preuves directes. La Section 3 est consacrCe B I'CnoncC des propriCtCs combinatoires des pavCs.

3. RCsultats Soient A, B, C trois intervalles de W; alors I'un d'entre eux contient I'intersection des deux autres.

Sous les pa&.

..

195

Une extension triviale est le suivant.

ThCoSme 3.1 (Folklore) (PropriCtC de Helly). O n a, a : h ( P n )= 2. (Autrement dit, des pave's en nombre fini ayant deux ci deux un point commun ont un point commun.) Une seconde extension est possible.

Theoreme 3.2 (PropriCtC de I'intersection). Duns une famille de 2n + I p a u b de W", il en existe un qui contient l'intersection des 2n autres. E n outre 2n + 1 est la meilleure borne possible. Parmi trois points de W", it en existe un qui est entre !es deux autres. Trois directions d'extension de cette propriCtC sont possibles.

ThCorbme 3.3 (Reay [17]) (Nombre de CarathCodory). On a ~ ( 9=,2; ) c ( P " , )= n pour n 2 2 . (Autrement dit un point qui est dans I'enveloppe convexe de A C Iw" est dans l'enueloppe convexe d'au plus n points de A.) Thbrbme 3.4 (Dimension convexe). 9'" est de dimension 2n. (Autrement dit, parmi 2n + 1 points de W" il en existe un qui est dans l'enveloppe convexe des 2n autres, et 2n + 1 est la meillewe borne possible.) Thkor&me 3.5. Parmi 22n-'+ 1 points de R" il existe 3 points 9",-de'pendants ( = un des points est entre les deux autres). E n outre, il existe efectivement 22" points de W" tel qu'aucun des pavks engendre' par deux points n'en contienne un troisikme. Nous avons enfin le suivant.

Theorbme 3.6 (Eckhoff [7]) (Nombre de Radon). O n a :

Ce rCsultat est B rapprocher du nombre de Radon du n-cube (convexit6 dCfinie sur (0,l)" par z E ( x ,y ) si et seulement si d (x, y ) = d ( x , z ) + d ( z ,y ) ; d dtsignant la distance de Hamming) qui est (cf. [ 5 ] ) log210grn+ 1 , ce qui correspond ii la version 'discrkte' du problbme.

ThCorkme 3.7 (Sierksma [HI) (Nombre d'kchange). On a e ( 9 " )= n

+ 1.


196

Ces thtorkmes peuvent s'Cnoncer sans grande modification dans le cadre des convexitCs d'ordre, oh les intervalles sont dtfinis de manikre naturelle. Ces rtsultats et des recherches sur les nombres de Radon et de Helly gtntralists feront I'objet d'un travail ulttrieur.

4. Preuves et commentaires

4.1. Preuve du The'orime 3.1 Evident par projection, en appliquant la propriCtC de Helly des intervalles de

w. 4.2. Preuve du Tht!ort?me 3.2 Soient Po,PI,...,P Z n , 2n + 1 pavts de R". D'aprbs (20), pour la ibme et tels que i ( P k )3 i(Pn(,)) f i(Pp,,)) l coordonnte, il existe deux paves Pa(,) pour tout k = 0,. . . , 2 n ; et ceci pour chaque i = 1,. . .,n. On a donc, pour un indice y E{O,. . . , 2 n } different de tous les a ( i ) et p ( i ) : " P, 3 fl (Pa(i1n PeciJ- 0 i=l

D'autre part, pour voir que 2n + 1 est la meilleure borne possible, il suffit de considCrer la famille des 2n pavts. P I , .. . ,P,, Q I , .. . , Q. dtfini par

i ( P k )= i ( O k )= ( - 1,l) si i# k, i(Pi)=(-l,O),

i(Qi)=(O,l).

4.3. Commentaires sur le The'orime 3.3 (Carathkodory ) Le rCsultat c ( P n ) sn est Cquivalent au thCorkme suivant qui se montre aistment par recurrence:

D ' u n recouvrement en p 3 n parties d'intersecrion vide d ' u n ensemble a n 2 2 e'le'ments, on peut extraire un recouvrement en n (21) parties d 'intersection vide. Pour en dCduire le ThCorbme 3.3, il s a t en effet de remarquer que si x, a l , .. . ,ap sont des points de R", avec p 3 n 3 2 et si i ( x )# i(a,) pour tout i et k, les parties A, = { i ; 1 s i s n, i ( a k ) > i ( x ) }

Sous les pavb . . .

197

forment un recouvrement de { l , . . . , n } et sont sans point commun si et seulement si x E ( a l , .. . ,up). Notons aussi une formulation duale de (21):

Pour toute famille ( A i ,Bi)i=l,...,n de n partitions ( n 3 2) en deux parties d'un ensemble X de cardinal m 2, il existe une partie Y (22) d 'au plus n points rencontrant chaque partie de chaque partition. 4.4. Preuve du Thiorime 3.4 (Dimension)

La vkrification est laisske au lecteur. Remarquons que I'optimum est atteint avec les points x k = ( S i k ) i = l . . . . . net - x k .

4.5. Preuve du Thiordme 3.5 (Premidre partie) Rappelons tout d'abord le suivant.

Theoreme d'Erdos-Szekeres [8]. Soient p, q deux entiers 3 0. Une suite b pq + 1 termes dans un ensemble totalement ordonne' contient une suite extraite croissante de p + 1 termes ou let une suite (23) dicroissante de q + 1 termes. ConsidCrons maintenant 2'"-' + 1 points de W" et rangeons les dans l'ordre lexicographique (indices des coordonnkes croissant). La suite des lbres coordonnkes de ces points est croissante. La suite des 2kmes 1. En appliquant (23), on peut coordonnkes est de longueur (22"-2)(22"-2)+ extraire de la suite initiale une suite de longueur au moins 2'"-'+ 1 dont les 2kmes coordonnkes des termes forment une suite monotone. En itCrant le procCd6 pour les 3kmes,. . .,nbmes coordonnkes on aboutit i une suite extraite d'au moins 3 termes, a, b, c, dont les coordonnies d'indice k quelconque (16 k S n ) forment une suite monotone. On a donc b E (a,c>. 0

4.6. Preuve du Thiordme 3.6 (Second partie) Le problkme est alors de construire effectivement 2'"-' points dans R" qui sont '3-indipendants', c'est-&-dire qu'aucun d'entre eux n'est entre d e w autres. I1 est clair que la propriktk ne dipend que de l'ordre des coordonnies de mime indice. Nous prockdons par recurrence sur n, en supposant que l'on ait construit dans w"-l = 22-2 points 3-indipendants xI,. . . ,xm dont les coordonnkes d'indice i sont toutes distinctes. Ceci est clairement possible dans W avec m =2.


198

Lemme. I1 existe dans W", m points avec les mimes propriktks : les points ont des coordonnkes d 'indice i distinctes et aucun d 'eux n 'est (24) entre deux autres. Notons x; (1 S p, q S m ) les points que nous allons dkfinir. L'ordre des m' coordonntes d'indice i ( i < n ) de ces points est obtenu par la rbgle suivante:

P o u r p Z p ' : i(x;)< i(x;:) ssi i(x,)< i(x,.). (25)

Pour q # q ' : i(x;)< i ( x $ ) ssi i(x,)< i(xq.). L'ordre des m 2 coordonnkes d'indice n est obtenu par:

Pour p # p ' : n(x;)< n(x;:) ssi l(x,)< I(x,,). Pour q # q ' : n ( x ; ) < n ( x ; : ) ssi l(x,)>I(x,.).

(26)

La construction, initialiske pour n = 1 par les rkels 1 et 2 donne: dans 02':

dans W':

Xf

XS

1 2 2 1

3 4 4 3

X f

XS

1 2 3 4 6 5 8 7 4 3 2 1

5 6 7 8 2 1 4 3 8 7 6 5

X4

X:

9 1 0 1 1 1 2 12141516 1 4 1 3 1 6 1 5 1 0 9 1 2 1 1 121110 9 1 6 1 5 1 4 1 3

etc. (27) Considkrons trois points x ; , x;:, x;'. l e i cas. Si p = p ' = p ' ' les n - 1 premibres coordonntes des trois points satisfont aux mCmes relations d'ordre que celles des points x,, x,., x,.. dans R"-'. Les trois points sont donc indkpendants. 2ime cas. Si p , p ' , p" sont tous distincts: m&meconclusion en considtrant les n - 1 premieres coordonnkes et les points x,, x,., x,". 3ime cas. p = p ' , p # p " ; on vkrifie alors aiskment en observant les coordonnkes d'indice 1 et d'indice n, qu'aucun des trois points n'est entre les deux autres. 0 Ceci achbve la dtmonstration du lemme et du thkorkme. 4.7. Preuve de l'inkgalitk

sz

dans le Tht!orime 3.6

La preuve donnte ici est conforme h celle d'Eckhoff [7]. Soit S un ensemble de p points de R" avec

Sous les p a v b . . .

199

Pour montrer que X admet une partition de Radon, considCrons pour i fix6 (1 C i c n ) les parties T de S vtrifiant les conditions:

(i(T)h n ( i ( S \ T ) h = 0.

(30)

Une partie non vide vCrifiant (30) conteint un ClCment de S dont la ikme coordonnCe est extrCmale dans i(S) (minimum ou maximum). Soit T, (respectivement T : ) une partie maximale (pour l’inclusion) de S vtrifiant (29) et (30) et aussi la condition min i ( T , )= min i ( S )

si 17;:Z0

(31)

(respectivement max i ( T : )= max i(S) si T :# 0). Montrons le suivant.

Lemme. T, et TI sont uniquement dkterminks par i.

(32)

I1 suffit, par symCtrie, de le prouver pour T,. Si 17;: est vide, c’est tvident. Sinon, soit T une partie de S vtrifiant (29), (30) et (31) et prouvons T C T,. On peut supposer T f 0. i ( T ) et i(T,) sont d’aprhs (30) des intervalles de i ( S ) (pour l’ordre induit par l’ordre des rCels) et ont mtme extrCmitC gauche; ils sont donc cornparables par inclusion. Si i ( T ) C i ( T , ) , un Clement t de T\T, vCrifierait i ( t ) E i ( T \ T , ) n i ( ? ; : ) , contradiction avec la condition (30) pour T,. Si i ( T ) C i ( T ) , un CICment t de T,\ T (non vide car T est maximal) vtrifierait i ( t ) E i(T,\ T ) i l i ( T ) , contradiction avec la condition (30) pour T. On a bien T C T , d’oii le lemme (32). Les T, et T : (1 s i d n ) sont donc en nombre S 2n et d’aprbs l’hypothkse (28), il existe une partie A de S avec IP/2J ClCments, distincte des 17;: et des Tf. A vtrifiant 1’tgalitC dans (29) ne peut vCrifier (30) sans Gtre Cgale A T, ou A T : . On a donc pour tout i : ( i ( A ) hn ( i ( S \ A ) h ZO. A, S \ A forment la partition de Radon cherchte. 0

4.8. Preuve de l’kgalite‘ du Thior2me 3.6 La preuve d’Eckhoff [7] est un peu plus detaillie ici: Soit p un entier vCrifiant:

(33)

M.Cochand, P. Duchef

200

Nous allons construire un ensemble S de p points de W" qui n'admettra aucune partition de Radon. Posons q = [p/2J, q' = r p / 2 ] + 1 et m = [(3/21; on a m s n. D'aprh un thCorkme de Baranyai [l], on peut partitionner I'ensembles des parties B q ClCments de {I,. . . , p } en m classes d'ensembles disjoints. Ecrivons donc les parties A q ClCments de { l , . . . , p } sous la forme: Ai, Bi ( i = 1,. . .,rn)

avec A, n B,= 0 (B, n'est dCfini que si (3 est pair). La construction s'appuie fortement sur le lemme suivant qui est une consCquence facile du thCorCme de Konig-Hall. (Voir par ex. [2], ThCorCme 7.5, Corollaire 2.)

Lemme. I1 existe pour k s q une injection cp de l'ensemble des parties a k e'le'ments de (1,. . . , p } duns l'ensemble des parties (35) k + 1 tltments telle que cp ( X ) 3 Xpour toute partie X a k e'le'ments. Dtfinissons les Cltments a f de {1,. . . , p } pour i = 1,. . . ,m et j = 1,. . . , p 2 partir des ensembles Ai et Bi par les rkgles de rCcurrence suivantes: aP est l'unique e'ltment de Ai \ A si A, = cp(A) sinon a9 est un

tle'ment quelconque de Ai.

(36)

Pour 1s j G q, a { est l'unique iltment de Ai \{afi}k=,+l. . . q \ Asi Ai \{af)r=j+l.....q= cp(A) sinon a { est un iltment quelconque de (37) Ai \ { a f}k

=,+I

,....q.

a:' est l'unique e'le'ment de Bi \B si Bi = cp(B) sinon a:' est un (38) tle'ment quelconque de B,.

Pour q ' s j s p , a : est l'unique e'le'ment de Bi \ { a ~ } k = 4 . . . . . . , si -I\B (39) Bi \{aflk=q......,-l = cp(B). Pour p impair a : + I est l'unique tle'ment de { 1, . . . ,p } \ Ai \ B, .

(40)

Les a ! sont alors tous distincts par construction (pour i fixC) et on a: Ai = { a : ,. . . ,a?},

Bi = { a ? , . . . , a : } .

(41)

Les coordonnCes d'indice i de p points de W" appelCs x (l),. . . ,x (p) sont alors dtfinies de manicre 2 ranger les a ! par ordre croissant:

i ( x ( a : ) ) =k

pouri = 1,..., m et k = 1 ,

..., p .

Les n - rn derni5res coordonnCes sont choisies quelconques.

(42)

Sous les pavb , . .

20 1

L'ensernble S = {x (l), . . .,x(p)} obtenu n'admet aucune partition de Radon. Raisonnons par I'absurde en supposant que A, S \ A est une telle partition; autrement dit pour i = 1,. . ,,n on a: ( i ( ~ ) )n* ( i ( s \ A ) ) , + 0 .

(43)

On peut supposer sans perte en gknkralitk que r = / A I < 4- I1 existe alors d'aprks le Lemme (35) et la construction des a{, un indice i tel que: A = {~(a:), . . . ,x(ai)} ou A ={x(aP-'"), ..., x(aP)}. On a alors i ( A ) = { l , ..., r } avec i ( S \ A ) = { r + l , ...,p} ou { p - r + 1 , ..., p} avec i ( S \ A ) = { l , ...,p - 2 ) . Dans les deux cas, contradiction avec (43). 0

i(A)=

4.9. Preuve d u 7'hkorime 3.7 (kchange) Le lecteur vkrifiera sans peine le lemrne:

Lemme (Sierksrna [181). Pour route convexirk,

e ( U ) s c ( U ) + 1.

(44)

L'CgalitC e(%'")= n + 1 rCsulte alors du fait que I'ensemble des n points (8ij)i=l,...,m, 1 S j S n, et le point 0 ne vkrifient pas la propriktk d'kchange.

Bibliographic [l] Zs. Baranyai, On the factorization of the complete uniform hypergraph, in: Infinite and Finite Sets (Kestbely, 1973), J. Bolyai Mat. Tdrsulat (Budapest & North-Holland, Amsterdam, 1975) pp. 91-108. [2] C. Berge, Graphs and Hypergraphs (North-Holland, Amsterdam, 2de td., New York, 1976). [3] J.R. Calder, Some elementary properties of interval convexities, J. London Math. SOC.2 (1971) 422428. (41 C. CarathCodory, Uber den Variabilitatsbereich der Keoffizienten von potenzreiken, die gegebene werke nicht annehmen, Math. Annalen 64 (1907) 95-115. [S] P. Duchet and H. Meyniel, Convex sets in graphs 11, VIth Inter. Hungarian Col. on Combinatorics, Eger (1981) preprint. [6] J. Eckhoff, Der Satz von Radon in konvexen produktstrukturen I, Monatshefte fur Math. 72 (1968) 303-314. [7] J. Eckhoff, Der Satz von Radon in konvexen produktstrukturen 11, Monatshefte fur Math. 73 (1969) 17-30. IS] P. Erdos and G . Szekeres, A combinatorial problem in geometry, Compositio Math. 2 (1935) 463470. (91 P.C. Hammer, Extended topology, domain finiteness, Indag. Math. 25 (1963) 200-212. [ 101 E. Helly, Uber Mengen konvexen korper mit gemeinschaftlichen punkten, Jber. Deutsch Math. Verein 32 (1923) 175-176.

202

M.Cochand, P.Duchet

(111 R.E. Jamison, A general theory of convexity, Doct. Diss., University of Washington, Seattle, Washington, 1974. [12] D.C. Kay and E.W. Womble, Axiomatic convexity theory and relationships between the Carathtodory, Helly and Radon numbers, Pac. J. Math. 38 (1971) 471-485. [13] F.W. Levi, On Helly’s theorem and the axioms of convexity, J. Indian Math. SOC.15 (1951) 65-76. [14) T.S. Motzkin, Linear inequalities, Mimeographed Lecture Notes, Univ. of California, Los Angeles, California (1951). [15] H.M. Mulder, The interval function of a graph, Math. Center Tracts 132, Mat. Centrum, Amsterdam (1980). [16] J. Radon, Mengen convexer Korper, die einen gemeinsamen punkt enthalten, Math. Ann. 83 (1921) 113-115. [17] J.R. Reay, Generalization of a theorem of Carathiodory, Amer. Math. SOC.Memoir 54 (1965). [18] G. Sierksma, Axiomatic convexity theory and the convex product space, Doct. Diss., Rijksuniversiteit Groningen (1976). [19] G. Sierksma, The minimal domain finite extension of a convexity space, Internat. Publ., Econometric Inst., Univ. of Groningen (1974). [20] G. Sierksma, Carathiodory and Helly numbers of convex product structures, Pac. J. Math. 61 (1975) 275-282.


ROTATION NUMBERS FOR UNIONS OF TRIANGLES AND CIRCUITS E.J. COCKAYNE Department of Mathematics, University of Victoria, Victoria, British Columbia, Canada V8W 2Y2

FANG Zu Yao Department of Mathematics, Shandong University, Jinan, Shandong, The People's Republic of China Let G be a simple undirected graph which has p vertices and is rooted at x. Informally, the rotation number h(G,x ) of this rooted graph is the minimum number of edges in a p vertex graph H such that for each vertex v of H, there exists a copy of G in H with the root x at u. In this paper we calculate rotation numbers for graphs which are the disjoint union of a triangle and a circuit C,.

1. Introduction A rooted graph is a pair ( G , x )where G is a simple undirected graph and x E V ( G ) .Let ( G , x ) and ( F , y ) be rooted graphs. We say, informally, that ( G , x )is a rooted subgraph of ( F , y ) if there is a copy of G in F with x at y. Formally (G,x ) is a rooted subgraph of (F, y ) if and only if there exists a one-to-one function f : V ( G ) + V ( F ) satisfying

(9 [a,bl E E [ G l I$ [ f ( a ) , f ( b )El E ( F ) and (ii) f ( x ) = y. We denote this property by (G,x ) < (F, y). If (G,x ) < (F, y ) for each y E V ( F )we say that (G, x ) is a homogeneous rooted subgraph of F and write (G, x ) < F. The rotation number h(G,x ) of the p vertex rooted graph (G,x ) is defined to be the smallest number of edges in a p vertex graph F such that (G,x ) iF. A graph F achieving this minimum is called an extremal (G,x) graph. In Fig. l(a) we depict a rooted graph (G,x ) and graph F such that (G,x ) < F. Fig. l(b) shows the copies of G, with the root x at each vertex of F. It is easy to verify that h(G,x ) = 5 . 203

204

E.J. Cockayne,Fang Zu Yao

(h)

Fig. 1. An example of (G,x ) < E

Some rotation numbers for complete bipartite graphs have recently been evaluated (see [l] and [ 2 ] ) .In this paper, we calculate this number for all graphs C3U C,,, ( m > 3) where the root x is a vertex of C,.

2. Results

Throughout the paper d F ( u )will denote the degree of the vertex u in the graph F, q ( F ) will denote the number of edges of F, G will denote the graph C3U C,,, ( m> 3) and x will be a vertex of the C3component of G. Suppose that the graph F has m + 3 vertices. x ) < F, then q ( F )2 :(rn Theorem 1. If (G,

+ 3).

Proof. Since (G, x ) < F, every vertex u of F is in a triangle with vertex set T, such that F - To (i.e., the subgraph of F obtained by deleting T,) is hamiltonian. We will use the notation T, for both the triangle and its vertex set. The context will settle the ambiguity. Let .T = {To u E V ( F ) } . Let T, ={a, b , c } be a triangle of .T having no vertex in common with any other triangle of .T. Then the degree sum of T, is at least nine. For otherwise say d F ( a )= 2 and if d is a fourth vertex, since F - Td is hamiltonian and rn z= 4, it follows that d F ( b )and d F ( c )are at least three. Let bz be a third edge incident with b. Since T, f l T, = 0, F- T, is hamiltonian and d F ( a ) = 2 , we have d F ( b ) 2 4 .Hence the degree sum is nine as asserted. It is now sufficient to show that a vertex u satisfying T,, n To# 0 for some other triangle T, E F, has degree at least three. Firstly suppose T. n To = {x}. If u = x, clearly d F ( u )2 4 . Otherwise since F - T, is hamiltonian, d F ( u )2 3.

I

Rotation numbers for unions of triangles and circuits

205

Secondly let T. fl T, = { x , y } . If u E {x, y}, clearly d F ( u )3 3. Otherwise, since F - T, is hamiltonian, dF(u)2 4. This completes the proof of Theorem 1. 0

Theorem 2. If m = 3n - 3 (n 2 4) then

Proof. From Theorem 1, we have h(G, x) 2 L(9n + 11/21. It remains to exhibit a 3n-vertex graph F with L(9n + 11/21 edges such that (G,x ) < F. The required graphs are depicted in Fig. 2(a) for n even and Fig. 2(b) for n odd.

ra I

(a)

n even

(b)

n odd

Fig. 2. Extremal graphs F for C, U C3n-3.

Let F* be the graph obtained by contracting each triangle of F to a single vertex. If the deletion of any vertex from F* leaves a hamiltonian graph, then the removal of any triangle from F leaves a hamiltonian graph, i.e., (G,x ) < F. The contracted graphs F* are shown in Fig. 3(a) for n even and Fig. 3(b) for n odd. Verification of the required hamiltonian property of F* is left to the reader.

Theorem 3. If m = 3n - 2 (n

2 2)

then

Proof. We use the notation of the previous proofs. In this case I Y 1 2 n + 1 and it is easy to show either (Case 1) there are two triangles of 9which intersect in two

E.J. Cockayne, Fang Zu Yao

206

(a)

n even

(b)

n odd

Fig. 3. Contracted extremal graphs F' for C, U C3"..,.

vertices or (Case 2) there are at least three triangles of 9-which intersect other triangles of 9 in precisely one vertex. Arguments used in the proof of Theorem 1 establish that in Case 1 the intersecting triangles have degree sum at least 14 and the average degree of the remaining 3n - 3 is at least 3 . Hence

For Case 2 the proof of Theorem 1 shows that there are seven vertices of the intersecting triangles of Y whose degree sum is at least 23, while the remaining 3n - 6 vertices have average degree at least three. Hence

2

d F ( x ) 3 3 ( 3 n- 6 ) + 2 3 = 9 n + 5 .

xEV(F)

The result h ( G , x ) sL(9n +6)/2] now follows by parity considerations.

+

Graphs with (G, x ) < F having L(9n 6)/2]edges are depicted in Fig. 4(a) for n even and at least 6 , Fig. 4(b) for n odd and Fig. 4(c) and 4(d) for n = 2 and 4. Verification of the hamiltonian property is omitted. As in the proof of Theorem 2, this task may be simplified by contraction to a single vertex of each triangle which does not intersect any other triangle. 0 Theorem 4. If m = 3n - 1 ( n 3 2 ) then

Rotation numbers for unions of triangles and circuits

*

I

\

-

(h)

(c)

207

11=

2

II

(d)

odd

n=4

Fig. 4. Extremal graphs for C, U C3n-2.

Proof. It is easy to show that there are two triangles of Y intersecting either in one vertex or in two vertices, while all remaining vertices have average degree at least three. In the former case the degree sum of the five vertices of the intersecting triangles is at least 16 and

In the latter case the degree sum of the four vertices of the intersecting triangles is at least 14 and

C

d F ( ~ ) 3 3 ( 3 n- 2 ) + 1 4 = 9 n +8.

xEV(F)

Hence h (G, x) 2 [(9n

+ 8)/2]. 17

E.J. Cockayne, Fang Zu Yao

208

(a)

n even

(b)

n odd

Fig. 5. Extremal graphs for C, U C?".,.

The graphs achieving this lower bound are depicted in Fig. 5(a) for n even and Fig. 5(b) for n odd. The following results complete the evaluation of the numbers h (C, U Cm,x ) for m >3.

Theorem 5. (a) h(C, U C5,x ) = 14. (b) h(C3U C6, x ) = 15. Proof. (a) Let G = C3U C5and (G, x ) < F where F has 8 vertices. Clearly 1 91 5 3 and there is at least one vertex of F in two triangles of Y. A simple case analysis using arguments similar to those used in the proof of Theorem 1 shows that if more than one vertex of F is in more than one triangle of 9, F has at least three vertices of degree 4. Therefore the degree sum is at least 28. If there is exactly one vertex in two triangles of F, then 9 consists of these two triangles and a third disjoint triangle. It is easy to verify that five additional edges are necessary to complete F with the required hamiltonian property. Hence the rotation number is 14. The extremal graph is drawn in Fig. 6(a).

(a)

m=5

(b)

Fig. 6. Extremal graphs for m = 5 and 6.

in= 6

Rorarion numbers for unions of triangles and circuits

209

(b) If G = C,U C,and (G,x ) < F where F has 9 vertices, one may show that if .T contains intersecting triangles then there are at least three vertices of degree four, hence the degree sum is at least 30. If 3 consists of three disjoint triangles, six additional edges are necessary for the hamiltonian property. Thus the rotation number is 15. The extremal graph is drawn in Fig. 6(b).

Acknowledgements This work was completed while the second author was a Visiting Scholar at the University of Victoria. The support of the Government of the People’s Republic of China, the Canadian National Research Council and the B.C. Science Council is gratefully acknowledged.

References [ I ] B. Boll6bis and E.J. Cockayne, More rotation numbers for complete bipartite graphs (submitted). [2] E.J. Cockayne and P.J. Lorimer, On rotation numbers for complete bipartite graphs, Proc. of 10th Southeastern Conf. on Combinatorics, Graphs and Computing (Utilitas Math., Winnipeg, 1980) to appear.


Annals of Discrete Mathematics 17 (1983) 21 1-217 0North-Holland Publishing Company

ON CLIQUES AND PARTITIONS IN HAMMING SPACES Gerard COHEN ENST, 46 rue Barrault, 75013 Paris, France

Peter FRANKL CNRS. Paris, France Soit H, = (Z/qZ)”, I’ensemble des n-uples d’entiers modulo q, muni de la mttrique de Hamming H: pour x = ( x , ) et y = (yq) dans Hq, H(r, y ) = ({i,xi# y,}]. Une R-clique S, R C 11.2,. . . ,n } ,est un sous-ensemble de H, tel que pour x et y quelconques dans S, H ( x , y ) prend ses valeurs dans R. Soit m ( n , R ) la cardinalit6 maximale d’une telle clique. Pour E C { 1 , 2 , ..., n},uneE-sphitredecentrex e s t E ( x ) = { y E H , , H ( x , y ) E E } . O n a I E ( x ) l = (q - 1)’(;). Un E-code C est un sous-ensemble de H, pour lequel les E-sphires centrtes sur ses 6liments sont d’intersection vide deux i deux. C est parfait si de plus ces E-spheres partitionnent H,.Quand E = {0,1,. . . ,e } , nous parlerons de e-sphere et de e-code. Nous montrons dans cet article 1es rCsultats suivants.

c,,,

Proposition 1. Pour q # 2 les seuls E-codes parfaits possibles sont les e-codes.

Proposition 2. m(n, R ) S inf(C.jR=b(q- 1 ~ ( : ) , qm(n - 1, R ) , ( l / q ) m ( n+ 1, R u R R+l={r+l, rER}. Proposition 3. Pour q = 2, R = {0,2,4,. . . ,2e} et n E-sphire, auec E = { e , e - 2 , e - 4 ,.... e - - 2 [ e / 2 ] } .

3 2e

+ 2,

+ 1))

oli

m(n, R ) est re‘alise‘ par une

Proposition 4. Pour q # 2, R = (0.1,. . . ,2e}, n assez ,grand, m(n, R ) est rialise‘ par une e-sphire.

1. Introduction We consider Hq = ( Z / q Z ) ” , the set of n-tuples of integers mod q, endowed with the Hamming metric H , defined for two elements x = ( x i ) , y = (yi) in H4 as the number of their different coordinates: H ( x , y) = I{i,xi# yi}l. Denote by 0 the all zero element and define the weight of x by w (x) = H ( x , 0). An R-clique S is a subset of Hq s.t. for any x and y in S, H ( x , y) E R C {1,2,. . . ,n}. The maximal cardinality of an R-clique is denoted by m ( n , R). For E C {1,2,. . . , n } , an E-sphere of center x is E ( x ) = {y E H,, H(x, y ) E E } ; obviously, IE(x)I = CICE(q - l)i(?). An E-code C is a subset of H, s.t. for any c I and cz in C, E(cl)n E ( c 2 )= 0. C is perfect if, moreover, { E ( c ) ,c E C} is a partition of H,, which implies 1 E ( x ) / .1 Cl = 9”.In the classical case when E = {0,1,. . . ,e } , we 211

212

G. Cohen. P. Frank1

shall speak of e-spheres and e-codes, which are R -cliques for R = {0,1,. . . ,2e} and R = {2e + 1,2e + 2,. . . , n}, respectively.

2. Inexistence results for E-perfect codes

Proposition 1. For q # 2, the only possible E-perfect codes are e-codes.

The proof will follow from two easy lemmas. Lemma 1. There is no triangle in H, with side lengths e l , e,, H(x, y), with e l , e2 in E and x, y in C. Proof. By translation and permutation of components, this would yield a triangle with vertices c I , c2, z , with cl, c2 in C and z E E ( c l ) n E(c2). Lemma 2. In H, there exists a triangle with sides a, b, c iff the triangle inequalities are satisfied. Proof of Proposition 1. First note that there is no e in E with e > Ln/2J, because for any c I , c, in Hqr H ( c l , c,) S n. So there would exist a triangle with sides e, e, H(c,, c2), violating Lemma 1 - hence E C {0,1,... , Ln/2J}. Suppose (which is always possible by translation) that 0 E C, and E is not an interval {0,1,. . . , e}. Let x be such that w(x) = min i, i e E, and m = max i, i E E. Then x e E(O), and there is a c in C for which x E E(c). Now w ( c ) S m + w ( x ) < 2 m , so there is a triangle with sides m, m, w(c), which is impossible. Remark. The case q = 2, which is more complicated because only a weaker form of Lemma 2 holds, is settled in [2]. If E C [0, Ln/2J], binary E-perfect codes, apart from e-codes, exist iff n = 2", E = {l},or n = 24, E = {1,3},or n = 2e + 2, E ={e,e - 2 , e - 4 ,..., e -2[e/2J}.

3. Bounds for R-cliques

Let us give now upper bounds on m(n, R). Proposition 2. m(n, R)Sinf(A1,A2,A3,A4),where A l = z!?:,(q - 1)'(;), A, = qm(n - 1, R), As = (l/q)m(n + 1, R U { r + 1, r E R}), A 4 = (n, R'), R C R'. Proof. m (n, R ) 6 A I is given by Delsarte [3]. Suppose S is a maximal R -clique.

213

Cliques and partitions in Hamming spaces

Let Si = {x E S, x, = i}. The {Si}, i = 1 , 2 , . . . ,q partition S, so one of them, say So, has cardinality greater than ( 1 l q ) m (n,R ) . Deleting the last component of the elements in So we get SA which is an R-clique in length n - 1, hence m ( n , R ) < q 1 Sol= q I SAl S qm ( n - 1, R ) = A Z .The proof of rn (n,R ) S A 3 is similar. Start from a maximal R-clique S and add an extra component in all possible ways. Then n + n + l , R + R U { r + l , r E R } and I S l + q l S I . r n ( n , R ) S A 4is trivial, We investigate now the following problem. For which E, R , is an E-sphere a maximal R -clique? 3.1. The binary case

In the binary case (q = 2), it is known that for R = {0,1,.. . , 2 e + E } , E = 0, 1, m ( n , R ) is realized by an e-sphere for E = 0 and a quasi ( e + 1)-sphere (cf. Definition 1) for E = 1 (Katona [7]). We now show the following. Proposition 3. For q = 2 , R = { 0 , 2 , 4 , .. . , 2 e } and n 2 2e + 2 , m ( n , R ) is attained by an €-sphere, with E = {e, e - 2, e - 4,. . . , e - 2 [ e / 2 ] } .But for n = 2e or 2e + 1 , m ( n , R ) = 2 " - ' . Proof. Define U,, : H,+ H, by

u,,(XI, xz,. . . , X I , . . . ,x,, . . . ,x,)

= (XhX2,.

. . ,o,.

if x,x, = 1 and U,(x) = x if x,xJ = 0, i.e. w(U,, (x)) also for i < j, V,J(XI,...,X,-r,oIX,+I,XJ

I = (XI,.

. .,o,. . . ,X")

= w(x)

or w(x)-2. Define

1 9 1 7 . . . ? x " )

. .,X8.-1,1, & + I , . . . ,X,-l,O,. . . ,X"),

and V, (x) = x in the other cases (when (1 - x,)x, = 0 or i > j ) . For a maximal R-clique S, set O , , ( S )to be 0, ( S ) = {O,,( v 1; u E s, ot,( u )

e S ) u { u ; u E s,0 ,

(0)E

s>,

where 0,)is U, or V,,. It is obvious that for R = { 0 , 2 , 4 , .. . , 2 e } , O,J( S ) is also an R -clique with of course I O,,( S ) l = IS 1 = m (n, R ) . Iterating for all i, j , one eventually gets an R-clique T stable by the O,J.Now we show that for n 2 2e + 2 , m (n,{ 0 , 2 , .. . , 2 e } )s m ( n - 1, { 0 , 2 , .. . , 2 e } )

+ r n ( n - 1 , { 0 , 2 ,..., 2 e - 2 ) ) . Let T be written T = Tn U TI as in the proof of Proposition 2 .

(1)

G. Cohen, P. Frank1

214

1 TAI = 1 To(s m ( n - 1 , { 0 , 2 , ... , 2 e } ) . We want to show that for any x', y ', in TI, H ( x ' , y ' ) S 2e - 2, or equivalently, for any x, y in T, with x, = y, = 1, H ( x , y ) < 2e. Suppose the contrary. Let x, y be in T , with H ( x , y ) = 2e. As n 5 2e + 2, there exists i, s.t. x, = y i . There are two cases: (i) x, = yi = 1. Then U,n(x) = z E T and H ( z , y ) = 2e + 2 which is impossible. (ii) xi = yi = 0. Then V,,(x) = z and we get also a contradiction, proving (1). Using (1) and

we see that if Proposition 3 holds for some no, it will hold for any n > no.

Lemma 3. Proposition 3 is true for no = 2e

+ 2.

Proof. For R = { 0 , 2 , .. . , 2 e } , it is clear that all elements in an R -clique T have weight of the same parity. Denote by ti the number of elements in T with weight i, and by f the complement of x. One has

w ( 2 )= no - w (x),

H(x, 2 ) = no> 2e,

so x E T j fe T, hence

Suppose e even and all weights in T even (always possible by translation). Then

But this bound is achieved by an E-sphere with E = {0,2,. . . ,e } . The case e odd is similar. Hence the lemma is proved, and also the proposition for n 3 2e + 2.

For n = 2e or 2e R -clique.

+ 1, take

all elements of even (or odd) weight for the

3.2. Quasi -spheres Definition 1. A quasi E-sphere of center x is a set Q E ( c ) such that 3i E E,

( E - { i ) ) ( x ) C Q E ( x )C E ( x ) . When E = {0,1,.

. . ,e } and

i = e, it will be called a quasi e-sphere Q e ( x ) .


215

The following extension of [8, 5.2.21 is easy to check.

A quasi E-perfect code will be a partition of H, by QE-spheres. Proposition 1 can be generalized to

Property 2. For q # 2, the only possible QE-perfect codes are Qe-codes. Example. A linear quasi-perfect code (n,K,2e Q ( e + 1)-perfect code.

+ 1) [9,

Chap. 1, $51 is a

By analogy with group theory, we now introduce the following. Definition 2. A t-stabilizator is a subset of H, constituted by elements having zeros in t given components. Of course a t-stabilizator is a {0,1,. . . , n - t}-clique of cardinality q"-'.

3.3. The general case

For any q, let us now prove the following. Proposition 4. ForqZ 2, R = {0,1,. . . ,2e + E } , E = 0,1, m (n, R ) is realized by a n - (2e + 8)-stabilizator for n small, by an e-sphere when E = 0 and n large enough, and by a quasi ( e + 1)-sphereof the following type when E = 1 and n large enough : the union of the e -sphere around 0 and the vectors of weight e + 1 with a '1' in a given position. Proof. The first part ( n small) is proved in [ 5 ] . Consider now the case E = 0, n large. Define, as in the proof of Proposition 3, for i, j, 1 C i C n, 1 S j s q - I , W,, H, H, by

-

WI,(XI,X2,.

*

. ,x,-l,j,x,+,,. . . ,x.)

= (XI,&,

. . . ,xl-l,o,x#+,,. . .

1

X")

and W, (x) = x if x, # j. Starting from a maximal R-clique with R = {0,1,.. . ,2e}, we apply all the W,, to get a stable maximal R-clique T. Now we show that every element in T has weight at most e. For suppose there is an x in T with weight e + 1. Call F ( x )the support of x, i.e., F ( x ) = { i , x , # O } , with I F ( x ) l = w ( x ) . For every V' in T, 1 F(u')\F(u)/S e - 1 . If not, for the element D" coinciding with u ' , except on F ( u ) f l F ( u ' ) where it has zero coordinates, one would have u" E T (obtained

G.Cohen,P. Frank1

216

from u' by applying some W i j ) and H(u, u") = I F(u) A F(u")l= IF ( o ) l + I F(u")I 2 2e + 1 ; which is impossible. Hence

is less than

2 (q

-ly

i =O

(I)

= O(n.>

which is the cardinality of an e-sphere, contradicting the maximality of T. Now let us turn to the case E = 1. It can be proved in the same way as above that w ( x ) S e + 1 for every x E T. Let us set

T* = { x E T, w ( x ) = e + l},

9 = { F ( x ) : xE T * } .

As I T / is maximal we have

On the other hand I T * 1 s 1 91 ( q - l ) e + l , i.e., 9 is a family of (e + 1)-subsets of {l,Z, ...,n} with F n F ' Z 0 for F, F I E 9 and I s 1 3 0 ( n e ) . By Hilton and Milner [ 6 ] we deduce 3i such that i E F for all F in 9.By symmetry we may assume i = n. Thus we deduce

which yields

From the proof it is clear that the only way to have equality is by the quasi-sphere. Proposition 4 settles a conjecture of [5] in the case n % 2e + E = n - 1. Namely that m (n, {0,1,. . . , 2 e + E } ) is obtained by a direct product of stabilizators and spheres. 3.4. An example

Propositions 2 , 3 and 4 can be combined to give bounds on m (n, R ). Delsarte's bound m ( n , R ) S A l is good when R is sparse, whereas m ( n , R ) S A , becomes


217

betterwhen R isdense(e.g., r m + M a x , ( r , , , - r , ) S 2 1 R I f o r R = { r l , r 2 ,..., m,} with rl < r2 < . . . < r,,,). Let us consider m (6, (0, 1 ,2,4}), q = 2.

rn (6, {0,1,2,4}) L rn (6, {0,2,4}) = 16 (Proposition 3). rn(6, {0,1,2,4}) s irn (7, {0,1,2,3,4,5}) = A,

(Proposition 2),

with A , = 22 (realized by a 03-sphere). Also

rn (6, {0,1,2,4}) s m (6, (0, 1,2,3,4}) = 22 (sphere of radius 2), so 16 6 rn (6, {0,1,2,4}) S 22.

Rosenberg notes that 16 is the exact value.

References C. Berge, Nombres de coloration de I’hypergraphe h-parti complet, Hypergraph Seminar, Colombus, Ohio, 1972 (Springer, Berlin, 1974) 13-20. PI G. Cohen and P. Frankl, On tilings of the binary vector space, Discrete Math. 31 (1980) 271-277. [31 P. Delsarte. Four fundamental parameters of a code and their combinatorial significance. Inform. Control 23 (1973) 407-438. [4] M. Deza and P. Frankl, Every large set of (0, + 1, - 1)-vectors forms a sunflower, Combin. (to appear). 151 P. Frankl and Z. Furedi, The Erdos-Ko-Rado theorem for integer sequences,,SIAMJ. Discrete Appl. Methods 1 (1980). [ h ] A.J.W. Hilton and E.C. Milner, Some intersection theorems for systems of finite sets, Quart. J . Math. Oxford Ser. (2) 18 (1967) 369-384. (71 G. Katona, Intersection theorems for systems of finite sets, Acta Math. Acad. Sci. Hungar. 15 (1964) 329-337. [8]J. van Lint, Coding Theory (Springer, Berlin, 1973) 201. 191 F. MacWilliams and N. Sloane, The Theory of Error-Correcting Codes (North-Holland, Amsterdam, 1977).



ORIENTED MATROIDS OF RANK THREE AND ARRANGEMENTS OF PSEUDOLINES Raul CORDOVIL Centro de Fisica da Matiria Condensada (lNlC'), A v . Prof. Gama Pinto 2, 1699 Lisboa codex, Portugal Centro de Matemurica da Univ. de Coimbra ( l N l C ) , Univ. de Coimbra, 3ooO Coimbra. Portugal Oriented matroids of rank 3 and arrangements of pseudolines in the sense of Griinbauni are equivalent structures. We present a new natural correspondence between them, which generalizes classical duality, and also several properties related to convexity (in particular the Hahn-Banach Theorem on oriented matroids of rank 3).

We suppose the reader familiar with the theories of oriented matroids [l], [ 5 ] , and [ 121, and arrangements of pseudolines (see [I 11 for an excellent survey of the subject up to 1971). Let A be a finite set of points in the Euclidian plane E', not all lying on a straight line. Let L be the set of lines joining any two points of the set A. Then A and L are, respectively, the points and the lines of a rank three matroid M ( A ) . The order of the field R determines a canonical orientation 0 of M ( A ) such that if X is a signed cocircuit of 0, then a and b, a, b E X,have the same sign if and only if the line generated by A - X intersects the line generated by a and b between a and b. This construction of oriented rank three matroids suggests the following generalization. Definition 1. A pseudoline in the Euclidian plane E 2 is the image of a straight f 9 13 2, and Y be, respectively, line, under a self-homeomorphism of E'. Let 9, finite sets of pseudolines and points of E'. We shall call ( 9 , Y )a configuration of pseudolines and points (or shortly a configurarion) if the following three conditions (a)-(c) hold: (a) for all p , 9 E 9'such that p # 9 there exists one and only one pseudoline P E 9 such that p , 9 E P ; (b) for all p E 9 there exists p , 9 E Y, p # 9, such that p , 9 E P ; (c) any two pseudolines of 9 have at most one meeting point and if they meet they must cross there. 219

220

R. Cordovil

Proposition 2. Let ( 9 , Y ) be a configuration of pseudolines and points. Let M = M ( 9 , Y ) be the rank three matroid on Y s u c h that three points are collinear if and only if there exists one pseudoline of 9 which contains the three points. Then M ( 9 , Y ) has a canonical structure of an (acyclic) oriented matroid. Proof. If X is a cocircuit of M ( 9 , Y ) there exists a pseudoline P, P E 9,such that X = {x : x E Y’, x$S P}. Let X and - X be the opposite signed cocircuits of support X such that for all a, b E X the sign of a and b is opposite if and only if the pseudoline P intersects between a and b the pseudoline generated by a and b. Then the signed cocircuits defined as above determine an orientation 0 of M ( 9 , Y ) .To prove this, it is enough to prove the signed elimination property for all pairs of modular signed cocircuits (see [13, Proposition 2.11). XEPI},X2= {x :x E 9, x e P,} be two modular cocircuits Let XI = {x : x E 9, of M ( 9 , Y ) . By definition of modular cocircuits the pseudolines P , and P2have one meeting point y and y belongs to 9. Let x E (Xt n X;)U (XI n X i ) and let P3 be the pseudoline of 9 such that x, y E P. Then if X3is an oriented cocircuit of 0 of support X3= {x : x E 9, XEP3} and a E ( X ; n X i ) , b E ( X ; f l X ; ) , the sign of a and b is opposite in X , and the signed elimination property is verified. 0 The following question is natural after the preceding proposition: Given an acyclic oriented rank three matroid does there exist a configuration of pseudolines and points (9,Y ) such that M = M ( 9 , Y)?

Theorem 3 [3]. If M is an acyclic oriented rank three matroid then there exists a configuration ( 9 , Y )such that M = M ( 9 , 9). The proof of this result is an application of Folkman and Lawrence’s Theorem [5] which establishes a connection between arrangements of pseudolines in the projective plane and oriented rank three matroids and the following characterization of these matroids:

Theorem 4 [3]. Let M be a simple rank three matroid. Then M is orientable if and only if there exists a total order of the points and a total order of the lines of M such a < b< c the three lines determined that for all non -collinear collinear three points by these points are in the lexicographical order ab < ac < bc or in the opposite order - be< ac 1 telle que K ( n ,3) c". Preuve. Soit x un mot sans c a r d sur I'alphabet {a, b, c} et soit k = Ix 1. On note x =xIxZ' * ' x k . Soit X = {y € { a , b, c, 6,b; f } k pour i = 1,. . . ,n, yi = xi ou f i } . L'ensemble X est donc constitub de mots sans carrC sur I'alphabet { a , b, c, 6,b; c'} et est de cardinal 2k.

1

Mots et morphismes sans carre'

239

On considkre alors les deux morphismes: h6:{a, b,c,ci, b;E}+-,{a, b,c,d}+, h4U) = abad,

h,(d) = UCad,

h6(b)= babd,

h,(i) = bcbd,

h6(c)= cacd,

h6(C)= cbcd.

h7:{a, b,c,d}++{a, b,c}', h7(a)= abcba cabc acb abc aba cb,

h 7 ( b )= abcba cbca bac bab cac bc, h,(c) = abcba cbca cba bca bac be, h,(d) = abcba cbca cba cab cac bc. En se servant du Corollaire 6, on vkrifie que h6 et h, sont sans carrk. La composition de ces morphismes est donc aussi sans carre, et en particulier injective. L'ensemble h70 h6(X) est constituC de mots sans carrt de longueur 80 k et de cardinal 2'. Ceci montre que si n est multiple de 80, K ( n , 3) 3 2"'*O. On obtient une minoration analogue lorsque n n'est pas multiple de 80. 0 La constante c obtenue par la preuve du thCor6me prickdent est approximativement 1.005; lorsque I'on calcule les mots sans carre jusqu'h la longueur 34, K ( u , 3 ) semble &tre voisin de (4/3)".

6. Caracterisation des morphismes sans carre Etant donne I'importance des morphismes sans c a d , il est interessant de pouvoir les caractCriser de fagon effective. Le premier resultat dans ce sens a CtC fourni par Berstel [5]. Pour un morphisme h de A' dans B' appelons kcart de h la quantitC: e ( h ) = m a x a E A(Ih(a)l)/min,,A (Ih(a)I). Le theoreme de Berstel s'knonce alors comme $a. Theorhme 8 [ 5 ] . Un morphisme h : A + - , B' est sans carre'ssi h(x) estsans carre' pour les mots x sans carre' et de longueur infkrieure u 2 + 2 re ( h ) ] (re (h)1 est le plus petit entier supkrieur i2 e(h)). Afin de prCciser ce rksultat, nous introduisons la notion de facteur rCpCtable pour un morphisme h : A + + B'.

240

M . Crochemore

Soit x un mot sans carre de A + et soit u E B + tel que h ( x ) = a u p pour a et p dans B * ; u est dit ripitable s’il existe x ’ dans A * tel que (i) x x ‘ soit sans carrC et u soit facteur gauche de p h ( x ’ ) ; ou (ii) x ’ x soit sans carrC et u soit facteur droit de h ( x ’ ) a . On peut alors Cnoncer le suivant. ThCoreme 9. U n morphisme h : A + + B’ est sans carre‘ ssi pour tout couple de lettres diffe‘rentes, a, b, de A, aucun facteur de h ( a b ) de longueur infkrieure a max(lh(a)), I h ( b ) l ) n’est re‘pitable. Preuve. On montre que si h n’est pas sans carrC il existe deux lettres a et b distinctes et un facteur rCpCtable dans h ( a b ) . La majoration sur la longueur de ce facteur s’obtient par un raisonnement analogue que nous ne reproduisons pas ici. Soit x de longueur minimale tel que h ( x ) = au’g avec a et y dans B *. On pose x = x t * * * x. ,. * x k avec h ( x t ) = a & , h ( x , ) = p p , h ( x , ) = y g , h ( x ) = a C u p p w y g et u = &up = pwy. La minimalit6 de x implique que (Y et y sont non vides. Quand on suppose que i > 2 (et symitriquement que i < k - 1) on peut alors donner un facteur rCpCtable dans I’image d’un mot de deux lettres. Dans le cas oij 1 C I # 1 fl I (ou 1 p I # I y I) on trouve un carrC dans I’image d’un mot de deux lettres distinctes. Supposons que I & / = l f l l et / p l = / - y l ; on a C = f l et p = y. S i x , est diffCrent ) a~pP;.jjqui contient le carri de x t et X k , xtx,xk est sans carri et h ( x I x , x k = ~p = Py. Si, par exempie, x , = x , , puisque x Iui-meme est sans carre, il existe j , 1 < j < i, tel que x l # x , + , - ~et que: de h ( x l ) et h ( x , + / - t I’un ) est facteur gauche de I’autre; ~ ( X ~ X , +ou/ - h~( )x , + l - , x , )contient donc un carrC. 0 Le thCortme prCcCdent amCliore la borne donnCe dans le thtorkme de Berstel. On peut encore ameliorer cette borne en donnant une condition assez proche de celle donnee par Thue, mais qui est nCcessaire et suffisante. ThCorhme 10. U n morphisme h :A -+ B’ est sans carre‘ ssi il satisfait les deux conditions : (i) h ( x ) est sans carre‘ pour tout x de A + sans carre‘ et de longueur 3 ; (ii) aucun h ( a ) ( a E A ) ne contient de facteur (interne) ripe‘tuble. +

La preuve se fait en utilisant le resultat du ThCorBme 9 et est plus technique que celle de ce thkorbme. On dCduit immidiatement une formulation moins algorithmique du ThCorkme 10.

Mots et rnorphismes sans carri

24 1

Corotlaire 11. Un morphisme h : A + + B' est sans carre' ssi h ( x ) est sans carre' pour tout x sans carre' de longueur infe'rieure a max{3,1+ ~(ma~.~~(~h(a)~}-3)/min.~A{Ih(a)J}l}.

Dks lors que I'alphabet ne possbde plus que 3 lettres, la borne que I'on peut donner est alors indtpendante des longueurs des h ( a ) pour a E A. Corollaire 12. Si [ A I = 3, un morphisme h : A + + B' est sans carre'ssi h ( x ) est sans carre' pour les mots x sans carre' de longueur 5 . Preuve. On suppose que I h ( a ) l a l h ( b ) l al h ( c ) l ; alors, pour des raisons de longueur, il est inutile d'exarniner les images des mots qui ne sont pas reprksentts dans les schtmas:

i i i

i t i b

a a

ce qui donne le risultat.

b

a c

i a

0

Nous donnons maintenant des exemples de morphismes qui ne sont pas sans carrk et pour lesquels il est ntcessaire d'examiner des mots de la longueur des bornes des corollaires 11 et 12 pour s'en apercevoir. Exernples. h, : { a , b, c } + + { a , 6, c, d , e } ' ,

hs(a)= deabcbda,

h,(b) = b,

Le facteur abcbd de h , ( a ) est ripitable car

h,(abcba) = d e p b c b r f 3 d e a b c b d a . a n } + + { # , a, , . . . , a n } + , h q : { a,..., , hu(al= ) # a i a 2 .. . anal pour i > 1, h,(ai) = a i .

h s ( c )= c.

M.Crochemore

242

Les images des mots de trois lettres sans carrC ne contiennent pas de carre. Le premier carrC est obtenu B I’aide de a l . . . a , qui a pour longueur 1 + “max{ I h ( a 1I) - 31/[min{I h ( a )1111 :

hu(ala2* a,;

=

-

# a l a 2 . . u n a l a 2. . a,.

7. Morphismes produisant des mots infinis sans carrC

Lorsque I’itiration d’un morphisme donne un mot infini sans carre, ce morphisme n’est pas nkcessairement sans carrC lui-meme. Le meilleur exemple de ce cas a CtC donnC par Istrail (191. Exemple. h t o: {a, b, c}++ {a,b, c}+,

hlo(a)= abc,

hlo(b)= uc,

h l o ( c )= b.

Ce morphisme n’est pas sans carrC car

hlo(aba)= a b m b c , et nianmoins le mot infini hYo(a) est sans c a d . Lorsque I’alphabet n’a que 3 lettres on peut decider, pour un morphisme h prolongeable en a, si h ” ( a ) est sans carrC. ThCoreme 13 [5]. Soit h un endomorphisme de {a, b, c}+ prolongeable en a. I1 existe un entier p effectivement calculable tel que : h ( a ) est suns carre‘ ssi h ( a ) est sans carre‘.

On considkre dans la suite des endomorphisme h, dit SCz, qui satisfont la propriiti: h ( a b ) est sans carrC pour chaque couple (a, b ) de lettres distinctes de A . Pour ces endormorphismes on obtient le mkme rCsultat de dCcidabilitC. ThCoreme 14. Soit h un endomorphisme SC, prolongeable en a de A. I1 existe un entier p effectivement calculable tel que : h ” ( a ) est sans carre‘ ssi h ( a ) est sans carre‘. Le thCorkme de Berstel en est une consiquence; en effet si I’alphabet n’a que trois lettres on peut virifier que tout mot sans carrt de longueur supCrieure B 14 contient comme facteur tous les mots ab avec a # b ; et donc pour que h ” ( a )soit sans carrC il est ntcessaire que h soit SC2. Le thCorkme se dkmontre B partir de deux lemmes dont les preuves sont analogues a celle du ThCorkme 9.

Mots et

morphismes sans carri

243

Lemme 15. Soit h un endomorphisme SC2.Soit x un mot sans carre' de A + tel que h ( x ) contienne un carre' et qu'aucun facteur propre de x ne satisfasse la mime proprie'te'. Alors, si Ix I > 2 + 2 re ( h ) ] on a : (1) 3a, b, c E A , 3 u E A + , a # b , b # c e t x = a u b u c ; ( 2 ) 3a, E € A * , 3p, Y E A ' , h(a)= cup, h ( b ) = y p , h ( c ) = y ~ . Lemme 16. Soit h un endomorphisme SC2.Soit x un mot sans curre' de A + tel que h ( x ) soif sans carre' et contienne comme fucteur dueuf ( d ,e, f E A et u E A *) et qu'aucune image d'un facteur propre de x ne contienne dueuf. A h , si I x I > 2 + 2 [ e ( h ) ] on a : ( 1 ) 3 a , b, c E A , 3 u E A + , a # b , b # c e t x = a u b u c ; (3) 3a, p, y, e € A * , h ( a ) = ad& h ( b ) = yep, h ( e ) = yfE. Preuve. Supposons que h " ( u ) contienne un carre; soit p le plus petit entier tel que h P ( a )contienne un carrC et soit w w un carrC de longueur minimale de h P ( a ) ;soit x le plus petit facteur de h P - l ( a )tel que h ( x ) contienne ww. En appliquant les deux lernmes prCcCdant on en dCduit une suite: (a,,b,, c,. v , ) , i = l , . . . ,k oh (al,bl,c,,u l ) satisfait (1) et (2) du Lemme 15, pour i > 1, (a,,b,,c,,u,)satisfait (1) et ( 3 ) du Lemrne 16, et a k v k b k v k c k est contenu d a m I'image de y de A + avec 1 y I S 2 + 2 [ e ( h ) ] ,et y facteur de h " ( a ) . Le point important dans la dCmonstration est le fait suivant: deux triplets ( a , ,b,, c , ) et (u,, b,, c,) ne peuvent Ctre Cgaux. En effet si (a,, b,, c,) = (a,, b,, c,) et i < j , on sait que a,u,b,u,c, est facteur de h P - ' ( u )et que son image par h' contient w w . D'aprks les propriCtCs de factorisation ( 2 ) et ( 3 ) h'(a,u,b,u,c,) contient un carre et donc h P - ' + ' ( acontient ) un carrC ce qui contredit la minimalit6 de p . Le second point a noter est que I'on peut calculer effectivement I'ensemble des facteurs de longueur infkrieure A 2 + 2 [ e ( h ) ] de h " ( u ) : comme I'ensemble des mots de longeur infCrieure Q 2 + 2 [ e ( h ) l est fini il existe un plus petit entier q tel que h q" ( a) a le mCme ensemble de mots de longueur infCrieur A 2 + 2 [ e ( h ) J que h q ( u ) .II est alors facile de voir que h q ( a ) contient tous les facteurs de longueur infkrieure A 2 + 2 [ e ( h ) ] de h " ( a ) . 0

8. Quelques questions non resolues

On a trait6 ici le cas des mots et rnorphismes sans carre, et obtenu la dCcidabilitC de certaines questions. On peut se demander si le mime travail peut 6tre reproduit pour d'autres motifs Cvitables. Par exemple, est-ce qu'on peut dCcider si un morphisme est sans cube, ou sans puissance k? Plus gCnCralernent pour un motif Cvitable w peut-on dCcider si un morphisme b i t e w ( h Cvite w ssi h ( x )Cvite w quand x lui-m&meCvite w ) ?

244

M . Crochemore

Pour les morphismes prolongeables, peut-on dicider s’ils engendrent un mot infini sans puissance k ou Cvitant un motif donnC?

RefCrences [I] S.I. Adjan, Burnside groups of odd exponents and irreducible systems of group identities, in: Boone and Cannonito Lyndon, ed., Word Problems (North-Holland, Amsterdam, 1973) pp. 19-38. [2] S. Argon, DPmonstration de I’existence de suites asymitriques infinies, Mat. Sb. 44 (1937) 769-777. (31 D. Bean, A. Ehrenfeucht and G. McNulty, Avoidable patterns in strings of symbols, Pacific J. Math. 85 (1979) 261-294. [4] J. Berstel, Sur la construction de mots sans carre, Skminaire de Thtorie des Nombres, Expose No 18 (1979). [5] J. Berstel, Sur les mots sans carre dkfinis par un morphisme, 6th ICALP Symp. Maurer, ed., Lecture Notes in Computer Science 71 (Springer, Berlin, 1979) pp. 16-25. 161 J. Berstel, Mots sans carri et morphismes iterts, Discrete Math. 29 (1980) 235-244. 171 J . Berstel et C. Reutenauer, Square free words and idempotents semigroups, in: Lothaire. ed., Combinatorics On Words (Addison-Wesley, Reading, MA) Chapter 2, to appear. [8]F.J. Brandenburg, On non-repetitive strings, prepublication. [9] C. Braunholtz, An infinite sequence of three symbols with no adjacent repeats, Amer. Math. Monthly 70 (1963) 675-676. [lo] J.L. Britton, The existence of infinite Burnside groups, in: W. W. Boone et al., eds., Word Problems (North-Holland, Amsterdam, 1973). [ I l l M. Crochemore, An optimal algorithm for computing repetitions of a word, Inform. Process. Lett. 12 (5) (1981) 244-250. [I21 F. Dejean, Sur un theorbme de Thue, J. Comb. Theory, Ser. A 13 (1972) 9@99. [I31 F. Dekking, On repetitions of blocks in binary sequences, J. Comb. Theory, Ser. A 20 (1976) 292-299. 1141 A. Ehrenfeucht and G. Rosenberg, On the separating power of EOL systems. R.A.I.R.O.. to appear. [15] R. Entringer, D. Jackson and Schatz, On non-repetitive sequences, J. Comb. Theory, Ser. A 16 (1974) 159-164. 1161 A.A. Evdokinov, Strongly asymmetric sequences generated by a finite number of symbols, Soviet Math. Dokl. 9 (1968) 536-539. 1171 W. Gottschalk and G . Hedlund, Topological dynamics, Amer. Math. SOC.Colloq. Publ. 36 (1 955). [I81 D. Hawkins and W. Mientka, On sequences which contain no repetition, Math. Student 24 (1956) 185-187. [ 191 S. Istrail, On irreducible languages and nonrational numbers, Bull. Math. Soc. Sci. Math. R.S. Roumanie 21 (1977) 301-308. 1201 J. Karhumaki, On cubic-free o-words generated by binary morphisms. prepublication (1981). [21] J. Leech, A problem of strings of beads, Math. Gaz. 41 (1957) 277-278. [22] M. Main and R. Lorentz, An O(nlog) algorithm for finding repetition in a string, CS-79-056, Washington State University Pullman, Washington, 1979. [23] M. Morse, A solution 3f the problem of infinite play in chess, Bull. Amer. Math. SOC.44 (1938) 632. [24] M. Morse and G . Hedlund, Unending chess, symbolic dynamics, and a problem in semigroups, Duke Math. J. 11 (1944) 1-7. [25] P.S. Novikov. On periodic groups, Dokl. Akad. Nauk SSSR 127 (1959) 749-752.

Mots et morphismes Sans carrt!

245

[2h] P.S. Novikov and S. I. Adjan, Infinite periodic groups, Math. USSR-Izv. 2 (1968) 209-36, 241-479, 665-685. [27] P.A. Pleasants, Non-repetitive sequences, Proc. Cambridge Phil. SOC.68 (1970) 267-274. [28] R. Ross and R. Winklmann, Repetitive strings are not context-free, CS-81-070, Washington State University Pullman, Washington, 1981. [29] A . Thue, Uber unendliche Zeichenreihen, Norske Vid. Selsk Skr., I. Mat. Nat. KI., Christiana 7 (1906) 1-22. [30] A. Thue, Uber die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Norske Vid. Selsk. Skr.. I. Mat. Nat. KI., Christiana 1 (1912) 1-67. I311 T. Zech, Wiederholungsfreie Folgen, Z. Angew. Math. Mech. 38 (1958) 206-209.



STRONGLY qTH POWER-FREE STRINGS L.J. CUMMINGS Universify of Waterloo, Waferloo, Ontario, Canada, N2L 3Gl It is shown that searching a string over a finite alphabet for ‘powers’ of a substring is made easier by searching for overlapping substrings instead. A proof is given that a string without overlapping substrings can always be extended with alphabet symbol provided only that it does not terminate in the square of a substring.

At the Second International Conference on Combinatorial Mathematics sponsored by the New York Academy of Sciences Read proposed the problem of enumerating all strings of length n over a finite alphabet which avoid repeated adjacent substrings [8]. The intimately related question of the existence of such strings and their construction has appeared in surprisingly diverse areas of mathematics. The Norwegian number theorist Thue wrote three fundamental papers on this topic [9], [lo], and [ l l ] early this century but his results were not widely circulated and have been frequently rediscovered. Thue’s first paper showed constructively the existence of an infinite string on 3 symbols which was ‘square-free’ [ll]. We remark that the construction of a square-free string a l . . a, over a finite alphabet 2 is equivalent to finding a colouring of M = (1,. . . ,m } with 2 colours such that not only will adjacent elements have different colours, but adjacent intervals will have distinct patterns of colours. This notion has been extended to obtain colourings of the integer lattice points in the Euclidean plane with the property that no two rectangles with a common edge will have the same colour pattern [l]. Indeed, it can be generalized to square-free and cube-free colourings of the class of all ordinals [8]. We consider only finite strings of the form

s = uoalu2.

* *

,

where the terms a, are taken from a finite set 2,called the alphabet. The length of I S I is denoted by I S I. For consideration of strings defined by ordinals other than the order type of the positive integers see [l] and [7]. The usual term ‘sequence’ is avoided here because we are interested in substrings of S which have the form B = &a,+,* * a i + k - l , 0 c i, 1 k,

-

247

L.J. Cummings

248

making the natural term 'subsequence' inappropriate. The substrings a. 0 s j are the initial substrings of S. We say that S contains B 2if, for some i,

* *

a,,

j = 0, 1,. . . ,k - 1.

&+, = U i + k + j ,

When q copies of B are adjacent in S we say that S contains B4. If S does not contain B 2for any substring B then S is square-free. A slightly weaker condition [6] has proved useful: S is strongly cube-free if S contains no substring of the form B2al . a, where a l . * a, is an initial substring of B. More generally, S is strongly (4 + 1)th power-free if it contains no substring of the form

-

B q a l . a,, r < I B 1,

(1)

with a l - - .a, a nonempty initial substring of B. Any strongly qth power-free string is (q + 1)st power-free. Fife [6] has given a method for generating all infinite strongly cube-free strings when r = 1. If A = a l * a, and B = bl * * b, are substrings of S then A overlaps B in k positions if

-

bl

= Un-k+l

'

* '

bk = a,.

If A = B then A is a self-overlapping substring which we call an (n,k)-substring. Theorem 1 proves that an (n,k ) substring induces a substring of the form (1). Proofs of the following three theorems appear in [3].

Theorem 1. If S is any string ouer a finite alphabet which contains an (n,k ) substring A = a l * * a, (1 < n ) then S contains the substring

B q a l . .* a,, where B = a ] ..

an-k

and

Here [XIdenotes the least integer strictly greater than x and 1x1 is the greatest integer less than or equal to x.

Corollary 1. Let S be a string ouer a finite alphabet containing an (n,k ) substring A = a l * a,. Then, (i) k = 1 implies S contains (al - U , - , ) ~ U ~ , (ii) k = n - 1 implies S contains a 7 +'. More generally than (ii) we have the following.

Corollary 2. Let S be a string ouer a finite alphabet containing an (n,k ) substring A = a l - * . a, then r = 0 if and only if n - k n.

1

Strongly qth power-free strings

249

Theorem 2. If a string S over a finite alphabet contains a substring P = B qa,. . . a, with 1 < 1 B 1, where a l * * a, is initial in B and 1 < q, then either (i) O < r, or (ii) O = r and 2 < q implies there is an (n, k ) substring inducing P. Theorem 3. If q = 2, then a string B 2 over a finite alphabet is induced by a self-overlapping substring if and only if B is periodic. Remark. If B 2 is induced by an ( n , k ) substring then B' is a*" for some a E C until n = 6 and k = 4. Theorem 3 shows that a square cannot in general be detected by searching for self-overlapping substrings. However, any strongly cube-free string can be detected in this way. If q = 2, then from Theorem 1 we see that p s n < 2 p where p = n - k. It follows that 2k < n. Accordingly, if we wish to search for substrings of the form (1) with q = 2 (and in light of Theorem 3, 0 < r ) we need only search for (n, k ) substrings with k s Ln/2J. More generally, for any q it follows easily that

Unfortunately, this is not immediately useful since we cannot assume, a priori, knowledge o f n, the length of the overlapping substring. If a string of length n is to be searched for a substring of the form (1) and only q is assumed to be given, the following theorem gives bounds on n and k for possible (n,k ) substrings.

Theorem 4. If a string of length m contains an (n,k ) substring then l U ¶ A c n Sr + m ( q - 1 ) 4 4

7

(2)

Proof. As in Theorem 1 , the occurrence of an (n,k ) substring in a string S of length m induces a substring

B4a1. - a,, 1 < q, 0 < r < n

- k,

-

where B = a , * a n - k .Since a string of length n overlapping itself in k positions creates a string of length 2n - k we have

2n - k = q ( n - k ) + r.

(4)

L..I Cummings .

250

Rewriting (4) we obtain (2 - q ) n

+ (q - l)k

= r.

Viewing (5) as a Diophantine equation in n and k, the general solution is given by the equations

n=r+a(q-l),

k

=T

+

(Y

(4 - 2),

I

ff

= 0 , + 1 , ... .

Now 1 S k < n implies a# 0. Since our string has length rn we have 3 s 2n - k S rn which implies

Using (6) we obtain (2) and similarly, 2n - rn

S

k G 2 n implies (3).

Suppose we wish to search an arbitrary string S = al * * a, for the occurrence of a pattern (1) where q is kept fixed. If (1) is induced by an (n,k)-substring of S then necessarily the affected portion of S has length t = 2 n - k = qb

+r,

(7)

where b denotes I B 1. The proof of Theorem 1 implies n - k = b. Accordingly (7) yields

n

= (q - 1)b

+ r.

(8)

Since k = n - b we further see that

k = (q - 2)b + r.

(9)

First, suppose we search the string S of length rn directly for the occurrence of (1). We must therefore test the first rn - t + 1 symbols of S as possible starting symbols of (1). For each successive choice of the starting symbol we can make the r(q + 1) comparisons &+j+w

= &+&+j,

j = 1,. . .,r ; w

=o,. ..,q,

(10)

-

involving the ‘remainder’ & + @ + I * * a+@+, first. The remaining (b - r)q comparisons necessary in the worst case are Ui+j+k

= Ui+wb+j,

i = r -k 1 , . . .,b, W = 0,. . ., q - 1.

Fig. 1.

(1 1)

Strongly qth power-free strings

25 1

The number of comparisons involved in (10) and ( 1 1 ) is therefore

(m-t

+ l ) [ r ( q+ 1 ) +

( b - r ) q ] = ( m - t + 1)t.

Now suppose we search S for the occurrence of an ( n - k)-substring where n and k have been determined by equations (2) and (3). Accordingly we consider the first m - t 1 symbols of S as possible starting symbols of the (n, k)substring and make the comparisons

+

a.,+, . = a ., + m - k + j ,

j = 1 ,..., m,

in each case. Now the total number of comparisons is just

(m-t

+ l)n,

and we see that there are ( n - t + l ) b fewer comparisons required in searching for an (n, k)-substring than in searching directly for ( 1 ) . Dekking [ 6 ] has shown that any infinite string over ( 0 , l ) without overlapping substrings is progressively repetitive; i.e., contains squares B Zfor arbitrarily long substrings B. In Theorem 5 it is shown that strongly qth power-free strings are easily constructed without considering the algebraic machinery of morphisms. Berstel [2] has shown that it is decidable whether a string of arbitrary length constructed by iteration of a morphism is square-free. Theorem 5 is further evidence of the limitations of constructive techniques which use morphisms. Theorem 5. Any string without self-overlapping substrings which does not terminate in a square can be extended by an arbitrary alphabet symbol to a string with the same property. Proof. Let S be such a string, but suppose that Sx has an ( n , k ) substring a , . . a. for some x E 2. By Theorem 1 , Sx contains a substring of the form ( 1 ) . If 1 < k it follows that a l * a,-l would be an ( n - 1 , k - 1 ) self-overlapping substring of S, contrary to assumption. But when k = 1 we have a, = a t and

-

S=*

*

+

al

* *

a,-lal * . * an-i,

contradicting the assumption that S did not terminate in a square. References [ I ] D.R. Bean, A. Ehrenfeucht and G.F. McNulty, Avoidable patterns in string of symbols. Pacific J. Math. 85 (1979) 261-294. [2] J. Berstel, Sur les mots sans carre definis par un morphisme, Proc. of the 6th Internat. Coll. on Automata, Languages and Programming, Lecture Notes in Computer Science, No. 71 (Springer, Berlin, 1979) pp. 16-29.

252

L.J. Cummings

[3] L.J. Cummings, Overlapping substrings and Thue’s problem, Proc. of the 3rd Caribbean Conf. in Combinatorics and Computing (University of the West Indies, Cave Hill, 1981) pp. 99-109. [4] F.M. Dekking, On repetitions of blocks in binary sequences, J. Comb. Theory, Ser. A 20 (1976) 292-299. [ 5 ) F.M. Dekking, Strongly non-repetitive sequences and progression-free sets, J. Comh. Theory, Ser. A 27 (1979) 181-185. [6] E.D. Fife, Binary sequences which contain no BBb, Trans. Amer. Math. SOC.261 (1980) 115-136. [7] J. Larson, R. Laver and G. McNulty, Square-free and cube-free colorings of the ordinals, Pacific J. Math. 89 (1980) 137-141. [8] R.C. Read, Problem no. 32,Znd Internat. Conf. on Combinatorial Mathematics, Annals of the N.Y. Academy of Sciences, Vol. 319 (N.Y. Academy of Sciences, New York, 1979). [9] A. Thue, Uber unendliche Zeichenreihen, T. Nagell, I. Selberg, S. Selbert and K. Thalberg, eds., Selected Mathematical Papers of Axel Thue (Universitetsforlaget Oslo-Bergen-Troms, 1977) pp. 139-158. [lo] A. Thue, Uber du gegenseitige Lage gleischer Teile gewisser Zeichenreihen, ibid., pp. 413-477. [ 111 A. Thue, Probleme uber Veranderungen von Zeichenreihen nach gegebenen Regeln, ibid., pp. 493-524.


CYCLIC CODES OVER GF(4) AND GF(2) Gilbert DELCLOS Universiti de Toulon et du Vur, UER Sciences et Techniques, 83130 La Garde, France

GF(p') is a vector space of dimension two over GF(p). We study in this paper the consequences of this fact. Necessary and sufficient conditions are given for a code on GF(p') to be a p-code (generator matrix consisting of elements from GF(p)),unreal code (code over GF(p2). the only codeword consisting of elements from GF(p) is 0). We study C, Re(C) (the real set of C), and C,, (the subfield subcode of the codewords of C which have components from GF(p)). We show how to find the generators of Re(C) and C, when we know the generator of the cyclic code C over GF(p2). The idempotents and quadripotents, - q 2 ( x ) # q ( x ) and q 4 ( x ) = q ( x ) - of binary cyclic codes are studied. The existence of quadripotents in a binary cyclic code is a necessary and sufficient condition for a binary cyclic code to be the real set of a code over GF(4) which is not a 2-code, namely, a code, a generator matrix of which consists of 0's and 1's.

c,

1. Linear codes

1.1. Terminology

We take the elements of GF(4) to be 0, 1, a, p with p = a ' = a + 1, a ' = p' = 1. A linear code C over GF(q) of length n and dimension k consists of 4 k vectors u = (uI,],. . . ,u"), u, is an element of GF(q), called codewords such that if u, u E C then u + u E C and au E C (see [4]). GF(4) is a vector space of dimension two over GF(2); { 1 , a } is a basis. If x E GF(4), then x = x I + a x 2 , where x I , xz E GF(2), and if u E C - C is a linear code over GF(4) of length n, then u = a + ab with a, b E GF(2)". An overbar denotes conjugation, that is, if x E GF(p2) then X = x p . A similar notation is used for vectors, codes, etc. Thus U = (GI,. . . , a,), and =

c

{ n ;u E C } .

The dot product is defined by u * u = C : = l ~ , u rwith the sum evaluated in GF(q). The dual code C ' = { u = ( u I , ..., u n ) ; u . u = O for all u EC} is an {n, n - k } code. Let C be a linear code over GF(q) with q = p'. We define the real part of C as: Re(C) = { a E GF(p)" ; u = a + ab with u E C}; { 1 , a }is a basis of the vector space GF@') over GF(p). 253

G. Delclos

254

We define a subfield subcode Co= C n GF(p)", where C, is a linear code over GF(p); C is an unreal code if Co={O}. A linear code over GF(p2)is a p-code if and only if C = Z + a 2 with 2 a linear code over GF(p); we denote C = C ( 2 ) .

1.2. Basic results The proofs are easy (see [l]). Let C be a linear code over GF(pZ).

Theorem 1

c n C = c(co), C+ =

= c(Re(C)),

c'.

1.2.1. The characterizations of p -codes Theorem 2. The following properties are equivalent: C is a p-code; dim,] C = dim, R e ( C ) ;C, = Re(C); a basis of C i s in GF(p)"; C' is a p-code ; C,l = (C')";

c = c.

1.2.2. The characterizations of unreal codes Theorem 3. The following properties are equivalent: C is unreal ; dim, Re(C) = 2dimp2 C ; if g1 is a basis of C and gi = R ( g , ) +a l ( g , ) then R ( g , ) , I(g,), i = 1,. . . ,k are independenf vectors over GF(p); C r l = (0).

1.2.3. The dimensions Theorem 4 dim, CoS dimpzC S dim,, Re(C) s 2 dim,^ C ;

dim, Re(C) = 2 dim,> C - dim,2( C n

c).

2. Cyclic codes

2.1. Terminology Let C be a cyclic code over GF(p2);we study Re(C) and Co. It is easy to prove that i f g ( x ) is a polynomial of GF(p2)(,,and is not a

Cyclic codes over GF(4)and GF(2)

255

polynomial of GF@),,, then g ( x ) = r ( x ) *s(x)- eventually r ( x ) is an element of GF@) - with ~ ( X ) € G F ( ~ )s(x)SIGF(p)(,) (~), and the degree of r ( x ) is the greatest one. We then prove that x " - 1 = g ( x ) . h ( x )= r ( x ) . s ( x ) . J ( x ) . p ( x ) with J ( x ) = Sx if s(x) = sixi and p ( x ) is either an element of GF(p) or an element of GF(P)(=)(see PI).

xr=,

x:=o

2.2. The results Theorem 5. Re(C) is a cyclic code over GF(p). If the generator of C,g ( x ) , is an element of GF(')(,, then the generator of Re(C) is g ( x ) . In the other case, g ( x ) = r ( x ) . s ( x ) and the generator of Re(C) is r ( x ) . Theorem 6. Let C be a cyclic code, the generator of which is g ( x ) . If C is a p-code, then the generator of Co is g ( x ) . In the other case, the generator of COis r ( x ) *s(x)* q x ) . Two consequences of the above are: (i) C is an unreal code if and only if r ( x ) *s(x). I(x) = 0 in the algebra GF,,~,(x)/x"- 1. (ii) If a cyclic code C over GF(,2) is unreal then Re(C) is the sum of minimal codes which have an even dimension.

3. Idempotents and quadripotents of binary cyclic codes 3.1. Some results We study the quadripotents, namely, g ( ~=)g (~x ) and g(x)'# g ( x ) , in GF,,2,(x)/x" - 1 with (n,2) = 1. We prove in [l] that a necessary and sufficient condition for a binary cyclic code to be the real part of a cyclic code over GF(4) which is not a p-code, is to have a quadripotent. The fundamental result is the following.

Theorem 7. Let C be a cyclic code over GF(pZ)which is not a p-code. The idempotent which generates C is e(x) = e l @ )+ Lye&), and the generator of C is e ( x ) = r ( x ) - s ( x ) . Then: ( 1 ) (e,(x),x" - 1 ) = r ( x ) . p ( x ) if x " - 1 = r ( x ) . s ( x ) - S ( x ) . p ( x ) . (2) (e,(x),x " - 1 ) = r ( x ) .

G. Delclos

256

Proof. We outline the proof of (1).We first prove that r ( x ) . p ( x ) divides e,(x). In fact, e 2 ( x )# 0 for C is not a p-code. Re(C) = ( r ( x ) )and the check polynomial of g(x) is p ( x ) - S ( x ) We . know that the code ( p ( x ) - s ( x )is) equivalent to C ' (see

[3]) and that the idempotent which generates it is 1 + e ( x ) = ( e l @ )+ 1 ) + ue2(x). If C*= (p(x).f(x)) then Re(C*) = ( ~ ( x ) ) . e2(x)ERe(C) so r ( x ) divides e2(x); e , ( x ) E Re(C*) so p ( x ) divides e z ( x ) . Now ( r ( x ) ,p ( x ) ) = 1 for ( n , p )= 1. We conclude by using Gauss's theorem. We now prove that (e2(x),x"- 1) = r ( x ) . p ( x ) . Suppose that (ez(x),x"- 1) = k ' ( x ) . r ( x ) . p ( x ) with k'(x)€GF@),,, and deg k ' ( x ) > O . The nth roots of unity which are zeros of k ' ( x ) are zeros of s(x)* S(x), and deg k ' ( x )< deg s(x). F(x) for deg e 2 ( x )< n. We now prove that k ' ( x ) = p a ( x ) . G ( x ) with a ( x ) l s(x), G(x) F(X) and /L E GF@). From the definition of s ( x ) , k ' ( x ) cannot divide s(x) and f(x). So k ' ( x ) is a product of two polynomials: vl(x) divides s(x) and v 2 ( x )divides S(x). It is easy to prove that a&) = p G l ( x )and finally that k ' ( x ) = p a ( x ) - G ( x )with p GF@). We now prove that a ( x ) e l ( x ) .Indeed since a ( x ) k ' ( x ) and k ' ( x ) e,(x), we conclude v ( x ) l e2(x).In fact, v ( x ) I s(x) and s ( x ) l e ( x ) ,so a ( x ) e ( x ) . We infer that a ( x ) l ( e f x )+ ae2(x))= e!(x). The proof that v ( x ) . G ( x ) Ie ( x ) follows the same lines. We can now conclude: e ( x ) does not have the same nth roots of unity as g(x) = r ( x ) - s ( x ) , namely those of G(x) which divide S(x). This is a contradiction (see [3]). Therefore the degree of k ' ( x ) is 0 and consequently,

I

I

I

I

I

(e2(x),x " - 1) = r ( x ) - p ( x ) . The proof of (2) follows the same lines. Now we only study the cyclic codes over GF(4) of length n, with (n,2) = 1 ; from the fundamental Theorem 7 we infer the following.

Theorem 8. Let C be a cyclic code over GF(4). The idempotent which generates C is e ( x )= e , ( x ) + ue2(x),eA(x) is the idempotent which generates C,,, and e ' ( x ) that which generates Re(C). Then, e,(x)+ e ' ( x ) +e;I(x)=O.

Remark. It is easy to prove that e l ( x )is a quadripotent, e z ( x )an idempotent, and e : ( x )+ e l ( x )+ e 2 ( x )= 0.

Cyclic codes over GF(4)and GF(2)

257

3.2. The quadripotents of a binary cyclic code of length n, with ( n , 2 )= 1

Theorem 9. Let C be a binary cyclic code. The following properties are equivalent : (i) q ( x ) is a quadripotent of C ; (ii) q ( x )is a solution in Cof an equation of the model X 2+ X + e = 0 with X a polynomial of GF(2)(,,/x" - 1 and e an idempotent of C, e # 0; (iii) if q ( x )= 2qiXi then q4i = qi and for j , qZj# qj ; (iv) if y is an n -th root of unity then q ( y) E GF(4) and for at least an n -th root of unity y', q ( y ' )= (Y or p. Proof. For the proof see [l].

Theorem 10. Let C be a binary cyclic code. If qo is a quadripotent of C, all the quadripotents of C are qo+ e', with e' an idempotent of C.

3.2.1. The existence of quadripotents in a binary cyclic code Theorem 11. ( 1 ) Only the minimal codes of even dimension have quadripotents, in fact, exactly two. ( 2 ) The quadripotents of a binary cyclic code are equal to the sum of the quadripotents of the minimal codes of the decomposition of the code into a sum of minimal codes. Remark. If the minimal codes are all of odd dimension then the code has no quadripotent. The proof is in [l].

4. Conclusion

We conclude with two applications: (1) Let C' be a binary cyclic code. W e construct cyclic codes over GF(4) which are not 2-codes and whose real part is C'. Let C' be a binary cyclic code, C" a subcode of C' which is the sum of minimal codes of even dimension, e'' is the idempotent which generates C", 8' is the sum of the primitive idempotents of C' which are not in C". We resolve in C" the equation X 2+ X + e" = 0. Let qo be a solution. (qo+ 8' $- ae") is the idempotent which generates a code on GF(4) which is not a 2-code and which has real part C'. If we add to qo+ 8' + ae" primitive idempotents which are in C"we still obtain

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G. Delclos

a code over GF(4) which has real part C’. If the primitive idempotents added are in the decomposition of 19’ then the real part of the code that we have constructed is a subcode of C’. (2) A n improvement of the BCH bound in the narrow sense of a class binary BCH code. Dumer and Zinoviev have proved (see [2]) that the cyclic codes over GF(4) of length n = (2”+’ + 1)/3 whose generator polynomial is the minimal polynomial m b of the n th root of unity y have the parameters k = n - 2s - 1; d = 5 if s 3 2 and ( s - 1 , 3 ) = 1 . Let C be such a code. We know that Co= (mb-tiib).Moreover C o C C, so dG 3 5 . We have proved (see [ 11) that Cois a binary BCH code, the designed bound in the narrow sense of which is 3. We can now conclude that the minimal bound of this class of binary BCH codes of length n = (22s+’+ 1)/3, dimension k = n - 4 s - 2 , and generator polynomial m tii 4 is at least 5 , namely, strictly greater than the BCH bound in the narrow sense of which is 3.

References [ l ] G . Delclos, Thtse de 3bme cycle. UniversitC de Provence, 1980. 121 1.1. Dumer and V.A. Zinov’ev, Problemy Peredati Informacii (14) 3 (1978) 24-33. [3] F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes (North-Holland, Amsterdam, 1977). [4] J.H. Van Lint, Coding Theory (Springer, Berlin, 1971).


SOME CONNECTIONS BETWEEN GROUPS AND GRAPHS. A SURVEY J. DENES Institute for Coordination of Computer Technique, Budapest, Hungary

1. Representations of group elements as products. Permutations as products of transpositions (see [2]) Hurwitz's theorem. If f o ( w )denotes the number of all distinct representations of a E S. ( S . denotes the symmetric group of degree n ) as products of w transpositions then fa ( w )= c If 7 + c2f ; + * + cef r, where k denotes the number of all distinct partitions of n (i.e. the number of all distinct conjugate classes of S"), f l , f 2 , . . . ,f k are integer numbers depending on n only, c l ,c2,.. . ,c k are rational numbers depending on (Y and n. It is known that f a ( n - 1)= fin-' if a is a cycle of degree n. Furthermore ( n - m ) = ( n - m)! ni"=, ii-2/(i- I ) ! where a E S,,is a permutation having m disjoint cycles, ai of which are of degree i (see [2]). For arbitrary w and a E S,,

fa

fa

c

(w) = i = l

si

*

f (w - 1 )

holds, where si depends on a, and ai(if ai,i = 1,2,. . . , n ) denotes a system of class representatives of S, such that a,E Ci, i = 1,2,. . . ,k, namely,

where I C, I and 1 Ci1 denote the cardinality of C, and Ci, respectively. pi = 1 if a = mi, and if such a transposition 7 does not exist, then p, = 0 and T ~ is, an~ integer which depends on ai and a (see [7]). The proof of the latter statement depends on the graph representation of a generating system of S. and on its adjacency matrix. A sequence of integers a l ,a 2 , .. . ,an-l,defined on the set H = [2,3,. . . ,n + 11, is called a major sequence if there exists at least one permutation defined on [ 1 , 2 , . . . , n - 11 such that 7r = ( b l ,b2,.. . ,bn-,) and ai > bi holds for every i = 1 , 2 ,..., n - 1 . 259

J. Dines

260

If y is a cycle of degree n, then

(allb1l)(al2bl2)* * * (a1.n-Ib1,n-1) = (a,n-2,l)(bnn-2.2).*

(a.--2.,-lbn--2.n-l)

where aij> bij, i = 1 , 2 , . . . ,n n - 2 ,j = 1,2,. . . ,n - 1. The following problem was posed in [4]. Prove or disprove that

are all the major sequences defined on [2,3,. ..,n ] . Ablonczy and Quattrocchi proved independently that there are cycles of degree n for which the reply is negative (see [l], [18] and [19]). They also characterized the 'good' cycles. Using Ablonczy's and Quattrocchi's result one can obtain the solution of a problem of the present author (see [2]), namely, how a direct correspondence can be set up between a tree and a minimal product of transpositions representing a complete cycle. One might consider a question that is more general than to represent an element of S. with the aid of transpositions, namely to enumerate the distinct representations of any given element of an arbitrary finite group as a product whose elements are taken from an arbitrary given set of generators. In [15] one can find the following result.

Let G be a finite group of order n and D a non-empty subset of G consisting of d elements. If H = GID is a subgroup of G then

where Na( r ) denotes the number of solutions of the equation x 1 * * x , = a, a E G , x , , x z , . . . ,x, E D ; [ G :HI denotes the index of H in G and y ( a ) = - 1 or [ G : HI - 1 according as a E D or a E D. As an application of this theorem one can find results in [15] on the number of representations of a given element in an Abelian group as a product of elements of maximal order of G. In [7] there are further results on this subject. For some problems on this subject see [5] and [8]. Very recently the author learned from Hamidoune that he considered the same problem as above using the diameter of the Caley graph of a group (see [20, 211); this idea was also used by the author (see [3]). A similar approach can be found in [22].

Some connections between groups and graphs

261

2. Sequenceable groups

Gordon called a group G of order n sequenceable if its elements can be arranged into a sequence a l ,az,. . . , a, in such a way that the partial products al,alaz,. .. ,a l a 2 ~a, are all distinct (see e.g. [lo]). For the connections between complete latin squares and seequenceable groups and Hamiltonian decomposition of complete digraphs, see [13]. A finite abelian group is sequenceable if and only if it is the direc! product of two groups A and B such that A is a cyclic group of order 2k, k > 0, and B is of odd order (see, e.g., [lo]). The following problem was posed by Gordon: Does any sequenceable group of odd order exist? Independently, Straus asked: Does there exist a complete digraph with 2k + 1 vertices such that it has a decomposition into Hamiltonian paths? (It is well known that if n is even then the complete digraph with n vertices has a decomposition into Hamiltonian paths. This can also be proved with the aid of Gordon’s result above, which implies that every cyclic group of even order is sequenceable.) In [13] it was pointed out that if there exists a sequenceable group of order n then the complete digraph with n vertices has a decomposition into Hamiltonian paths. A sequenceable group G of order 21 has been constructed, see [13]. (G is generated by two elements a and b with the defining relations a’ = b 3 = e, ab = ba2.) Independently, N.S. Mendelsohn gave another example of a sequenceable group of order 21. Soon afterwards it became known that there exist sequenceable groups of order 27, 39, 55 and 57 on two generators. This adds strength to an earlier conjecture of Keedwell that every finite non-abelian group of odd order on two generators is sequenceable. This conjecture was partially proved; namely, if p and q are distinct odd primes with q = 2ph + 1 for some integer h and if p has 2 as a primitive root, then the non-abelian group of pq is sequenceable. There is another conjecture of Keedwell (a special case of the former one), namely, the dihedral group D, of order 2n is sequenceable for all odd integers n. Keedwell has obtained sequencings for all such dihedral groups up to and including D2,(see (161).

3. C- and P-groups

Let G be a finite group of order n and G‘ its commutator subgroup. If every element of G’ is a commutator, then G is called a C-group. If every product of n distinct elements of G covers a coset mod G‘, then G is called a P-group (see [lo]).

J. Dtnes

262

It is known that there exist finite groups which are not C-groups. Rodney gave a simple example of a group G of order 21°, in which some product of two commutators is not itself a commutator, and Guralnik gave an example of such a group G of minimal order 96 (see [17]). If G = S., then G' = A,,, the alternating group. It is easy to see that every a in A,, is the commutator of two elements from S.. If n 3 5, then A,, is simple and A A = A.. In this case, every element a of A,, is the commutator of two elements from A. itself. Several classes of non-abelian finite simple groups are C-groups. This suggests the well-known conjecture that every finite simple group is a C-group (see [17]). If the group in question is not a C-group the question arises as to how many commutators should be used to obtain the elements of the commutator subgroup as their products (see [12] and [23]). Under certain conditions the C property implies the P property (see [lo]). There was also a conjecture (see [lo]) that every finite group is a P-group. This was proved quite recently by Hermann and the present author (see [9]). The proof of this theorem uses the Feit-Thompson theorem (see [141). The problem arose of proving that every finite group is a P-group without using the Feit-Thompson theorem (see [6]). It is not a new idea to give a simple proof of the Feit-Thompson theorem using the fact that every finite group is a P-group (see [13]). DCnes and Hermann's theory implies that if a finite group of order n has not a proper cyclic 2-sylow subgroup then its identity element can be represented as a product of n distinct elements. This is closely connected with transversals of latin squares and 1-factorizations of complete bichromatic graphs (see [lo]). Very recently, Keedwell defined a super P-group G of order n as one in which, if gi E hG', then every element of hG' can be written as a sequencing of G (except if h is the identity element). If h is the identity, then every element of G' is an R-sequencing (see [24]). (A finite group G is R-sequenceable if it is possible to list the nontrivial elements of G in a sequence a ] ,a 2 , .. . ,a,,-l such that the.quotients a ;la2,a;'a3,. . . ,a ;!lal are all distinct. The existence of an R-sequencing of a finite group G is equivalent to finding a Hamiltonian circuit in a complete directed graph whose vertices are labelled by the nonidentity elements of G.) One can easily carry out the graph-theoretical interpretation of super Pgroups using Cayley graphs. Another combinatorial approach to simple groups is to study the lengths of the products which represent the identity element of a simple group. There are results and problems in doing this (see [8] and [ll]),which are closely connected with the adjacency matrix of the Cayley graph of simple groups. It is of some interest to study how group-theoretical results can be applied to other branches of mathematics, e.g., to graph theory (see [3]).

n:=,

Some connections between groups and graphs

263

Acknowledgement

My sincere thanks go to Dr. A.D. Keedwell for some helpful comments. References [ l ] P. Ablonczy, On the connection of complete cycles and major sequences, Lecture given at the Conf. on Diskrete Mathematik und ihre Anwendungen in der Mathematischen Kybernetik, 1981. [2] J. Denes, The representation of a permutation as the product of a minimal number of transpositions and its connection with the theory of graphs, Publ. Math. Inst. Hungar. Acad. Sci. 4 (1959) 63-71. [3] J. DCnes, Some applications of group theory to other branches of mathematies, Publ. du Centre de recherches Math. pures, Neuchatel, Serie (1) 12 (1977) 7-9. 141 J. DCnes, Research problem No. 25, Period. Math. Hungar. 10 (1979) 105. [5] J. DCnes, Research problem No. 27, Period. Math. Hungar. 1 1 (1980) 177-178. [6] J. DCnes, Research problems, Europ. J. Combin. 1 (1980) 207-209. [7] J. DCnes, A generalization of a result of A. Hurwitz, in: Algebraic Methods in Graph Theory, Coll. Math. SOC.Jdnos Bolyai 25 (North-Holland, Amsterdam, 1981) pp. 85-93. [8] J. Dtnes, Research problem, to appear. [9] J. DCnes and P. Hermann, On the product of all elements in a finite group, Ann. of Discrete Math. 15 (1982) 107-111. [lo] J. DCnes and A.D. Keedwell, Latin Squares and their Applications (Academic Press, New York, English Universities Press, London, AkadCmiai Kiad6, Budapest, 1974). 111) J. DCnes and K.H. Kim, Simple groups and Boolean matrices, An. Fac. Ci. Univ. Porto 52 (1980) Fascs 1 a 4, 4-7. 112) J. DCnes and K.H. Kim, On a problem of P. Erdos and E.G. Straus, Studia Sci. Math. Hungar. 14 (1979) 189-191. [13] J. DCnes and E. Torok, Groups and graphs. Cembinatorial Theory and Its Applications, Proc. of the Conf. at Balatonfiired, 1969 (North-Holland, Amsterdam, 1970) pp. 257-289. [14] W. Feit and J. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963) 775-1 029. [15] D. Jacobson and K.S. Williams, On the number of distinguished representations of a group element, Duke Math. J. 39 (1972) 521-527. [I61 A.D. Keedwell, Sequenceable groups. A survey, in: B.R. Hughes, P.J. Cameron and J.W.P. Hirschfeld, eds., Finite Geometries and Designs, London Math. SOC.Lecture Notes, No. 49, 205-2 15. 1171 R.C. Lyndon, Equations in groups, Bol. SOC.Brasil. Mat. 11 (1980) 79-102. [IS] G . Quattrocchi, Su un problema di J. Dines inerente certe successioni finite, Atti Sem. Math. Fis. Univ. Modena 29 (1980) 1-9. (191 G. Quattrocchi, Relazione fra major sequences e cicli compfeti di grado n, Atti Sem. Mat. Fis. Univ. Modena 29 (1980) 34-47. (201 Y.O. Hamidoune, Connectivity of transitive digraphs and a combinatorial property of finite groups, Combinatorics 79, Proc. Colloq. Univ. Montreal, Montreal, Quebec, 1979. Part 1, Ann. Discrete Math. 8 (North-Holland, Amsterdam, 1980) pp. 61-64. [21] Y.O. Hamidoune, An application of connectivity theory in graphs to factorizations of elements in groups, Abstracts of papers presented to the Amer. Math. SOC.(10) 3 (1981) 364, T-05-235. [22] H.H. Teh and S.C. Shee, Algebraic theory of graphs, Lee Kong Chian Institute of Mathematics and Computer Science, Nayong University, Singapore, 1976. [23] R.M. Guralnick, On a result of Schur, J. Algebra 59 (1979) 302-310. [24] A.D. Keedwell, Lecture given to 8th British Combin. Conf., July 1981.



ALGEBRAIC CODES ACHIEVING THE CAPACITY OF THE BINARY SYMMETRIC CHANNEL Ph. DELSARTE and Ph. PIRET P h i l i p Research Laboraiory, Ao. Van Becelaere 2, Box 8,B-1170 Brussels, Belgium

Based on Justesen’s idea of variable concatenation, an algebraic construction is given of a family of easily decodable binary linear codes achieving the capacity of the binary symmetric channel. The probability of erroneous decoding is shown to tend exponentially to zero when the block length tends to infinity, for any encoding rate less than the capacity.

1. Introduction

The present communication deals with the transmission of information via the binary symmetric channel BSC(p) with crossover probability p . More than 30 years ago Shannon proved that this information can be reliably transmitted provided it is encoded with a rate less than a quantity C = 1 - X ( p ) that is called the capacity of BSC(p). Here X ( p ) denotes - p logp - (1 - p)log(l - p ) and logarithms are to the base 2. Moreover, it turns out that ‘most’ encoding procedures are actually good, but this is only a probabilistic result and does not lead to explicit constructions of encoders nor to efficient decoding methods. In this paper we describe an explicit algebraic construction of a family of binary codes of increasing length No and rate 2 Ro such that the error probability after decoding becomes arbitrarily small for large No when Ro is < C. This construction is completely deterministic, without any random selection at any step. It is based on the concatenation method introduced by Forney [l]. This is a 2-step encoding procedure. In the first step, a sequence of k K binary digits is represented as a K-tuple over GF(2k),which is encoded into an N-tuple u = (u,,.. . ,u N ) over GF(2k) by means of an (N, K ) Reed-Solomon encoder called the outer encoder. Each ui (1 C i S N ) is then considered as a binary k-tuple and encoded into the n-tuple wi by means of a ‘good’ linear (n,k ) binary encoder called the inner encoder. The resulting binary linear code has length No = nN and dimension KO= kK. This procedure is able to achieve the capacity of BSC(p) but is not really explicit because the inner encoder cannot be a priori specified. More recently Justesen [2] has introduced a variable concatenation scheme in which different inner codes are used to encode all symbols ui produced by the 265

Ph. Delsarte, Ph. Pirer

266

Reed-Solomon encoder. This idea has led him to construct long codes with a good minimum distance Do. More precisely, for fixed Ko/Noand for arbitrarily large No, the ratio Do/Nois lower bounded by some strictly positive number. A better ratio Do/Nowas obtained later by Weldon for small values of Ko/No[3]. The present paper applies a variable concatenation scheme to provide an algebraic construction of encoders achieving the capacity of BSC(p). Moreover, the complexity of the corresponding decoders turns out to be low (of the order of N;). In Section 2 we describe the properties of ‘good’ sets of encoders for BSC(p) and in Section 3 we give a specific good set. A variable concatenation scheme built from this set is presented in Section 4 and the performances of the resulting codes are discussed in Section 5. Some comments on the constructive character of the codes are made in Section 6.

2. The mean error probability of a set of encoders Let X = (0,l) be the binary alphabet endowed with the structure of GF(2). An (n, k ) encoder g is a mapping g : X k - + X ” : a + u =g(a).

The rational number I = k/n is called the encoding rate. We emphasize that the n-tuples u produced by distinct information k-tuples a are not necessarily distinct. We suppose that maximum likelihood decoding (MLD) is used to estimate the information k-tuple a from the received n-tuple y, which is the noise-corrupted version of the n-tuple u. We now consider any set G of (n, k ) encoders g, and relative to this set G we define vl(a, u) as the number of g E G satisfying u = g(a). Similarly we define vZ(u,a’,u,u’) as the number of g E G satisfying both u = g(a) and u’ = g(a’). Here a and a ‘ are distinct but u and u’ may be identical. Finally we define Qg(u) to be the error probability in the evaluation of the information a encoded by the encoder g when MLD is used at the decoding. This error probability satisfies, for all p E [0,1]:

I

where P ( y u) denotes the probability of receiving y when 1) has been sent on BSC(p). This bound follows from replacing probabilistic arguments by counting arguments in the proof given by Gallager [4] to the channel coding theorem. The expression (1) may seem to be formidable. However, suppose that v2 is equal to a constant h for all a # a ’ and all u, u’. In this case v1 is also a constant (equal to

Algebraic codes

267

A2") and the cardinality of G is A22". Then G is called a balanced set of encoders. Using the fact that BSC(p) is memoryless, we finally obtain

where r = k / n and

This function E ( r ) can be shown to be > 0 for r < C. The average error probability over a balanced G thus satisfies the upper bound of the celebrated random coding theorem [4] applied to the channel BSC(p). 3. Construction of a good set of encoders

We construct a balanced set of ( n , k ) encoders as follows. We endow the space = kt + m with t = ( ( n - l ) / k J and 1 S m 6 k. As usual 1x1 stands for the largest integer S x. Then with any choice of elements cro E X"'and yo, yl, crlr.. . , y,, cr, E GF(2k),we associate the encoder g : X' + X" given by

X kwith the structure of the Galois field GF(2') and we write n

g ( a ) = [p(yna)+ cro, ylu

+ U l , . . ., yfa +

Cf],

(3)

where p ( b )denotes the m first coordinates of b. The set G of all encoders (3) is a balanced set with v 2 = 2'-" so that it satisfies (2). Let us now define Goas the set of g E G having yo = 1 and a, = 0 for all i. This set has cardinality 2'' and can be shown to satisfy:

The reader is referred to [ 5 ] for further details. As a result one has:

will now be used in an explicit concatenated encoding scheme that This set G,, achieves Shannon's capacity. 4. Concatenation scheme Suppose we have constructed the Galois field GF(2k") for some u 2 t = [ ( n - l ) / k J . Let us specify an ( N ,K ) Reed-Solomon code over GF(2'") by the following generator matrix,

Ph. Delsarre, Ph. Piret

268

(Yo

1 a1

(Y;

f f ;

1

%=

fft-1

...

1

. * .

(YN-I

...

a;-, ,

. ..

a N-l

K-l

5. Performances on BSC(p) First it is important to remember that when g is a linear encoder used on BSC(p) the error probability Qg ( a ) is constant for all a E X k .Now let E be an arbitrarily small (but fixed) positive number and define the integer z to be z =

"z-'"],

with N = 2". From (4)it follows, in view of Chebyshev's inequality, that there exist at most z 'bad encoders' gi characterized in terms of their error probabilities Qi by the inequality Qi > Q = (1 - 2-k)-12-n[EV)-cI (6) Hence there are at least N - z 'good encoders' gi characterized by Q, S 0.Let P be the probability of erroneous estimation of any given information message a EXKii.The principle of the decoding method is to use MLD to decode the inner codes and to use an algebraic procedure (as Berlekamp's algorithm) to perform bounded distance decoding of the outer RS code. So, by definition, P is bounded from above by the probability that the inner decoders produce an incorrect estimation of at least (N - K)/2 + 1 symbols u, in the corresponding codeword u of the outer code. The error probability on the ui corresponding to the N - z good encoders gi is upper bounded by 1 - (1 - Q)" and hence by u Q . so that P satisfies:

269

Algebraic codes

Using (5), (6) and No = unN, we finally obtain N i l l o g P s - ( l - ( R o / r ) ) [ E ( r ) - ~ ] / 2+ua ( n ) ,

(7)

with Ro = r K / N the overall encoding rate, and where a ( n )goes to zero for large n and fixed u, r and Ro. Since E ( r ) is positive for all r less than C, it is obvious that E ( R o ,E ) = (1 - ( R o / r ) ) [ E ( r )~- ] / 2 is u also positive whenever R o < Cprovided E is sufficiently small. As a result, the probability P of erroneous decoding tends exponentially to zero when N o + m provided R o < C.

6. Constructivity of the encoding scheme

The codes described above need the construction of two Galois fields, namely, GF(2k)and GF(2"k),where u is a fixed integer 2 t = ( ( n - l)/k] . For any value of t we should thus be able to define an integer u 5 t with the property that there exists an infinite sequence of integers k , < k2< k , < * . such that irreducible polynomials of degree k, and uk; (both over GF(2))can be directly specified for all i. A solution to this problem is given by the set of polynomials

-

p m ( a ) =1+a3"'

rn = l , 2 , 3 ,...,

that are irreducible over GF(2) [ 2 ] .The requirements above are indeed satisfied by choosing ki = 2.3' and u = 3' for a fixed but sufficiently large j .

References 111 G.D. Forney, Concatenated Codes (MIT Press, Cambridge, MA, 1966). [2] J. Justesen, A class of constructive asymptotically good algebraic codes, IEEE Trans. Inform. Theory 1T-18 (1972) 652-656. [3] E.J. Weldon, Justesen's construction - the low-rate case, IEEE Trans. Inform. Theory IT-19 (1973) 711-713. [4] R.G. Gallager, Information Theory and Reliable Communication (Wiley, New York, 1968). [5] Ph. Delsarte and Ph. Piret, Algebraic constructions of Shannon codes for regular channels, IEEE Trans. Inform. Theory IT-28 (1982) 593-599.



PARTITIONING SUBWORDS OF LONG WORDS Walter DEUBER Uniuersitai Bielefeld. Fakultai fur Maiheinatik, 4800 Bielefeld I, Germany

Section 1

I n [ 3 ] Hindman proved the following theorem. Theorem 1. Let A be a finite alphabet and W : w -+ A a word 0ver.A of length w. Then there exist infinitely many io < i , < . . such that W is constant on the set

{ 2 ik I K c

w,

K # 0, K finite

kEK

Definition. Let W , W ' be words over an alphabet A. W' is a subword of W iff there exists i,,< i , < iz < . - .such that W ' ( k )= W ( L )for all k < length( W ' ) .Let Sub(W) be the set of all subwords of W. Definition. Let A be an alphabet. A word W over A with a parameter A is a word W over the alphabet A U {A} in which A occurs (i.e., W - ' ( A ) #0). If W ( h )is a word with a parameter and a E A then W ( a )is the word obtained by replacing all occurrences of A by a. Main Theorem. Let A be a finite alphabet and k, 6 < w . Every word W : w -+ A which contains each letter infinitely often has the following property : For every mapping A :Sub( W)+ S there exist k words with parameters Wo(Ao), . . ., Wk-,(Ak-,) such that A is constant on the set

-

{W,,(a,,)x. . * x W , . , ( a , , ~ , ) l O ~ i*c. l 2). We prove that for every E > 0 there exists a 6, so that one can have (1 + S ) d points in Ed all the angles of which are < ( d 3 ) + E. Erdos and Szekeres proved that 2" points in E , always determine an angle > n(1- l / n ) and Szekeres proved that this result is best possible in the following sense: One can give, for every E > 0,2" points in E 2no angle of which is greater than n(1- (l/n)) + E . It does not seem easy to get a result of the same precision for higher dimension but we will get inequalities giving estimates for the number of points in Ed which give an angle > I T - E .

2. Strictly antipodal polytopes

Let us denote by Ed the d-dimensional Euclidean space. Let 9 C E d be a pointset, and let a, b E'9. We shall say that a and b are an antipodalpair of 9 275

276

P. Erdos, Z. Fiiredi

provided there exists a pair of parallel (distinct) supporting hyperplanes of 9’ such that a belongs to one of them and b to the other. A pointset 9’ is said to be antipodal provided each two of its points forms an antipodal pair of 9. (Clearly, in this case 9 is the vertex set of the polytope conv 9, the convex hull of 9.) Let us denote by G ( d ) the maximal number of vertices in an antipodal d-polytope. Danzer and Grunbaum [4] showed the conjecture of Klee [ l l ] that G ( d ) = 2d. Theorem 2.1 [4]. If the pointset 9C Ed is an antipodal, then 1 9 I S 2d. Equality holds if and only if 9 consists of the uertices of a d-dimensional parallelotope. This result implies also that F ( d ) = 2d is the answer to the following problem of Erdos [5,6]: What is the maximal possible number F ( d ) of points in E,, such that all angles determined by the triples of them are less than or equal to OW? Now let the notion of an antipodal pair be modified by defining a pair a , b E 9 as k -antipodal provided there exist parallel (distinct) supporting hyperplanes of 9, each of which intersects conv 9 in a set of dimension at most k, such that a belongs to one of the hyperplanes, and b to the other. In analogy to the above, we define k-antipodal polytopes and the numbers G, ( d ) . Clearly, a d-polytope is antipodal if and only if it is ( d - 1)-antipodal. A number of interesting problems concern 0-antipodality, which is called strict antipodality. While it is easy to show that G42) = 3, the proof of G,,(3)= 5 is rather involved (Grunbaum [ 1I]). For d 2 4, it is known that G , , ( d )2 2d - 1, and it has been conjectured that G , , ( d )= 2d - 1 (Danzer-Grunbaum [4], Grunbaum [ I l , 121). In this section we disprove this conjecture, giving a construction with more than 1.15d points. (See Theorem 2.2.) As in the case of antipodal pairs, G , , ( d )may be considered as the affine variant of the following Euclidean problem due to Erdos [6]: Determine the maximal possible number f ( d ) of points in Ed such that all angles determined by triples of them are acute. Examples show that f ( d ) s 2d - 1, and clearly G , , ( d ) zf ( d ) . In contrast to the situation in the case G ( d ) ,it is not known whether G , , ( d )= f ( d ) for d 2 4. (Direct proofs of f ( 3 ) = 5 were given by Croft [ 3 ] and Schiitte [IS].) The following theorem implies that f ( d ) > 1.1Sd. Theorem 2.2. There exists a pointset 9in Ed of cardinality 1.15* such that all angles determined by triples of 9 are acufe. Proof. We select the points of 9 from the vertices of the d-dimensional cube. As usual, the d-dimensional 0-1 vectors correspond to the subsets of a d-element set X. More precisely, if a E (0;1)” then let A = A ( a )= {i: ai = l}, X:={l,2,. . . , d } .

The greatest angle among n points

277

Lemma 2.3. The points a, b, c E (0; l}d determine a right angle at the point c if and only if

AnBCCCAUB,

(2.4)

where the sets A , B, C C X are associated with the vertices a, b, c.

(This lemma is a trivial consequence of Pythagoras’ Theorem.) As the angles determined by the triples of the cube are less than or equal to n/2, the construction of the desired 9 will be completed if we find a set system 9 over X no three different members of which satisfy (2.4), and whose cardinality is greater than 1.15d. Let h ( d ) denote the greatest cardinality of such 9,i.e., h ( d ) : = m a ~ ( 1 9 1 : 9 C 2 ~for , all A # B # C € 9 , A flB!ZC or C l t A UB}. Lemma 2.5. h ( d ) > (2/d3)d-’ Proof. Let us choose independently the coordinates of the d-dimensional 0-1 vectors a l ,a?,. . .,a 2 , with probability Prob(a,, = 0) = 1/2, Prob(a, = 1) = 1/2, 1 G i s 2m, 1 S j 6 d and m = [ ( 2 / ~ ) ” - ’ ]Then, .

Prob(a, b, c hold for (2.4)) = (3/4)d.

(2.6)

Indeed, (2.4) means that for all 1 S i S d neither a, = b, = 1, c, = 0 or a, = b, = 0, 1 hold. The independency of the coordinates yields (2.6). Hence the expectation number c, =

E(the number of the triples (a, b, c ) satisfying (2.4)) = = 2m(2m - 1)(2m - 2)(3/4)d < m.

Hence the expected number of vectors to remain after the omission of the points which are vertices of a right angle is greater than 2m - m = m, and for that set of vectors the conditions are already satisfied. 0 Finally, since f ( d ) a max(2d

-

1, h ( d ) } , (2.2) follows. 0

A bit more complicated random process gives

-

h ( d ) > ( T h - ~ ( l ) ) 1.189.. ~ .d.

(2.7)

Instead of Lemma 2.5, we can use the following theorem due to Frank1 and the authors [9] to prove G o ( d )3 f ( d ) > (1 + c ) ~ . Theorem 2.8 [9]. There exists a set-system 9 over a d-element underlying set in 9 1> 1.13”. which no set is covered by the union of two others and 1

P. Erdos, 2. Fiiredi

278

We have the following.

Conjecture 2.9. There exists an absolute constant c > O (not depending on d ) such that

f ( d ) Go(d)< (2 - c ) ~ . (?) The following fact says that it is not possible to choose more than 2 V 5 d points from the vertices of a cube such that all angles among them are less than d 2 . Even more, denote by Gn({O;l}d)the greatest cardinality of a strictly antipodal 9 c (0;

Fact 2.10. h ( d )G Go({O;

< fi(\5)d.

Proof. If {a, b } ,{ c , d } C 9 are distinct pairs, then a + bf c + d (since a + b = c + d implies that the points a, c, b, d form a parallelogram which is contrary to the strict antipodality of 9).Thus

(I9!+

')

=/{a

+ b : a, b E 9}1S I{O;

1;2jd I = 3 d .

0

Finally, we mention some more problems. Slightly generalizing the question about G ( d ) one is led to the problem of determining e ( d , n ) , the maximal number of antipodal pairs among the vertices of a d-polytope with n vertices. It is not hard to show that e ( 2 , n ) = [3n/2J (Grunbaum [ll]), and that

e ( 3 , n)> Ltn] \f(n + l)]

+ LlnJ + [i(3n + 1)J.

Conjecture of Grunbaum [12]. Does the relation

(2.11) hold for all d

3 2.

In analogy to the above, we define e k ( d , n ) , the maximal number of k antipodal pairs. Regarding e o ( d , n ) , it is easy to prove that e o ( 2 , n ) = n (Grunbaum Ill]), but even the 3-dimensional case seems to be very complicated. Clearly, eo(d,n ) 2 d ( d , n ) , the maximal number of diameters of an n-element set in E d . It is known that d ( 2 , n ) = n (Erdos [7]), d ( 3 , n ) = 2n - 2 (Grunbaum [lo], Heppes [13] ana Straszewicz [16]), but for d'd4 lim d ( d , n ) / n 2= t - 1 / ( 2 [ d / 2 ] ) (Erdos [7]).


279

Theorem 2.2 yields a new lower bound of the following problem: What is the order of the number c d defined as follows: For a d-dimensional convex body K , denote by c ( K )the minimal number of translates of K the union of which covers K. Then c d is defined as the maximum of c ( K ) for all d-dimensional convex bodies K . It is easily seen that c d 2 G o ( d ) ,but it is not known whether c d = G , , ( d )for d 2 3 . Corollary 2.12.

cd

3

1.15d

The following conjecture would extend Theorem 2.1. Conjecture 2.13. There exists a positive constant E ( E independent of d ) for which the following is true : If the pointset 9 C Ed and I 9 1 3 2d + 1, then 9 contains an angle greater than ~ / +2E .

3. Pointsets with all angles small Denote by a (9)the greatest angle determined by the triples of the pointset 9, and ad(n)=:inf{a(y): 1 9 1 = n, 9 c E d } .We can write Theorems 2.1 and 2.2 and Conjecture 2.9 as follows: (Yd

(zd + 1)> n/2,

(Yd

( 2 d )= 7f/2

(cf. 2.1)9

( ~ ~ ( 1 . 1 5n/2 ~ ) < (cf. 2.2),

4 5 ) < n/2, 3 c > 0:

ad

( ~ 4 6=) d 2

(3.1) (3.2)

(Croft [3] and Schiitte [15]),

((2- C ) d ) = n/2 (?)

(cf. 2.9).

(3.3) (3.4)

We were not able to establish an even weaker version of (3.4). Conjecture 3.5. For each ad

But for

E

E

> 0 there exists a

C(E)

> 0 with

((2 - c ) d ) > n/2 - E . large enough we can prove the following.

Theorem 3.6. If 0 < c < 1 then a r ( ( l + cY)> 7~/3 + ~ / -0(1). 4

(3.6a)

Further, there exists a construction showing that (Yd

((1 + c ) d )< d 3 + &.

(3.6b)

P. Erdiis, Z . Fiiredi

280

Clearly 1 9 123 implies a(9)b 7r/3. What Theorem 3.6 says is that there are exponentially many (1 + c ) ~points in E d with all angles less than 61", but that 1.4d points always determine an angle larger than 72".

Proof. We start with the construction. Lemma 3.7. There exists a k-uniform set system 9 ooer d elements such that 1 F, f l F, 1 < ~k for each distinct F,, F2 E 9 and 191 > (1 + 0 . 4 ~ ~ ) ~ . Proof. We choose F,, F2,. . . ,F , , . . . recursively. Suppose F , , . . . ,E are already chosen. Let 2& = : { F C X : IF1 = k, IF n F, 1s x } for all 1 S j S i. We can select an E+IG % if I % I < (f).Thus this process can be continued for at least as many as

steps. Putting k = de/4 and formula. 0

x = ke

we prove (3.7) using Stirling's

a,

Returning to the proof of (3.6b), put E = and let 9 be the set system defined in (3.7). For the distances of the two vertices f l , f 2 corresponding to FI,F2E 9 we have d 2 k (1 - E ) < If, - f 2 1 S

V%,

i.e., the distances defined by 9are almost equal. Now a simple calculation gives (3.6b). To prove (3.6a) let 9 = {PI,P 2 , .. , ,P,} C E d be a pointset with every angle less than 7r/3 + x. Hence the ratio of the smallest and largest sides of a triangle PiPjPk is greater than sin(7r/3 - 2x)/sin(r/3 + x ) 2 1 - 2x. So if the largest distance in 9 is 1, then the smallest is at least (1 - 2 x ) ' 2 1 - 4 x (x S 1/4). Let S be the smallest ball containing 9 with center 0. By the Yung theorem [12], the radius of S is less than 1/-< l/d?.Project the points of 9' from 0 to the surface of S. The image of Pi is Q i . It is easily seen that 1 PipjI < l / f i implies I QiQj1 2 1 Pipj1. So if x is small enough (x < 0.07) then any two point QiQj have a distance at least 1 -4x. So we can apply Boroczki's quite sharp estimation [ l ] about the density of a packing of the d-dimensional sphere by congruent balls, to get 1 9 1 = 19 I < (1 - 4 ~ ) - ~ 2However d. we can give a simple straightforward proof of this last step.

Lemma 3.8. If 9 is a pointset on the surface of the sphere S of radius l / f i , and for all O,, Q2E 9 we have 1 Q I Q 2 > J 1 - y then 19 1 < d * 2d (1 - Y ) - ~ .


Proof. Any ball with radius (1 - y)/2/2 contains at most d Taking averages,

"

'

28 1

+ 1 points

of 9.

surface area of S ( d -t ') maximal surface area of the intersection of S with a sphere of radius (1 - y)/2/2

< ( d + 1)(1- ~ ) - ~ d .0

4. The greatest angle among n points

Erdos and Szekeres proved [8] that 2" points in the plane always determine an and Szekeres [17] proved that this rtsult is best possible in angle > ~ ( 1 (l/n)) the following sense: One can give, for every E >O, 2" points in E z no angle of which is greater than ~ ( 1 (l/n)) + E. I.e., a2(2")=7F

(1-- 9

([8] and [17]). So in general,

In this section we are concerned with the d-dimensional version of this fact. Theorem 4.3. We have

The proof shows that one cannot hope to get an essentially better result without a new estimation for the sphere-packing problem in d-dimension, which is far better than the existing ones. The proof used adapted the proofs in [8] and [17]. We need two facts. (4.4). In the d-dimensional space there exist more than lines going through the point 0, such that any two of them determine an angle greater than P.

(4.5). In the d-dimensional space there exist fewer than ( 4 / ~ ) ~ -lines * going through 0, such that any other line going through 0 determines an angle less than p / 2 with some of them.

P. Erdos, Z. Fiiredi

282

The facts (4.4) and (4.5) are in fact equivalent to the well-known spherepacking and sphere-covering problems. There are much better estimations, but (4.4) and (4.5) are very easy to check, and the better known estimations would not eliminate the difference between the constants in (4.3). The proof of the upper bound. Let e l ,e 2 , .. . ,em,with m > ( l / ~ ) ~be - ' a, system of lines determined by (4.4). We are going to construct 2" points recursively such that every angle in the triples is less than I T - p . Let PI= { A ,B } where A, B E e l . Translating in the direction e2 we get A ', B' and if A ' and B' are far enough then AB' and A 'B are almost parallel to e2. Then translate the parallelogram A A 'BB' in the direction e3 far enough . . . and so on. After m - 1 translations we get a construction showing a d (2'")< IT - p . The proof of the lower bound. Let f l , .. . , f m , with m < ( 4 / ~ ) ~ -be ' , a set system of lines determined by (4.3, and let 9 C Ed be a pointset with more than 2" points. Consider the complete graph with vertex set 9 and colour its edges in the following way. For U, V E 9 the edge UV gets colour i, 1 C i C m, provided the angle between UV and fi is less than p / 2 . Lemma 4.6 (see [8]). If the edges of the complete graph with more than 2" vertices are coloured with m colours then there exists an odd circuit whose edges are of the same colour.

Lemma 4.6 implies that there are U, V, W E 9 and an fi such that the angles ( U V , f ) ,( V W , f )and (WU,f ) are less than p / 2 . But then the greater angle in the 0 triangle UVW is at least I T - - . One more open problem: Let 9 be a pointset in Ed and 0 < a < IT. Define f(9,< a ) (resp. f(9,> a)) as the number of angles in 9 smaller (resp. greater) than a. Put

f d (n, < a ) := minu(8, < a):1 8 I = n, 9 c Ed}, and f d ( < a ) : = l i "m-mf d ( n , < a ) / ! n ( n- 1 ) ( n - 2 ) . ( < a ) and f d ( > a ) show at least what percent of the angles are smaller (or greater) than a for a pointset 9 in E d . Convay, Croft, Erdos and Guy in [ 2 ] investigate f 2 and f 3 . They show, for instance, that

fd

b s f2( > :IT)

A

(4.7)


([2]). In higher dimensions no estimation is known for f d ( < a ) is known for only a finite number of a.

283 fd.

Even for d 6 3 ,

References [I] K. Boroczki, Packing of spheres in spaces of constant curvature, Acta Math. Acad. Sci. Hungar. 32 (1978) 243-261. [2] J.H. Convay, H.T. Croft, P. Erdos and M.J.T. Guy, On the distribution of values of angles determined by coplanar points, J. London Math. SOC.(2) 19 (1979) 137-143. 131 H.T. Croft, On 6-point configurations in 3-space, J. London Math. SOC.36 (1961) 289-306. [4] L. Danzer and B. Griinbaum, Uber zwei Probleme beziiglich konvexer Korper von P. Erdos und von V.L. Klee, Math. Z . 79 (1962) 95-99. [5] P. Erdos, Problem 4306, Amer. Math. Monthly 55 (1948) 431. [6] P. Erdos, Some unsolved problems, Michigan Math. J. 4 (1957) 291-300. [7] P. Erdos, On sets of distances of n points in Euclidean space, Magyar Tud. Akad. Mat. Kutatd Int. Kozl. 5 (1960) 165-169. [8] P. Erdos and G . Szekeres, On some extremum problems in elementary geometry, Ann. Univ. Sci. Budapest, Eotvos Sect. Math. III/IV (1960-61) 53-62. [9] P. Erdos, P. Frank1 and Z. Fiiredi, Families of finite sets in which no set is covered by the union of two others, .I. Comb. Theory, Ser. A 33 (1982) to appear. [lo] B. Griinbaum, A proof of VBzsonyi’s conjecture, Bull. Res. Council Israel 6A (1956) 77-78. 1111 B . Griinbaum, Strictly antipodal sets, Israel J. Math. I (1963) 5-10. [ 121 B. Griinbaum, Convex Polytopes (Interscience, London, 1967); Pure and Applied Mathematics 16. (131 A. Heppes, Beweis einer Vermutung von VBzsonyi. Acta Math. Acad. Sci. Hungar. 7 (1057) 463-466. [ 141 V.L. Klee, Unsolved problems in intuitive geometry, Mimeographed notes, Seattle, 1060. [IS] K. Schiitte, Minimale Durchmesser endlicher Punktmengen mit vorgeschriebenem Mindestabstand, Math. Ann. 150 (1Y63) 91-98. [ 161 S. Straszewicz, Sur u n p r o b l h e geometrique de P. Erdos, Bull. Acad. Polon. Sci. CI. (111) 5 ( I 957) 39-40. [I71 G . Szekeres, On an extremum problem in the plane, Amer. J. Math. 63 (1941) 208-210.



EDGE-RECONSTRUCTION OF GRAPHS WITH TOPOLOGICAL PROPERTIES S. FIORINI and J. LAURI The University of Malta, Malta

1. Introduction

A graph G is said to be edge-reconstructible if G is uniquely determined (up to isomorphism) from the collection (called the d e c k ) 9 ( G ) = {G - e : e E E ( G ) } of edge-deleted subgraphs of G. In this paper we deal with two kinds of graphs having certain topological properties. In Section 2 we outline how all 4-connected planar graphs are edge-reconstructible. In the last section we deal with the edge-reconstruction of certain graphs which triangulate surfaces.

2. 4-Connected planar graphs

In this section, we let G be a 4-connected planar graph. When the minimum valency 6 ( G )is 5 , reconstruction follows by [3], so that we can assume 6 ( G )to be 4. Since G - e is 3-connected for every edge e, then by Whitney’s well-known theorem, G - e has a unique embedding in the sphere. It follows that the sequence of face-valencies in non-increasing order in each G - e is uniquely determined. In fact, more can be said. Proposition 1. The face-valency sequence of G is reconstructible from 9 ( G ) . Corollary. Let G - e E 9( G ) .If e is incident in G with faces F, F’, then the pair of face -valencies p ( F ) , p ( F ‘ ) is reconstructible from 9( G ) . The main result in this section is to prove the following. Theorem A. If G is a 4-connected planar graph, then G is edge-reconstructible. 285

286

S. Fiorini, J . Lauri

To this end, we let u be a 4-vertex incident to consecutive faces F, with face-valencies p ( E ) = a, + 2 ( i = 0,1,2,3). Then the wheel-sequence W ( u )of u is (ao,a l ,a 2 ,a3).The conclusion of Theorem A hinges on certain auxiliary results on wheel-sequences, the central one being the following. Proposition 2. Let G have a 4-vertex which has a repeated term in its wheelsequence. Then G is edge -reconstructible.

3. Two reconstruction techniques In establishing the results in Section 2, use is made of the following two techniques. (i) It is easily shown (using wheel-sequences) that if a 4-connected planar graph G is not edge-reconstructible, then there exists exactly one edgereconstruction H of G not isomorphic to G. The graph H can be represented as G - vx + uy, where u is a 4-vertex. Given a fixed 4-vertex u and using successive representations of this type, we generate a sequence of graphs (Go,G I ,Gz,. . .) such that Gz, = G and G21+I =H. Under certain conditions we obtain the contradiction that G is isomorphic to G21+Ifor some j. (ii) In trying to establish that a graph G (no longer necessarily planar and 4-connected) is edge-reconstructible, it is possible sometimes to consider certain ‘configurations’ such that if G contains any one of them then G is edgereconstructible. Often, such a set of configurations is ordered in a sequence (C,, CI,C,,. . . ), called a reconstructor sequence, in such a way that the proof that the configuration Ci gives reconstruction of G depends on the fact that previous configurations Cj (j< i ) also give reconstruction of G. The configurations which we use to prove the results in Section 2 are wheel-sequences. This technique, introduced by Hoffman [7], was used by Caunter [2] and by Swart [9] in the different context of the edge-reconstruction of bidegree graphs. These same two techniques, mutatis mutandis, are also the principal tools we use in establishing the results in the following section.

4. Graphs which triangulate surfaces In this context, we deal with the following two theorems. Theorem B. If a graph has connectivity 3 and triangulates some surface, then it is edge -reconstructible.

Edge-reconstruction of graphs with topological properties

287

Theorem C . Any graph which triangulates the real projective plane is edgereconstructible. The conditions of 4-connectedness and planarity in Section 2 allow us to make heavy use of embedding properties and of Kuratowski’s characterization of planar graphs. In general, these are no longer available for surfaces of higher genera. Hence, to prove Theorem B we have to use methods which do not involve considerations of embeddings. We do this by actually edgereconstructing a class % of graphs wider than the class of graphs which triangulate surfaces and have connectivity 3. The class %’ is the class of graphs with connectivity 3 and such that if G E % and a, b, c are three vertices whose deletion disconnects G, then a7 b, c induce a 3-circuit in G. The only connecting link between the graph theoretic properties of the graphs in % and the topological properties of graphs which triangulate a surface is that for any vertex v of a graph G which triangulates some surface, the subgraph of G induced by the neighbours of v is Hamiltonian. Whereas to prove Theorem B we are able to bypass all considerations of embeddings by working with the class %, to prove Theorem C an important part is played by the topological properties of the graphs under consideration. For example, to solve the problem of edge-recognition (i.e.7 to show that we can determine from 9(G) whether or not G triangulates the real projective plane), we make use of the Kuratowski-type theorem of Archdeacon [l]and Glover et al. [6] which characterizes projective graphs in terms of a set of 103 ‘forbidden subgraphs’. A simple example of a reconstructor sequence for graphs G with minimum valency 6 used to prove Theorems B and C is shown in Fig. 1, where labels correspond to valencies of the vertices in G. 6

6+1

s

6+1

Fig. 1

The above results appear in full detail in [4] and [5]. During the course of this conference, Lovisz [8] pointed out that our main theorem of Section 2 also follows from the following result he had just proved.

Theorem (Lovisz). If a graph with average valency greater than 4 and with a sufficiently large number of vertices has a Hamiltonian path, then it is edgereconstructible.

288

S. Fiorini, J. Lauri

References [l] D.S. Archdeacon, A Kuratowski theorem for the projective plane, Doctoral Thesis, Ohio State University (1980). [2] J. Caunter, Private communication. [3] S. Fiorini, On the edge-reconstruction of planar graphs, Math. Proc. Cambridge Phil. SOC.83 (1977) 31-35. [4] S. Fiorini and J. Lauri, Edge-reconstruction of 4-connected planar graphs, J. Graph Theory 6 (1982) 33-42. (51 S. Fiorini and J. Lauri, On the edge-reconstruction of graphs which triangulate surfaces, Quart. J. Math. Oxford 33 (2) (1982) 191-214. [6] H.Glover, J. Huneke and C. Wang, 103 graphs that are irreducible for the projective plane, J. Comb. Theory, Ser. B 27 (1979) 332-370. [7] D.G. Hoffman, Notes on edge-reconstruction of bidegree graphs, unpublished. (81 L. Lovisz, Private communication. [U] E. Swart, The edge-reconstructibility of planar bidegree graphs, University of Waterloo Res. Rept. Corr. 78-44 (1978).


CONSTRUCTING FINITE SETS WITH GIVEN INTERSECTIONS Peter FRANKL CNRS, Paris, France We show via a counter-example that Theorem 1 below cannot be extended to composite moduli.

1. Introduction

Let X be a finite set of cardinality n and 9 = { F l , . . . , F m ] a family of k-element subsets of X. For a subset L = { I , , . . ., I,} of (0,. . . , k - 1) the family 9 is called an (n, k, L)-system if for all 1 S i < j S m we have 1 F, r l F, 1 E L. The maximum cardinality of an (n, k,L)-system is denoted by m(n, k, L ) . This function was investigated by many authors. For an extensive list cf. [l]. Frank1 and Wilson [l] proved the following. Theorem 1. Suppose p is a prime, a ! ,. . . , a, and a . are s + 1 different residues modulo p such that k = a,, (mod p ) , every 1 E L is congruent to one of u I , .~., a, modulo p . Then:

In [l] the problem was raised as to whether (1) extends to composite moduli. For the case I L 1 = 1 a positive answer was given in [ 2 ] . In this paper we give a general construction which will yield in particular m(n,55,{3,6,10,15,21,28,36,45)) >

(y)

for n sufficiently large, i.e., Theorem 1 does not extend to p a x= 4, and a. = 1.

289

(2) = 6,

a l = 0, az = 3,

P. Frank1

290

2. Constructions

Let

p ( x ) = b o + b l x + b 2( i ) + * * - + b , ( T ) be a polynomial of degree t, where the b,’s are non-negative integers, b, 2 1. For a fixed positive integer n let r ( n ) denote the maximal integer r such that p ( r ) 6 n. We use the notation ( y ) = { F C Y :1 F I = i}. Let FI = {ff(l),. . . , f ; ( r ) } be pairwise disjoint sets, each of cardinality r, for i =O,. . . , t ; j = 1,. . . ,b,. For a subset D of I , ..., r we define

where Fj(D) = Cf;(d): d E D}. Thus G ( D )is a subset of cardinality p ( I D G({1,. . . 3 r } ) . By the construction it is evident that

I) of

G ( D ) n G(D’) = G(D n D’).

(3)

Proof. Let 9 = { D , ,. . . , D,,,} be an (n, k, L)-system consisting of k-subsets of {I,. . ., r } . Then %(9) = {G(D,), . . . , G(D,,,)}is a family of p(k)-element subsets of the p(r)-element set G({l,.. . , r } ) , and, in view of (3), %(9)is a (p(r),p(k),{p(l): 1 E L})-system of cardinality m, proving (4).

Now to show (2) let 9 consist of all the 11-element subsets of (1,. . . , r } containing 1, 2 and 3, i.e.

Let us choose the term (i)for p ( x ) . Then an application of Theorem 2 gives that for (5) s n < (‘>I)

( r - 3 ) - 16n4/8!> (i)

m(n,55,{3,6,10,15,21,28,36,45})a for n sufficiently large.

Constructing finite sets with given intersections

291

One can get counter-examples to (1) for prime-square moduli p = q 2 by applying Theorem 2 for

and p(x)=x’=x+2

(3

.

Then 1 G ( D ) J= l(q2), while the intersections G ( D )n G ( D ’ ) have cardinality belonging to (4’ - q ) / 2 other residue classes modulo g’. This construction will furnish a counter-example to (1) for large n whenever q 2.5,r = r ( n ) . It would be very interesting to know whether (1) remains true for composite moduli in the caSe I L I = 2. Singhi [3] observed that the methods of [l] yield a positive answer for prime-power moduli.

References [l] P. Frankl and R.M. Wilson, An intersection theorem with geometric consequences, Combin. 1 (1981). [2] P. Frankl and I.G. Rosenberg, A finite set intersection theorem, Europ. J. Combin. 2 (1981) 127-1 29. [3] N.M. Singhi, Personal communication, 1981.



S Y S ~ M EDE REFERENCE DE TREMAUX DWNE REPRESENTATION PLANE D’UN GRAPHE PLANAIRE H. DE FRAYSSEIX Laboratoire de Physique Marhkmatique. ColUge de France, I 1 Place Marcelin Berthelor, 7.500.5 Paris, France

P. ROSENSTIEHL Ecole des Hautes Etudes en Sciences Sociales. 751 70 Paris. Cedex 06, France The embedding of a planar graph G is geometrically determined for a given TrCmaux tree of G,by the choice, left or right for each half-line of the cotree, of a side of embedding according to a tree-chain. A characterization of the circular order at each vertex allows an effective embedding in linear time. It provides a new outlook concerning automatic network-design. of electrical networks in particular.

1. Introduction

L’objet du prCsent article est de montrer que I’implantation plane (sans intersections) d’un graphe planaire 3-connexe, est gComCtriquement dCterminCe, pour un arbre de TrCmaux donne du graphe, par la connaissance d’une fonction binaire, prenant les valeur ‘gauche’ et ‘droite’ dCfinie sur I’ensemble des brins d’aretes du coarbe associk. La valeur prise par cette fonction pour un brin donnC, reprtsente le c6tC d’implantation de ce brin par rapport 2 une chaine de I’arbre qui se dCfinit trks naturellement. En fonction de cette premiire implantation sommaire “Gauche-Droite” du coarbre par rapport A I’arbre on explicite totalement en chaque sommet I’ordre circulaire de tous les brins incidents, qu’ils soient d’arbre ou de coarbre. I1 s’ensuit que la recherche d’une implantation plane d’un graphe planaire 5 m arGtes et n sommets consiste simplement a fixer la valeur de 2(m - n + 1) variables binaires. (I1 en est effectivement ainsi de notre ‘algorithme Gauche-Droite pour le test de planarit6 et I’implantation dans le plan en temps lintaire’ [5]’.) Le systtme- de rCfCrence de TrCmaux fait apparaitre que le choix mEme de I’arbre de Tremaux est une question importante pour les techniques de t r a d

I L’algorithme est programmi en langage Fortran et PLI. II effectue sur calculateur IBM 370/168 I’implantation d’un graphe de 1O.OOO arites en un temps de I’ordre d’une seconde.

293

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H. de Fraysseix, P. Rosensriehl

automatique des plans, des schCmas Clectrique notamment. On propose. pour finir, une conjecture sur les traces de plans par segments parallkles. Pour les dtfinitions gtntrales sur les graphes, on renvoie Berge [l],pour I’ttude de I’implantation circulaire des brins aux travaux de Cori [2], et Fontet [3], e t pour le recours aux arbres de T r h a u x pour I’implantation plane 2 Hopcroft et Tarjan [6], et nos articles [4]et [9]. Dans la suite pour simplifier I’exposC, G dCsigne toujours un graphe 3connexe simple, G son implantation plane unique sur la sphere S ’ . V dCsigne I’ensemble des sommets et E I’ensemble des arktes de G et G ?I la fois.

2. Arbre de Tremaux d’un graphe 3-connexe

On dCsigne par T un arbre maximal de G, par V o ( T )les sommets incidents a une seule artte de T, dits sommets terminaux, et par VI(T) les autres sommets, dits sommets inte‘rieurs. On dCsigne par r,, un sommet arbitraire de G appelC racine, et on note < I’ordre partiel des sommets de G induit par I’arbre plantt (T.r0) dont ro est I’ClCment minimum. Pour tout sommet x # ro, on note alors PRE(x) I’arite unique de T incidente a x et a un sommet inftrieur a x. U n arbre de Tre‘maux est un arbre plant6 (T,rll)tel que, pour toute arite a du coarbre associC a T, les deux sommets incidents a a sont comparables (de tels arbres sont engendrts par l’algorithme de labyrinthe de Trtmaux [8], qui a pris aussi le nom de procedure Depth-First-Search [lo]). Toute arkte de I’arbre ou du coarbre a ses sommets incidents comparables, notts B ( e ) et H ( e ) , avec B ( e )< H ( e ) . Le sommet B ( e ) est appelC sommet bas, et H ( e ) sommet haut de l’ar2te e. Et selon la terminologie de A . Jacques [7] I? toute arkte e on associe deux brim (ou demi-arttes), le brin bas B B ( e ) et le brin haut B H ( e ) ,respectivement incidents 2 B ( e ) et H ( e ) . Par la suite on suppose toujours que (T,rl,)est un arbre de TrCmaux. Pour une ar@teu de I’arbre, on appelle: - fuseau de a l’ensemble F o ( a )des ar&tesa du coarbre telles que B ( a )= B ( a ) et H ( a )3 H ( a ) ; - fuisceau de a I’ensemble F’(a) des arites a du coarbre telles que B ( a )< B ( a ) et H ( a ) B H ( a ) . On note L la fonction ‘LOW’ dtfinie par Tarjan [lo] qui associe B toute ar&tede G un sommet, de la fagon suivante: pour e E T : L ( e ) = min{B(a) a E F 1 ( e ) j ; pour e c Z T : L ( e ) = B ( e ) (voir Fig. 1). On sait que I’hypothkse de 2-connexitC implique d’une part que la racine ro est terminale, c’est-ti-dire incidente ?I une arkte unique a. de I’arbre, et d’autre part que pour toute artte a de I’arbre, autre que ao, on a : L ( a ) < B ( a ) .

1

Systime de rkfirence de Tre‘maux

295

Fig. 1. Le Haut et le Bas definis par I’arbre de TrCmaux (ligne droite pour une arite d’arbre et ligne courbe pour une arCte de coarbre). P ( b )= g}; F ’ ( b )= {e. i, h } ; PRE(x) = b.

u,

L’hypothkse de 3-connexitC implique d’une part que I’unique ar&ted’arbre a,) incidente a rtl a une incidence commune avec une unique arkte d’arbre que I’on note a l ; et d’autre part le lemme suivant. Lemme 1. Pour route arkte d’arbre a, autre que aa et a t , il existe au moins deux arites d u coarbre a, p E F’(a), auec B ( a )< B ( P ) < B ( a ) (voir Fig. 1).

3. Systeme de refirence de Trimaux d’un graphe planaire

Afin de dkfinir ‘le cbtC dimplantation’ des brins, on va considkrer une premiere famille constituke des brim bas de coarbre, puis une seconde farnille constituie des brim hauts de coarbre et des brins bas d’arbre. Pour implanter la deuxikme famille, on dCfinira le systime de rifirence de Tre‘maux. Pour la premikre famille, la definition du c6tt d’implantation est immediate: tout brin bas de coarbre B B ( a ) , incident a un sommet autre que la racine, appartient i un fuseau F”(a),et B B ( a )est dit impfante‘ugauche (resp. i droite) si B B ( a ) est a gauche (resp. 6 droite) de la chaine constituee de I’arite PRE(B(a)) suivie de a. Ainsi a sCpare F”(a) en brins gauches et brins droits. Par convention, les brins bas incidents a la racine sont tous considCrCs conime Ctant implantis i gauche. Le Thkorkme T ci-aprbs dkfinit des conditions nkcessaires et suffisantes sur les c6tCs d’implantation des brins bas de coarbre pour qu’un graphe soit planaire. II

ff. de Fraysseix, P. Rosenstiehl

296

est une gtntralisation de notre thtorkme de caractkrisation des graphes planaires par un arbre de Trtmaux [4]. Sa dtmonstration sera donnie ulttrieurement.

Theorhe T. Etant donnks un graphe G, un arbre de Tre'maux, et l'ordre de demi-treillis < associe', une condition ne'cessaire et sufisante pour que G soit planaire est qu'il existe une partition des arites du coarbre en deux classes compatibles avec les trois relations ci-dessous, dkjinies sur l'ensernble des arites d u coarbre : Etant donnkes deux ar2tes a et P, (i) s'il existe y telle que: alors a et P sont dans une m&me classe ; (ii) s'il existe y telle que:

alors a et p sont dans des classes difkrentes ; (iii) si B ( a )= B ( P ) et si H ( a ) et H ( P ) sont non comparables. et s'il existe y e t S telles que: B ( y )= B ( 6 ) B ( a )= B ( P )< H ( a )A H ( P ) ,

alors a et P sont dans des classes difkrentes. De plus, si G est une reprksentation plane d ' u n graphe planaire, les deux classes dkfinies c i - d e w s repre'sentent la partition 'Gauche-Droite ' des brins bas dans chaque fuseau (voir Fig. 2).

(i)

(ii)

(iii)

Fig. 2 . Les trois types de relations pour deux brins bas (ligne droite pour une chaine d'arhre d i n e arete au mains, ligne courbe pour une artte de coarbre, petit rond pour un sommet, gros rond pour un sous-arbre de TrCmaux. Exception: dans (i), B ( a ) et B ( p ) peuvent etre confondus).

Sysdme de rifkrence de Trkmaux

297

En termes d e faisceaux le ThCorCme T implique les deux remarques suivantes: vu le Lemme 1, il ne peut exister trois arCtes d’arbre de mCme sommet bas et mCme LOW; si parmi les arites d’arbre de meme sommet bas l’arite a n’a pas le plus petit LOW, alors tous les brins bas du faisceau de a sont implant& d’un meme c6tC. II est facile de verifier que I’hypothirse de 3-connexitC implique que s’il existe parmi les ar&tesd’arbre de m&mesommet bas x, deux arites a et b de plus petit LOW, disons u, alors tous les brins bas du faisceau de a et du faisceau de b incidents B u sont implanth d’un mime c6tC. Maintenant, en vertu du thtoreme T. il apparait que I’un des deux faisceaux et pas I’autre, est implant6 des deux c6tCs: I’un a tous ses brim bas par exemple implantis B droite, et I’autre a tous ses brins bas incidents B u A droite et ses autres brins bas A gauche (voir Fig. 3). La prCcCdente distinction suggkre la definition naturelle du systirme de rCfCrence qui va permettre de dCfinir le cBt6 d’implantation des brins autres que les brins bas de coarbre. Le systime de rifirence associC B I’arbre de TrCmaux (T,T O ) est la donnCe, en tout sommet intCrieur x, d’une arCte REF(x) dite arite de rkfkrence, qui par dtfinition est I’arite d’arbre de plus petit LOW, parmi les arites d’arbres de sommet bas x, et s’il en existe deux, celle dont le faisceau n’est pas implant6 d’un seul c6tC. Le c6tk d’implantation d ’ u n brin haul de coarbre ou d’un brin bas d’argte d’arbre non de rCfCrence, incidents B un sornmet intCrieur x, est son cBtC d’irnplantation par rapport B la chaine constituie de I’arkte PRE(x) suivie de I’arite REF(x). Par convention, les brins hauts de coarbre incidents B un sornmet terminal sont tous considCrCs comme Ctant implant& B gauche.

Fig. 3. Implantation Droite et Gauche des brim (ligne droite flCchCe issue de x pour REF(x) G pour une implantation B gauche et D pour une implantation B droite). e et f sont des arttes basculies.

79x

H. de Fruysseix. P. Rosensriehl

Une ar2te de coarbre incidente A deux sommets inttrieurs et dont les deux brim sont implant& de c6tts opposCs est dite basculie pour le systitme de rtftrence choisi. Du thtorkme sur le croisement des faisceaux de deux aretes d’arbre de m2me sommet bas, dCmontrC en [9], dCcoule le Lemme 2.

Lemme 2. Tout brin bas d’arite d’arbre, non de rkfkrence, a mPme cdti d’implantation que les brins bas de son faisceau.

4. Explicitation de I’ordre circulaire

On dCduit les propositions suivantes du Lemme 2 et de la propriCtC tlkmentaire de l’ordre circulaire des rayons d’une roue implantte dans S’ (voir Fig. 4), h savoir: les brins incidents au centre de la roue ont un ordre compatible avec i’ordre des brins correspondants sur la jante.

Proposition 1. En un sommet inte‘rieur x de G les brins bas d’arttes d’arbre implantis h gauche (resp. h droite) apparaissent, entre PRE(x) et REF(x), dans le sens des aiguilles d ’une montre (resp. dans le sens contraire) selon les LOW strictement dkcroissants. Pour situer, dans I’ordre circulairc en x, les brins hauts et bas de coarbre incidents a x, il faut distinguer: (i) le paquet des brim hauts d’arites bascultes; (ii) les brins hauts d’ar&tesnon basculCes; (iii) les fuseaux de brim bas.

Fig. 4. Implantation plane de la roue (I’ordre des brins incidents au centre correspondant a I’ordre des brins incidents a la jante).

Sysrime de rifirence de Trimaux

299

Proposition 2. E n un sommet inte'rieur x, les brins hauts d 'arttes bascultfeesde G forment un paquet de brins place's tous Ci la suite, qui ne peut &re se'pare' du brin bas de R E F ( x ) que par des brins bas du fuseau de R E F ( x ) . Proposition 3. En un sommet x de G, les brins bas d'arttes d'arbre autre que R E F ( x ) et les brins hauts d'arttes de coarbre non bascule'es, implantis Ci gauche (resp. d droite), apparaissent 6 partir de P R E ( x ) dans le sens des aiguilles d'une montre (resp. dans le sens contraire) selon les LOW de'croissants, auec prioritk, d LOW e'gal, d u brin d'arbre sur celui de coarbre (voir Fig. 5).

Dans la proposition suivante, on entend par parcours de I'arbre T dans G selon la rkgle 'premiere ii gauche', un parcours des aretes de T une fois dans chaque sens avec, en chaque somrnet, le choix du premier brin A gauche de celui prkckdernent parcouru. Un tel parcours rencontre chaque brin de coarbre une fois. Proposition 4. Entre un brin bas du fuseau F " ( a ) et le brin bas de a , il ne peut apparaitre dans G que d'autres brins bas de F"(a). Deux brins bas B B ( a ) et

Fig. 5. Ordre circulaire des brins en x selon les LOW dicroissants partir de PRE(x) (chaque ar&te d'arbre est entourie de son fuseau. les autres plages hachurkes correspondent au faisceau de PRE(x)).

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B B ( a ’ ) d’un mtme fuseau F ” ( a )apparaissent au sommet B ( a ) a la suite dans le sens contraire des aiguilles d’une montre, si B H ( a ) et B H ( a ’ ) sont les brins hauts du fuseau rencontris a la suite dans un parcours de l’arbre selon la rigle ‘premiPre a gauche’. Pour a = ao, la Proposition 4 dCfinit I’ordre circulaire h la racine (voir Figs. 4 et 5).

Conclusion Les Propositions 2, 3 et 4 correspondent bien h I’objectif vise: dtfinir I’ordre circulaire des brins incidents B chaque sommet du graphe 3-connexe G en fonction de la simple donnCe de I’implantation ‘gauche’ ou ‘droite’ des brins de coarbre incidents B un sommet inttrieur. Si G n’est pas 3-connexe, la definition du systkme de rCfkrence demeure, mais les rhgles d’ordre circulaire ne donnent que les implantations dites de TrCmaux [!I L’algorithme ]. [ 5 ] en exhibe toujours une. Les autres implantations font infraction h la r&glede la Proposition 4 concernant I’implantation du fuseau Fo(a) autour de BB(a). La Proposition 1 met pour sa part en valeur que I’implantation de l’arbre se diduit immkdiatement des c6tCs d’implantation des brins bas du coarbre, fait fondamental pour les algorithmes d’implantation plane en temps lineaire recourant B un arbre de TrCmaux ( [ 6 ] ,[S] et [lo]). Pour terminer on Cvoque un problkme concernant le track des plans qui sera I’objet d’un prochain article.

ProbKme (Marcel Dassault)2: Comment frouuer dans tout graphe planaire un systime de rifirence de Trimaux vis-a-vis duquel aucune artte n ’est basculke. Le choix de I’arbre de TrCmaux semble jouer un rble essentiel dans le trace des plans. En particulier I’absence de bascules autorise l’implantation effective d’un reseau Clectrique dont les noeuds et connexions sont repr6sentks par des segments parall5les du plan (voir Fig. 6). Nous avions pose en conjecture le sous problkme suivant.

Probleme des segments. Tout graphe planaire esf reprisentable par deux familles de segments du plan, tous disjoints, paralliles dans chaque famille, chaque segment de la seconde famille (les arktes) &ant incident ci deux segments de la premiire famille (les sommets). Cette propriitk des arbres de Tremaux nous est apparue en observant la clarte des traces empiriques de circuits ilectriques de Mirage B I’Usine Marcel Dassault.

301

Sysrirne de rifkrence de Trimaux 3

3

Fig. 6. Reprisentation d’un graphe planaire par deux familles de segments parallkles: (a) donnie initiale du graphe; (b) implantation plane avec systkme de reference de Trimaux sans ar@tebasculie; (c) reprisentation du meme graphe par des segments horizontaux pour les sommets, et des segments verticaux pour les arktes.

Cette derniere conjecture a CtC rCsolue par Aa, Hamidoune, Las Vergnas et Meyniel [12]. Nous en donnerons une construction effective en temps IinCaire dans un prochain article.

References [ 11 C. Berge, Graphes et Hypergraphes (Dunod, Paris, 1970). 121 R. Cori, Un code pour les graphes planaires et ses applications, Astirisques (27) 1975. 131 M. Fontet. Connectivirt des graphes et automorphismes des cartes: propriitis et algorithnies, Thbse, Universitt Paris VII, no 79-23, ma; 1979. [4] H. de Fraysseix and P. Rosenstiehl, A depth-first-search characterization of planarity, in: Ann. Discrete Math., Proc. Conf. on Graph Theory at Cambridge University (North-Holland, Amsterdam, 1982) to appear. [5] H. de Fraysseix and P. Rosenstiehl, The left-right algorithm for planarity, testing and embedding in linear time, to appear. [6] J.E. Hopcroft and R.E.Tarjan, Efficient planarity testing, J. Assoc. Comput. Mach. 21 (1974) 549-568.

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171 A . Jacques, Sur le genre d'une paire de substitutions, C.R. Acad. Sci. Paris 267 (IYhti) 625-627. IS] P. Rosenstiehl, Les mots de Labyrinthe, in: Colloque sur la Thiorie des Graphes, Bruxelles, Cahiers du C.E.R.O. 15(3) (26-27 avril 1973) 245-252. [9] P. Rosenstiehl. Preuve algibrique du critere de planariti de Wu-Liu. Ann. Discrete Math. Y (1980) 67-78. [lo] G. Penaud, Algorithmes de planaritt. in: J.C. Bermond et R. Cori, eds.. Journkes de Combinatoire et Informatique, Universiti de Bordeaux, IY75. [ I t ] R.E. Tarjan. Depth-first-search and linear graph algorithms. SIAM J. Comput. 2 (1Y72) 146-160. 1121 P. Aa, Y . Hamidoune, M. Las Vergnas et €3. Meyniel, Communication au seminaire de C. Berge-P. Rosenstiehl du 11.01.82.

Annals of Discrete Mathematics 17 (1983) 303-306 0 North-Holland Publishing Company

SUR UN PROBLEME D~ERDOSET HAJNAL Guy GIRAUD Bellevue A I , 142 Bd Paul-Claudel, 13010 Marseille, France We present two graphs with no K,, of orders 16 and 15, such that any 2-colouring of their edges displays a monochromatic triangle.

1. Introduction

Nous disons qu’un graphe G posskde la propriCtC 9 ou nous Ccrivons G + ( K 3 ,KJ, pour exprimer que tout bicoloriage de ses ar&tesfait apparaitre un triangle unicolore. w ( G ) , nombre de clique de G, est I’ordre maximal d’un sous-graphe complet de G. xp( G )est le nombre minimal de classes entre lesquelles on peut rCpartir les sommets de G de fagon qu’aucune classe ne contienne un K, ;x 2 ( G )= ,y(G) est le nombre chromatique de G. En riponse i un problkme d’ Erdos et Hajnal [l], Graham a obtenu en 1968 le plus petit graphe sans K , vCrifiant 9, soit lejoint C5+ K,, Folkman a prouvC dans un article publiC en 1970 I’existence de graphes sans K , vkrifiant P, et ce resultat a CtC Ctendu en 1976 par NeSetiil et Rod1 au cas d’un nombre fini quelconque de couleurs. Pour ce qui est des graphes sans K , vCrifiant 9, Irving [3], retrouvant une idCe de Pbsa e t la mettant en oeuvre, en a obtenu un d’ordre 18.’ Nous montrons que I’ordre minimal d’un tel graphe est compris entre 11 et 15, et indiquons qu’on ne peut descendre en dessous de 13 en suivant la mCthode de P6sa et Irving.

2. Un graphe sans

K5,d’ordre 16, vkrifiant 8

Dans Cs+ K,, dCsignons par a, b, c les sommets du K,. Au moins 3 arites joignant a au C, sont de m&me couleur, bleues par exemple. Si ab est bleue, I’argument classique montre qu’il y a un triangle unicolore. Pour I’Cviter, il faut que ab (et ac pour la meme raison) soit rouge; mais alors le triangle abc est rouge car ce qui est vrai pour a I’est pour b. I

Nous apprenons que Nenov et Hadiiivanov [4] en ont trouvi un d’ordre 16.

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304

Considirons maintenant les trois chemins P = abb‘, P’ = b‘bcc‘ et P“ = c’ca, et les joints P + Cs, P ’ + C{ et P“+ C;’dont I’union (les trois Cs sont disjoints; voir Fig. 1) est un graphe sans K5d’ordre 20 ayant 70 arites. Comme il a CtC expliquC, pour Cviter d’zmblCe un triangle unicolore, il faut que les trois chemins soient unicolores; mais alors le triangle abc est unicolore. I1 est ensuite possible de choisir deux points inddpendants dans chaque Cs, soient x et y, x ’ e t y‘, ~ “ ey”, t et d’identifier x, x‘, ~ ” a i n sque i y, y ‘ , y”, sans faire apparaitre un K s dans le graphe d’ordre 16 qui en rCsulte et qui virifie toujours

9.

Fig. I .

3. Un graphe sans K,, d’ordre 15, virifiant 9

La construction est ici bas& sur I’argument de P6sa2 que si x3(G)= 3 , alors G + K , vCrifie 9,et sur la remarque que dans toute (xl = 2)-coloration des sommets de C, les sommets du ont nkcessairement meme couleur. Le graphe d’ordre 19 ayant 48 argtes constitut par trois Csdisjoints, joints (au sens du joint) h a et respectivernent aux sommets b, b’, b” d’un K , (voir Fig. 2) a clairement w = 3 et x1= 3 . II est ensuite possible de choisir comme il est indiquC sur le figure des sommets x, y, z, t sur C,, x’, y’, u sur C;. z‘, t ‘ , u’sur Cg, et d’identifier x et x’, y et y’, z et z ’ , t et f’, u et u’ sans faire apparaitre un K4 dans le graphe G d’ordre 14 qui en rCsulte et qui a toujours x;= 3. Par suite, en joignant ce graphe 2 un 15‘ sommet, on obtient un graphe G + K , sans K , qui vtrifie 9.

+E.

Argument non publit, retrouvt indtpendamment par Irving.

Un problime d’Erdos et Hajnal a

b’

Fig. 2

4. Perspectives

Tout graphe vkrifiant B doit Cvidemment avoir commode d’introduire la suivante.

x 3 6 et

,y7* 3. II est

Definition. Quels que soient les entiers p , k , c tels que p, k 2 2 et p + 1 c c s ( p - I)k + I , f(xpd k , w 3 c ) est le plus grand entier n tel que tout graphe d’ordre n ait, soit ,yp d k , soit w 3 c. L’existence de ces seuils dkcoule pour p = 2 des constructions de Tutte, de Zykov, de Mycielski, et pour p > 2 d’un th6orirme d’Erd6s et Rogers [ 2 ] ou d’une construction de Sachs et Schauble [ 5 ] .Sous une forme voisine, le problirme de leur estimation pour p = 2 a Ct6 signal6 par Erdos en 1969 (voir The Art of Counting, p. 115). Etant donne que f(xlS 2, w 3 5 ) = 7 + K , ) = 3), que f ( x 5, w 3 5) = 9 et que le seul graphe d’ordre 10 ayant ,y = 6 et w = 4 est C, + C, pour lequel ,yl = 2, tout graphe sans K , vCrifiant 9’ doit i3re d’ordre au moins Cgal 2 11. P6sa, alors i g k de 13 ou 14 ans, s’Ctait limit6 2 I’existence de f(xl G 2, w 3 4); dans le paragraphe pricedent, nous avons major6 ce nombre par 13. Nous

(,y3(c

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G. Giraud

appuyant sur le risultat f ( x 6 4,w 3 4) = 10, de dimonstration pCnible quant la minoration, nous avons Ctabli que f(x3=z2 , w 2 4)3 11, ce qui signifie que tout graphe sans K4 ayant x3 = 3 a au moins 12 sommets; la mithode de P6sa et Irving ne permet donc pas d e construire un graphe sans K s virifiant 9’et ayant moins de 13 sommets.

RCferences [ l ] P. Erdos et A. Hajnal, Research problem 2-5, J. Comb. Theory, 2 (1967) 105. 121 P. Erdos et C.A. Rogers, The construction of certain graphs, Canad. J. Math. 14 (1Y67) 702-707 (reproduit dans The Art of Counting, 418-323). [3] R.W. Irving, On a bound of Graham and Spencer for a graph-coloring constant, J. Comb. Theory, Ser. B 15 (1973) 2GU-203. [4] N.D. Nenov et N.G. Hadiiivanov, The Graham-Spencer number, C.R. Acad. Bulgare Sci. 32 (2) (1979) 155-158 (en russe). [ 5 ] H. Sachs, Finite graphs, in: W.T. Tutte, ed., Recent Progress in Combinatorics, Proc. 3rd Waterloo Conf. Combinatorics, (1968) 175-184.


LA k-ARBORICITE LINEAIRE DES ARBRES M. HABIB and B. PEROCHE C.N.R.S. G r o u p de Recherche 22 associC a I’VniversitC Pierre et Marie Curie, 4 place Jussieu, 75230 Paris, Cedex 5, France We denote by a linear k-forest a graph whose connected components are chains of length less than or equal to k. We define the linear k-arboricity of a graph G as the minimum number of linear k-forests that partition the edges of G. We study this notion for the trees and ask some questions for other classes of graphs.

1. Introduction

Dans ce qui suit, nous considCrons des graphes finis, non orientis sans boucles ni arites parallkles. Les termes non dCfinis ici sont dCfinis dam Berge [2j. Harary, dans [4],a introduit la notion de forit linCaire comme Ctant un graphe dont les composantes connexes sont des chaines CICmentaires. Cette notion conduit B un invariant, not6 la(G) et appelC arboricitk linkaire, se dkfinissant comme le nombre minimum de forkts IinCaires qui partitionnent les arites de G. L’arboricitC IinCaire a CtC CtudiCe en particulier par Akiyama, Exoo et Harary dans [I]. Un problkme classique de la thCorie des graphes est celui de la coloration des arites d’un graphe. Or si nous considirons les arites du graphe d’une couleur donnee, nous obtenons ainsi une forit IinCaire particulikre dont toutes les chaines Cltmentaires ont pour longueur 1. Cette remarque nous a conduit A reprendre I’Ctude des forits IinCaires en ajoutant des contraintes aux composantes de ces forits: si k E N’ nous appelons k-forit line‘aire une forkt de chaines ClCmentaires dont la longueur est infCrieure ou Cgale B k. Ainsi pour un graphe G = ( X , E ) avec IX I = n, nous dCfinissons n - 1 invariants, notis lak(G), 1 s k d n - 1 , appelCs k-arboricitt IinCaire de G, et dCfinis comme le nombre minimum de k-forkts qui partitionnent les arites de G. Lorsque k = 1, nous retrouvons I’indice chromatique et pour k = n - 1 I’arboricitC IinCaire. Dans ce papier mis 2 part quelques propriCtCs gCnCrales ClCmentaires nous nous intkressons essentieliement aux arbres. Nous montrons que si le degrC maximum A ( T ) est impair, alors lai ( T )= [A(T)/21pour i 3 2. La situation est 307

M. Habib, B. Piroche

308

plus complexe lorsque I’arbre considirk T a un degrk maximum A(T) pair. Nous montrons dans ce cas:

En outre, parmi ces arbres nous caracthisons ceux pour lesquels nous avons: la2(T)=

do .

Signalons que dans [3] avec Fouquet nous donnons des risultats sur ces invariants pour d’autres classes de graphes, par exemples les cubiques.

2. Definitions et proprietes Clernentaires

Soit G

= ( X ,E

) un graphe simple non orient6 fini, nous notons

A(G)=max{d(x)}, XEX

n = J X I et

rn = ( E l .

Par q ( G ) nous notons I’indice chromatique de G, c.a.d le nombre minimum de couleurs nicessaires pour colorier les arCtes de G sans que deux arites de mime couleur soient adjacentes. Enonqons tout d’abord quelques risultats CICmentaires, sans dimonstration. Proposition 1. Si G a pour composantes connexes G , , . . .,G p alors

Vi 3 1, l a i ( G ) = max {la, (Gi)}. IS,SP Si H est un sous-graphe partiel de G,alors Vi 3 1, la, (H)s la, (G). Dans la suite nous ne considirons donc plus que des graphes connexes. Proposition 2. La suite des lai est monotone de‘croissante

q(G) = la,(G)a la,(G)s

*

3 Ian-,(G) = la(G).

Rappelons deux rksultats concernant q ( G ) et la(G): A(G)Sq(G)SA(G)+ 1 ( t h e o r h e de Vizing [6]) et

La k-arborkit6 liniaire des arbres

309

Akiyama, Exoo e t Harary ont pose dans [l] la conjecture suivante:

Ces deux rksultats e t la proposition 2 fournissent donc des bornes pour les invariants lai (G), mais nous pouvons amCliorer la borne infhieure: Proposition 3. V i > 1,

la,(G)z=max[[

{’[m]}.

do 2

i+l Corollaire. Si G est k-re‘gulier, V i > 1,

3. RCsultats sur les arbres Nous allons distinguer deux cas suivant le degrC maximum de I’arbre CtudiC.

Theoreme 1. Soit T = (X, E ) un arbre ve‘rifiant les deux conditions : (i) il existe au plus un sommet xo E X , de degre‘ 2k + 2 , (ii) V x EX, x # x o , d ( x ) G 2 k + I . Alors l a z ( T ) s k + 1. Preuve. Nous allons raisonner par induction sur k. Pour k = 0, T est alors une chaine d e longueur infCrieure ou Cgale A 2 e t le rCsultat est trivial. Supposons k > 0. Pour Ctablir I’induction nous allons utiliser une procedure qui partitionne les aretes de T en une forkt F d e chaines d e longueur c 2 et un for6t d’arbres A , , . . . , A , sur lesquels nous pouvons appliquer I’hypothkse d’induction au rang k - 1. II nous faut donc maintenant Ctudier trks prCcisemment cette proctdure avant de revenir sur la dimonstration. Procedure BIFORET( T, xo)


310

- tant qu’il existe x E P tel que f ( x ) n (X- P u M) # 0 faire dCbut C + T ( x )n ( X - P U M ) (a) Si 1 C I* 2 alors faire dCbut choisir y, z E C,y # z , M + M u {Y, 2 ) F+FU{[YJl,[xJll B +{x, y , f }

fin (b)

c ={Y}

Si f ( y ) n (X- P u M ) # 0 alors faire dkbut choisir z E T(y) n ( X - P U M) M + M u {x, 2 ) F + F U { [ L y l , [Y,21) B +{x, 2) Y 7

P+PU{Yl

fin Sinon faire dCbut

M +M u { x , y ) F + F U {[x,Y 1)

L

fin Pour Vt E B faire dCbut pour Vz E r(t)fl ( X - P U M ) P+PU{z} A + A u { [ t ,211 fin B +0 fin fin.

Soit T = (X, E ) un arbre vCrifiant les conditions du T h C o r h e 1, avec k 3 1. I1 existe donc un sommet x o E X , tel que d ( x o )= A ( T )a 3, examinons dans le detail ce que fait cette procedure si nous l’appliquons au couple (T,x,)). (1) La procCdure utilise un double marquage des sommets par P (plus) et par M (moins). La partie centrale de la procCdure est constituke d’une boucle ‘tant que’. A chaque itiration de cette boucle un sommet au moins est marqui, et les nouveaux sommets mhrquCs sont empiles dans une pile B,qui est vidCe avant de passer h I’itCration suivante. La proctdure se termine donc nCcessairement en au plus n = IX I itirations; remarquons qu’une artte de T est considCrCe au plus deux fois au cours de la

L a k-arboricirk liniaire des arbres

311

procedure et donc celle-ci utilise un nombre d’opkrations Clkmentaires linCaire en n. En outre, T Ctant connexe, A la fin de la prockdure tout sommet est marque par P ou M . (2) Vx EX, x est incident h au plus une chaine de F. En effet si x est incident a une chaine de F, x a CtC marqut au cours d’une itkration de la boucle ‘tant que’ et au cours de cette iteration toutes les arites incidentes A x ont leur autre extrCmitC marqube. Nous avons alors T ( X ) ~ ( X - P U M ) = 0 et ainsi nous ne pouvons plus choisir de mettre dans F une autre chaine incidente I? x. (3) A et F partitionnent les ar&tesde T. Supposons qu’il existe une arite e = [x, y ] telle que e G! A U F. Comme T est un arbre, e sCpare T en TI et T2avec par exemple x E TI et y E T2.Supposons que x soit marqut avant y ; tous les sommets de T sont marquis au cours de la procidure, donc aussi y, mais l’unique chaine de rnarquage emprunte nkcessairement l’arite e et donc e E A U E Les arites de T sont donc partitionnkes en une forit de chaines de longueur < 2 ( F ) et une autre forit ( A ) . (4) Vx EX, si x est non pendant alors il est incident au rnoins une arite de F. Supposons qu’il existe x, d ( x ) s 2 , non incident Q une arite de F et soient y l # y , deux voisins de x, d’aprbs 3, [x, y l ] et [x, yz] appartiennent A A . Comme un sommet n’est considCrC qu’une fois comrne sommet ‘x’ de la boucle ‘tant que’, car aucun marquage n’est enlevC, nous pouvons affirrner que les deux arites prCcCdentes ont CtC affectkes a A, au cours de I’itCration concernant le sommet y l (ou un de ses voisins different de x ) ou celle concernant le sommet y? (ou un de ses voisins different de x). En effet au cours de l’ittration concernant x, soit T ( x )n ( X - P U M ) = 0 et donc dans ce cas aucune arite n’est affectie a A, soit x adrnet un voisin non marquC, dans les deux cas de la procedure, on affecte au moins une arEte incidente I? x, a la forit F, et ceci est exclu par I’hypothbse sur x. Ainsi il existe deux chaines distinctes joignant x0 A y~ et xo a y ~ ce . qui est absurde car cela implique un cycle dans l’arbre T. Notons A o , A , ,..., A, les arbres qui constituent la forCt A, en supposant x,,E A,,. D’aprbs 4, Vx E X , da,(x)C2 k . ( 5 ) Si A, contient un sornmet x, non pendant (dans A,), extrimit6 d’une chaine de F, alors: xoE A, (i.e., i # 0) et il n’existe pas dans Ai d’autre somrnet possedant cette propriCtC. Montrons tout d’abord que AO ne peut contenir de somrnet non pendant extrCmitC d’une chaine de F. En effet s’il existait un tel sornmet y, considdrons p I’unique chaine de A,, joignant xII a y. Posons p = [x0,xt,. . . ,xh = y ] , au cours de la proctdure les sornrnets de cette chaine sont marques P, en outre I’ordre de marquage est celui des indices croissants. Ainsi le sornmet y a CtC marque P lors


312

de I’itCration concernant le sommet X h - I . Au dCbut de I’itCration concernant y, nous avons If ( y ) f l ( X - P U M)I 3 2 , cornme y est par hypothkse non pendant dans A,,et qu’il existe une arete de F incidente ii y . Mais alors dans un tel cas (a) la proctdure choisit une chaine de longueur 2 dont le milieu est y. qu’elle affecte ii F, ce qui contredit I’hypothkse initiale. Etudions maintenant le deuxieme point et supposons qu’il existe x , y deux sommets de A , , i # O , tous deux extrCmit6 d’une chaine de F et non pendants dans A , . Si x est marque avant y dans la procedure (i.e., x est marquC dans une itCration prCcidant celle ou y est marquC). La chaine de marquage est donc I’unique chaine de T n A, joignant x et y. Donc y est marque avant tous ses voisins, except6 celui appartenant 6 la chaine mentionnee ci-dessus. Comme y est non pendant dans A, et extrkmitt d’une chaine de F. lorsque y est marque nous avons alors: If (y ) n ( X - P U M)I a 2 . Mais alors nicessairement au cours de la proctdure y est choisi comme le milieu d’une chaine de F. d’oh la contradiction. Si x et y sont marques au cours de la meme itCration de la procedure et comme ils appartiennent au meme arbre A , , il existe donc un voisin commun z a x et y. re1 que [ x , z ] E A, et [ y , 21 E A , . x et y sont donc marques P. Mais alms ils ne peuvent itre extremite d’une chaine de F, qu’ii la condition: f ( x ) = {x’} et T ( y ) = ( y ’ } e tcela entraine [ x , x ’ ] f Fet [ y , y ’ ] E F et d A , ( X ) = d A , ( y ) =1, d’ou la contradiction. (6) V i 3 0 , A, admet au plus un sommet de degrC 2k. C’est une conskquence immkdiate de 5: en effet, x # X I , et d A t ( x=) 2 k implique que x est extrimit6 d’une chaine de F. ( 7 ) Parmi les somrnets marquCs M de A, il y a au plus un sommet y tel que dA, (Y ) 1. En effet un sommet marqut M est nicessairement extrCmitC d’une chaine de F.

’

Fin de ta preuve du Theoreme 1. Si T posskde un sommet xn teI que d ( x o )= 2 k + 2, alors nous appliquons la proddure BIFORET(T, x,) qui nous fournit une forct de chaines de longueur S 2 : F, et une forst d’arbres A[),..., A,. D’apres 6, les arbres Ai vCrifient I’hypothese de rCcurrence et nous pouvons donc partitionner les aretes de T e n k forits de chaines de longueur s 2 . 0 Nous avons en fait obtenu un rCsultat ICgkrernent plus fort que celui annonce au ThCoreme 1. Ce rksultat nous est utile par la suite et s’Cnonce comme suit.

Corollaire 1. Soit T un arbre ue‘rifiant les conditions du The‘orZme 1, il existe une partition des argtes de T e n au plus k forits de chaines de longueur S 2 telle que tout sommet soit extre‘mite‘ d’au plus deux chaines.

La k-arboriciti liniaire des arbres

313

Preuve. Remarquons que lors de l’application de la procCdure BIFORET( T,xo) les sous-arbres A, qu’elle engendre posskdent au plus un sommet non pendant marqut M et qu’en outre si A, posskde un sommet de degrt 2 k , ce sera ce sommet. Ainsi, nous pouvons rCutiliser la procCdure pour les couples (Ao,xo) et (A,, y, ), i > 0, ou y, est l’unique sommet marquC M de A, lorsqu’il existe et n’irnporte quel sommet sinon. Nous appliquons donc de manikre recursive la proctdure comme indiqut ci-dessus, aux arbres qui constituent les for& TI = T, T2= TI - F , , T,, .. ., T = T,-I- F , - l , .. . (oa T, - E - , est le sous-graphe partiel obtenu a partir de T, en enlevant les arites de la forit E-l). Montrons qu’un sommet x de T est au plus extrCmitC de deux chaines des forits E. Soit E la premikre forit pour laquelle x soit extrCmitC de chaine, deux cas sont a considCrer: (1) x est marquC P et M dans la procCdure BIFORET(?;) qui permet d’obtenir E . Donc nous avons dT, (x) = 2 , donc dT,+I(x) 6 1et x ne peut plus i t r e extrtmitt d’une chaine de F,, j > i. (2) x est marquC M. Si dT,(x) = 1, alors x f f T,,, et donc ne peut &re extrCmitC de chaine pour F,, j > i. Sinon x sera le seul sommet de sa composante connexe de T,,, B 6tre marquC M. D’aprks la procidure, il sera au plus une fois encore extrtmitC d’une chaine (s’il existe j 5 i 1 tel que dT,(X)= I), d’ou le rtsultat. 0

+

Corollaire 2. Soit T un arbre ayant au plus un sommet de degre‘ 2 k + 2 et au moins un sommet de degre‘ 2k + 1. Alors V i 2 2 ,

la,(T)= k

+ 1.

Preuve. I1 suffit d’appliquer le ThCorttme 1 et les Propositions 2 et 3. 0 Corollaire 3. Soit T un arbre de degrk maximal 2 k . Alors V i 2 2,

k s lai (T) S k

+ 1.

Preuve. I1 suffit d’ajouter une ar6te pendante un sommet de degrt 2 k de T et d’appliquer le thCorkme prCcCdent 6 I’arbre ainsi obtenu. Corollaire 4. Soit T u n arbre posse‘dant au plus un sommet de degre‘ 2 k + 2 et au moins un sommet de degre‘ 2 k 1, il existe un algorithme utilisant O ( k n ) ope‘rations e‘le‘mentaires qui calcule une partition optimale des arites de T e n forits de chaines de longueur G 2 .

+

Preuve. I1 suffit d’appliquer rkcursivement la prockdure vu qu’elle Ctait lintaire en n. 0

BIFORET,

car nous avons

314


Nous allons maintenant pricker un peu la situation pour les arbres T = (X, E), tels que A (T) = 2k et possCdant plus d’un sommet de degrC 2k. Parmi ces arbres nous caractkrisons ceux pour lesquels la2(G)= k.

Definitions. xo E X est dit extrimit6 d’une chaine k-chargke s’il existe une chaine [ x o , x I , . , ,Xh], h 2 1 vCrifiant: dT(xh)= 2k et Vi E [l, h - 11, d r (x,) = 2k - 1. xlI E X, est appelC un k-centre, s’il est extrimiti de p chaines k-chargies et virifie: dT(x) = p + 9 avec p + [qD] > k. Nous sommes maintenant en mesure d’inoncer le deuxikme thiorkme.

Theoreme 2. Soit T un arbre de degre‘ maximum 2k. Alors a2(T)= k si et seulernent si T n’admet pas de k-centre. Preuve. (1) On suppose que la2(G)= k. Supposons que T contienne une chaine k-chargCe d’extrCmit6 x,,, p = [x,,, . . . ,x,]. Par hypothkse, d(x,) = 2k, donc toutes les forits de chaines F, partitionnant T traversent x,. En particulier, il existe une forit de chaines Fh qui p. Comme d(x,-,) = 2k - 1, contient [xI-,,xA,y ] ou y est un voisin de xAet les k - 1 forits autres que Fh traversent XA-1. Donc il existe une forit de chaines Fh’ qui contient [ x A - 2 , x r - ] ,z], oh z est un voisin de x,-~ et z P p. De proche en proche cette propriCtC se transmet jusqu’en xu, qui est donc extrimiti d’une chaine [ X , ~ x, I , t ] d’une forEt F h . , avec t voisin de x I et t t f p. Ainsi, i chaque chaine k-chargie d’extrimiti x,,, on peut associer une forit E contenant une chaine de longueur 2 i laquelle appartient la dernikre ar2te d’extrimitk xlI de la chaine k-chargCe. Si on suppose que T admet xII pour k-centre, avec p chaines k-chargCes d’origine xII, il y a p forits utilisies par ces p chaines. Les 9 autres arites incidentes i xo nicessiteront au moins 14/21 autres for&, donc la2(T)z p + [9/2] > k, puisque x,, est un k-centre, d’oh la contradiction avec I’hypoth6se la2(T) = k. (2) On suppose maintenant que T ne posskde pas de k-centre. On va faire un raisonnement par rkcurrence sur le nombre p de sommets de degrC 2k de T. La propriCtt est vraie si p = 1 (c’est le ThCorkme 1). Soit T = (X, E ) un arbre ayant p > 1 sommets de degrC 3k, et soit xIIun tel les voisins de xll. Dans ce qui suit, on va chercher sommet. Appellons x,, , . . , h construire un sous-arbre de T, A, n’ayant que xlI pour sommet de degrC 2k. Soit i donne, 1 C i s 2k. On retire l’arkte xI,x,, crCant ainsi deux arbres; on note T celui qui contient x,. Zer cas. Si T ne contient aucun sommet de degrt 2k, on remplace T par

ye

La k-arboricite‘ IinPaire des arbres

31s

T - T, oh T - T, est le sous-graphe partiel obtenu a partir de T e n supprimant a T les ar2tes de T,. Ztme cas. T contient 9 3 1 sommets y l , . . . , y4 de degrt 2 k . T Ctant un arbre, pour tout j , 1 S j S q, il existe une chaine unique p, = [XI,, . . . , y,] dans T. Pour chaque chaine p,, on dCfinit un sommet t, comme Ctant le sommet de p, de degrC S 2 k - 2 , le plus proche de x0. Comme T n’admet pas de k-centre un tel sommet existe pour tout j , mais naturellement les sommets t, ne sont pas nicessairement tous distincts. Pour j allant de 1 a 9, on effectue la procCdure suivante, admettant deux sous-cas: (a) Pour un indice j tel que V j ‘ E [1,9], ti # t,, on remplace alors le sommet t, par deux sommets t i et t‘,!,distincts. On joint a t’,!le sommet de la sous-chaine [ t , , . . . , y,] de p,,qui Ctait adjacent a t,. On joint 2 r: tous les autres sommets qui Ctaient adjacents 5 t,. T, donne donc naissance ti deux sous-arbres Ti et T;’qui contiennent respectivement t ; et t ; ’ , et I’on remplace T par T - T ; . (b) Pour un indice j tel que J, = { m E [l, 91 t,,, = t,} vCrifie IJ, 13 2 . Posons .. J, = {jl,= 1 . ~.~. ..,j,}, r 2 1. Deux cas sont possibles: (a) Sur la chaine P , ~0, S m s r, il existe a u moins un sommet z,, de degrC S a 2 k - 2 entre t,, et y,,. On remplace alors z,, par deux sommets zimzet z’,!, en joignant a z’,’ le sommet de la sous-chaine [z,, . . . , y,,] de p,, qui Ctait adjacent a z,, et tous les autres voisins de z, sont affect& 2 z ;,. Soient Ti, et TYmles deux sous arbres de T, ainsi obtenus, ils contiennent respectivement zin,et z;’,. On remplace T par T - T’,!,,,, (p) II n’existe pas de sommet de degrC S 2 k - 2 entre ti, et y,, sur la chaine p,,. Soit s le nombre d’indices jm pour lesquels on a cette propriCtC. On rernplace alors t, par s + 1 sommets t : , . . . , r ; + ’ . Le sommet t r est reliC au sommet de la sous-chaine [ t , , . . . ,y,,] de p,, qui Ctait adjacent a t,, et ceci pour V w E [ l . s ] . Tous les autres voisins de t, sont affectis au sommet t : ” . T engendre ainsi s + 1 sous-arbres T : ,. . . , Ti+’(oh Tycontient le sommet t,”). On remplace alors T par T - ( U,,,,,T:). Aprks avoir effectuC les opCrations prCcCdentes pour tout i E [ I , 2 k ] , I’arbre T ainsi obtenu (notons le A pour Cviter toute confusion) ne posskde qu’un sommet de degrC 2 k , xCl. Le complCmentaire de A, dans I’arbre initial, est constituk d’une for2t. Chaque arbre de cette for& poss6de au plus p - 1 sommets de degrC 2 k , nous pouvons donc appliquer 5 chacun de ces arbres I’hypothttse d’induction: il existe k forits de chaines de longueur S 2 , partitionnant leurs aretes. En outre d’apres le Corollaire 1 du Thtorkme 1, il existe une partition des aretes de A en au plus k for& de chaines de longueurs2 et telle que tout sommet soit au plus extrCmitC de deux chaines. Nous allons maintenant montrer qu’il est possible de raccorder routes ces

1

M . Habib, B. Piroche

316

forits pour obtenir une partition des arktes de I’arbre initial en au plus k forits de chaines de longueurS2. Considirons un sommet t, (resp. 2,) de I’arbre initial, iI y a 2 cas possibles: ler cas. t, (resp. 2,) a CtC remplaci par deux sommets t i et t‘,‘ (resp. z ; et 2;‘). Par construction d, ( t ; ) < 2k - 3 (resp. d, ( 2 ; ) 2k - 3), posons d, ( t : ) = tf (resp. d, ( 2 : ) = d ) . D’apr6s le Corollaire 1 du ThCorbme 1, il existe au plus: 2 + [ ( d - 2)/2] chaines incidentes 2 t i (resp. 2;) dans A , si d est pair. 1 + [ ( d - 1)/2] chaines incidentes a t i (resp. 2:) dans A , si d est impair. Comme d 6 2k - 3 , en considirant les deux cas d = 2k - 3 et d = I k -4, nous remarquons qu’il existe au plus k - I chaines incidentes a r ; (resp. 2 ; ) . II existe donc au moins une forkt inutilisie en t i (resp. 2;). il suffit d’identifier la forit de chaines contenant I’ar6te adjacente 5 t’,’ (resp. z ’,’) avec une de celle5 qui ne figurent pas en t i (resp. zi). 2Pme cas. t, a ete remplace par s + 1 sommets t i , 1 i 6 s + 1, s 2 2. d T ( r , )= 1 + s + h, oh s + 1 est le nombre de chaines k-chargees d’extrkniite r,. Comme r, ne peut Ctre un k-centre, nous avons: 1 + s + [h/2] 6 k. Or dA (tf”) = h 1, et d’aprbs le Corollaire 1 du T h e o r h e 1, il y a au plus [(l + h ) / 2 ] + 1 forits adjacentes 8 r,’”. Ainsi il est possible de raccorder les arbres sans avoir besoin de forits supplimentatires, car 1 + [(l + h)/2] + s k, d’oh le risultat. 0

+

4. Une famille d’arbres

Pour terminer ce rapport, now alons Ctudier les arbres de Moore a un centre, qui se difinissent comme suit.

Definition. Nous appelons H : arbre de Moore un arbre a ( -1y-1- 2

4-2 sommets dont tous les sommets non pendants ont pour degri q,

L a k-arboricite‘ liniaire des arbres

Propriete. Lorsque q = 2q’, et pour p

317

2 , nous avons

Vi < 2p, lai ( H : ) = q ’ + 1, Vi

3 2p,

la, (Hg)= 4’.

Preuve. Remarquons qu’un tel arbre posskde deux types de chaines: 8 celles qui ont pour extremitCs deux sommets pendants de Hf;et 8’les autres. S’il existe i entier tel que lai (Hg) = q’ alors une partition des arCtes de l’arbre en q’ forits de chaines de longueur S i, n’utilise necessairement que des-chainesde 8.Sinon au noeud de Hg extrimit6 d’une chaine de 8’ il faut alors q ‘ + 1 forCts pour partitionner les arites adjacentes i ce noeud, et donc i 2p. D’oij pour V i < 2p. lai (Hg)3 q r+ 1, d’aprbs le Corollaire 3 du ThCorbme 1, nous avons lai ( H ; ) < q ’ + 1 d’oi~la, (Hfj) = q r+ 1. Montrons maintenant la2, (Hf;)= q’ par induction sur p . Pour p = 2 , H i est une Ctoile et le rtsultat est trivial. Supposons p > 2 . D’aprks I’hypothkse d’induction, H:-’ posskde q r forits de chaines de longueur S 2 ( p - 1) qui partitionnent ses arites. Pour passer de Hg-l i H : il suffit d’ajouter une couche de sommets pendants. En prolongeant les chaines de Hf;-l jusqu’aux sommets pendants de Hfj et en associant deux B deux les arites pendantes restantes nous obtenons bien q‘ forits de chaines de longueur s 2 p qui partitionnent les arites de Hf; et donc la,, ( H ; ) s q’. L’Cgalitt provient de la Proposition 3 en remarquant A (Hfj) = 2q’. 0

References [I] J. Akiyama, G . Exoo and F. Harary, Covering and packing in graphs 111, Cyclic and acyclic invariants, Math. Slovaca 30 (4) (1980) 405-417. [2] C. Berge, Graphes e t Hypergraphes (Dunod, Paris, 1973). [3] J.L. Fouquet, M. Habib and B. Peroche, On k-linear arboricity, en priparation. 141 F. Harary, Covering and packing in graphs I, Ann. N.Y. Acad. Sci. 175 (1970) 198-205. [S] B. Peroche, On partitions of graphs into linear forests, soumis h J. Graph Theory. 161 V.G. Vizing, On an estimate of the chromatic class of a p-graph, Diskret. Analiz. 3 (1964) 25-30,


Annals of Discrete Mathematics 17 (1983) 319-326 Publishing Company

0 North-Holland

DESIGNS AND CODING THEORY Marshall HALL Jr.* California Institute of Technology, USA

1. Introduction Coding theory has been valuable in communications and in various areas of engineering. Recently it has become a valuable tool for investigating block designs. In 1973, MacWilliams, Sloane, and Thompson [5] used the binary code of the plane of order 10 to investigate its properties. It has been shown by Anstee, Hall and Thompson [ l ] that a plane of order 10 cannot have a collineation of order 5 . It has been shown by Z . Janko (personal communication) that there is no collineation of order 3. Together with earlier results it now follows that a plane of order 10 can have only the identity collineation. This leaves the code as the main tool for investigating a plane of order 10 or showing that it does not exist. Section 2 gives some general results on the code of a design. Section 3 discusses some of the applications, in particular the plane of order 10, construction of a (41, 16, 6) design and very recent work on the code of a plane of order 12 over GF(3). 2. The code of a design. Some general results

Let A be the incidence matrix of u, b, r, k , A incomplete block design D so that A is a u by b matrix. Then it is well known (see (2.1)) that A A ' = B = ( r - A ) ] + hJ and det B = ( r - A)"-'rk. By definition, the code C of the design D over a finite field F , = GF(q) is the subspace of V = Ff;spanned by the rows of A regarded as vectors in Ff;.Let q = p ' . p a prime. If p r - A then B is of rank 1 or 0 (the latter in case p also divide r and A ) over GF(q). This is a very useful case:

I

Theorem 2.1. If C is the code of D a ( v , b, r, k , A ) design over F,. q = p ' and p r - A , then C n C' is of codimension at most one in C.

1

Proof. Let ri, r,, r, be any three rows of A. Then * This research was supported in part by NSF Grant No. MCS 7821599. 319

M.Hall, Jr.

320

(r,, r , ) = r = (r,, r , ) = A (mod k ) , and (r, - r,, rr) = 0 (mod k ) .

Hence r, - r, E C'. It is particularly useful, for a symmetric design, if p divides r - A = k - A to exactly the first power. In this case we can determine the exact dimension of C and C'. Theorem 2.2. Let D be a symmetric ( u , k , A ) design and let C be its code over Fq where q = p' and the prime p divides k - A to exactly the first power. Then if p % k , p kA, d i m C ' Z 2u - 1 dim C =and C 3 C'. 2 '

u-1 dim C =2 '

u+l dim C' = 2

and

C C C'.

Proof. Note that u must be odd since if u is even k - A is a square. Let A be the incidence matrix of D so that from (2.1)

det A

= &k(k -

Add columns 1 , 2 , . . . , u - 1 to the last column of A and then subtract the last row from the others. This gives

with det B2= ? ( k -A)"')''. diagonal form. This gives

With elementary operations we can put B, into

32 1

Designs and coding theory

U, V, unimodular, b l b 2 . b,-, = ? ( k - A ) ( v - 1)/2. Thus at most ( u - 1)/2 of br . * b,-I are multiples of p . Hence over GF(p) rank A, = rank A 3 ( v - 1)/2. Hence dim C 3 ( u - 1)/2 and if p Y k,dim C 3 ( v + 1)/2. Thus in the case p ,'j k , k Y A we have dim C 3 ( u + 1)/2 and from Theorem 2.1 dim C' 3 (v - 1)/2, but as dim C + dim C' = v these inequalities are in fact equalities. If p k, p A, then C C C' and the result follows.

I

I

A knowledge of the dimension of C is of course particularly valuable for use of the MacWilliams Identity. MacWilliams Identity. Zf

is the weight enumerator of a code C over Fq and C is of dimension s, then 1 Wc1( x , y ) = 7w c ( x

4

+ (q - l ) y , x

- y ).

A collineation group G of a design D will in a natural way act on the code C of D. We assume here that I G 1, the order of G, is not zero in Fq, the field of the code. Thus if q = p', we assume that the prime p does not divide 1 G 1. Let C , ,. . . , c h be the orbits of G on the b coordinates of V. Let the orbit C, have length m,,and define a vector as that vector of V which has l/dG as its entry for every coordinate belonging to C, and zero for every other coordinate. As m, divides ( G1, m, is not zero in Fq and 1 / 6 exists either in Fq or in the extension Fq2. Then, since distinct orbits C,, C, are disjoint, we have

_ -

(C,,C,)=l,

_ _

(C,,c,)=O,

izj.

c, -

Hence . . . , c h are an orthonormal basis for a subspace W of V (possibly over Fq-.). We wish to consider the orbit space W and in some way C under the action of G as lying in W. For this we define an operator 8,

Here 8 is an idempotent operator and W = V8 is the subspace of V with the orthonormal basis . . , Ch.Also if xi y are vectors of V lying in W, then (x, y ) " = ( x , y ) W, the first being the ordinary inner product, - the second the inner product with respect to the orthonormal bases . . , c h . The following is a theorem of Hayden generalizing a result of Thompson [2].

c,.

c,.

M.Hall, Jr.

322

Theorem 2.3. Let V be an m-dimensional vector space over a field F. Assume that G is a permutation group on the coordinates of V and that 1 G 1 has an inverse in F. Suppose that C C V is a G module then with the operator 8 defined for v E V by

putting W = VO and H = C8 we have (C8)',= H L = (CL)8.

Thompson [ l ] proved the special case in which C = C'

3. Some applications of codes to designs

An important application of codes to designs was made in 1973 by MacWilliams, Sloane, and Thompson [5]. If it exists, a projective plane of order 10 is a symmetric (111, 11, 1) design. From Theorem 2.2, the code over F z , the binary code C,is of dimension 56 and C' consists of the words of even weight (in fact of weights multiples of 4) in C.From this and the MacWilliams Identity the entire weight distribution is determined once the number of words of weights 12, 15, and 16, viz., A , * ,A l s , and A16,are known. In this paper it was shown that Ais=O. If Wls is a word of weight 15, these being 15 points of the plane, every line of the plane intersects Wls in an odd number of points. With appropriate numbering of the points 1 to 15 we must have the following configuration of the 15 points on 21 lines, 1 2 1 6

2 6 3 7 4 8 5 9

3 7 10 10 11 12

5,

4 8 11 13

12, 14.

13

IS,

14

15.

9,

1 1 1 2 2 2

10 15, 11 14, 12 13, 7 15, 8 14, 9 13,

3 3 3 4

6

IS.

8

12. 11, 14, 12,

9 6 4 7 4 9

6 13, 5 7 If, 5 8 10.

5

(3.1)

10,

A computer search showed that this configuration cannot be completed to a full plane, so that the conclusion is that A I s= 0. It has been announced by M. El Ghabbach that he has shown that A l h= 0. A word of weight 12 would be an 'oval' of points such that every line intersects the oval in two points or more. The code C of the plane ii-of order 10 can be extended by adjoining a further coordinate x and taking it as a 'parity check', taking 1 in position x if the word in C has odd weight and 0 in position x if the word in C has even weight. The larger code C ,in V t l 2has dimension 56 and is self dual, i.e., C: = C,.

Designs and coding theory

323

If r has a collineation u of order 5, it is easily shown that the only case which requires consideration as that in which u fixes exactly one point P and exactly one line L. Numbering the points 1,. . . ,110,P we have u =(1-**5)(6.**10)-.*(101*** 105)(106*.* 11O)(P)(x),

and the fixed line L in the code CI has the form

(3.2)

\

L = (101,. . . ,105)(106,.. ., llO)(P)(x).

(3.3) Here Theorem 2.3 applies, since the orbit space W of CI is of dimension 24 consisting of 22 five cycles and fixed P and x. Since C: = C , from the theorem, it follows that with H = CIB,the words of C , fixed by u,then H ' = H in W. But Pless and Sloane [6]have determined all self dual binary codes in dimension 24. The Golay code G,,contains no words of weight 4 but in our code H the line L is a word of weight 4.From their listing H must contain at least 5 further words of weight 4 disjoint from L. Let one of these be (1* * *5)(6... 10)(11 *

* *

15)(16..-20).

This corresponds to a word of weight 20 in C fixed by u.One possibility involves the configuration of 20 lines I 2 3 4

2 3

4 5 5 1

6 7 8 9 10

1 3 2 4

I1

8, 9, 10, 6,

3 5

12 13, 13 14,

4

1

14

7.

5

2

15

12,

15, 11,

11. 12, 13, 14, 15,

13. 14, 15, 11, 12,

16 17, 17 18, 18 19, 19 20, 20 16,

8 Y 16 18, 9 10 17 19. 10 6 18 20, 6 7 19 16. 7 8 20 17.

(3.4)

There will also be 70 lines containing exactly two of the points and 21 lines containing no points. A computer search showed that (3.4)cannot be completed to a full plane. Thus is up to isomorphism one of 92 starts to be considered and computer search showed that none of these can be completed. The conclusion, published in [l], is that a plane of order 10 cannot have a collineation of order 5 . For a (41,16,6)symmetric design a similar assumption led to a very different conclusion. Assuming a collineation a of order 5 and fixing exactly one point and one block led to the construction of the design. Let the fixed point be x and the fixed block be Bo. Then a = (x)(l,.. . ,5)(6,.. . , 10)(1I,.. . ,15)(16,. . . ,20)(21,.. . ,25)

(26... . ,30)(31,.. . ,35)(36,.. . ,40)

(3.5)

on points and similarly on blocks. It is possible to find a table showing how many points of each cycle x, C , ,. . . ,C, as above lie on a representative block. One of 15 possibilities is

M. Hall, Jr.

324

The structure of this table suggests the possibility of a further collineation /3 of order 3 such that on point cycles

p

= (x)(C,,

C,)(C,, cs,C,)(C,)(CX).

c2,

In full on points

p

= (X)(1 6 11)(2 7 12)(3 8 13)(49 14)(5 10 15)

(16 21 26)(17 22 27)(18 23 28)(19 24 29)(20 25 30) (31)(32) (33) (34) (35)(36)(37)(38)(39) (40).

(3.7a)

With a little trial this led to the full design D. Representative blocks are: B,,

B, B, B,, B,, B2,

B,, B,, B,,

,y ,y ,y ,y 6

3 1 1 1

1 2 4 6 3 5 1 2 7 9 4 5 2 4 3 6 2 6

3 7 Y 8 13 11 8 8 7

4 13 11 10 I4 12 9

11 11

5 15 12 14 15 14 10 13 12

6 16 17 16 18

16 17 18 17

7 21 18 22 19 23 20 19 20

8 27 19 23 22

24 21 20 22

9

I0

28 20 24 25 27 28 23 25

29 21 25

26 30 29 24 27

11 30 26 26 31 31 31 25 30

I2 31 31 31 32 32 32 28 36

13 32 32 32 34 34 34 29 38

14 36 36 36 39 39 3Y 30 39

15 38 38 38 40

(3.7b)

30 -10 32

40

This construction is given in detail in [2]. For the projective plane of order 12, or symmetric (157, 13, 1) design, Theorem 2.2 applies for p = 3 but not for p = 2. Here for the code C of the plane over GF(3) we have dim C = 79, dim C' = 78 and C' consists of the words whose weights are multiples of 3. All wieghts are = 0, 1 (mod 3) so that A3r+2 =0 for all s. The MacWilliams Identity is applicable but its direct application involves heavy calculations with very large numbers.

Designs and coding rheory

325

By considerations of invariants of a finite group of order 2592 Mallows and Sloane [4] have found an expression for the weight enumerator W c ( x ,y ) initially in terms of 27 parameters. Let us write

44=

(X 3

+ gY

31,

(p4

= (x -

3),

713

=Xy

h(X

-

3,

(2x

+

'). (3.8)

Their formula is wc-(X,y

) = X~I($;,

(pi)

+ TnPz($;, (pi).

(3.9)

Here Pl(u, u ) is a homogeneous polynomial of degree 13, Pz(u,u ) of degree 12, thus giving initially 14 coefficients in PI and 13 in P 2 . Showing that A " = 1, A 1 3= 314, and A l = A, = A, = Ah= A 7 = A, = A l o =A12 = A I S= Alh=A19= A z l= Azz= 0 reduces the number of parameters from 27 to 12 which may be taken as A I s ,Az4,Azc, A,,, A28, A3",'A31, A33, ,434, A36, A37, A N . These calculations will be published in a joint paper by the author and Wilkinson [3]. A word of weight 18, Wls, will correspond to an interesting configuration. For any line L we will have (Wig, L ) = 0 (mod 3). Let there be Ci lines with exactly i points in common with Wlx.Note that CI = 0 since a line L with precisely one point in common would have ( W18.L ) = ? 1. Then Co+ Cz+ C,+ C,+ 2C2 f 3C3 + 4C4

+

C s + * * * + Ci,=157

SCs + . *

+ 13C13 = 234 = 13.18

(3.10)

The first of these counts the total number of lines, the second the incidences of the 18 points on lines, and the third the incidences of pairs of the 18 points. Subtracting the third from the second gives

C2 - 2C4 - 5C5 + *

* *

- 65C13 = 81.

(3.1 1)

It now follows that C23 81. But a line L with exactly 2 points of Wls must have P, Q in Wlx,with the P coordinate + 1 and the Q coordinate - 1. If of the 18 points there are m with coordinate + 1 and n with coordinate - 1 then Czs mn. But as m + n = 18 it follows that C 2 681. Thus C, = 81, rn = n = 9 and from (3.1 1) C, = C, = . . = C1,= 0. Now from (3.10) Co= 52, Cz= 81, C, = 24. A line L with three points of Wlxmust have all three + 1 or all three - 1 in Wlx. Thus the 9 points which are + 1 lie three at a time on lines to form a Steiner triple system and the same holds for the 9 points which are - 1 in W18. Mallows and Sloane also found a much more complicated formula for the complete weight enumerator of the plane of order 12 which initially has 196 parame ters.

326

M. Hall, Jr.

References [ I ] R.P. Anstee, M. Hall Jr. and J.G. Thompson, Planes of order 10 do not have a collineation of order 5, J. Comb. Theory, Ser. A 29 (1980) 39-53. [2] W.G. Bridges, M. Hall Jr. and J.L. Hayden, Codes and designs, to appear in J. Comb. Theory, Ser. A. [3] M. Hall and J. Wilkinson, The ternary code for the plane of order 12, to appear. [4] C.L. Mallows and N.J.A. Sloane, Weight enumerators of self-orthogonal codes over GF(3), to appear. [5] F.J. MacWilliams, N.J.A. Sloane and I.G. Thompson, On the existence of a projective plane of order 10, J. Comb. Theory, Ser. A 14 (1973) 6 6 7 8 161 V. Pless and M.J.A. Sloane, On the classification and enumeration of self dual codes, J . Comb. Theory, Ser. A 18 (1975) 313-335. 171 2. Janko, Personal communication.


LE DECODAGE RAPIDE DES CODES DE REED-SOLOMON ET LEUR GENERALISATION S. HARARI Universiii de Paris VI. Laboraioire Probabiliies, 4 place Jussieu, 7.Q30 Paris, Cedex 05. France

1. Introduction

Les codes de Reed-Solomon sur le corps F, oh q = p ’ , p entier premier, existent pour beaucoup de longueurs et de rendements. Ces codes sont optimaux (maximum distance separable [4]). 11s font I’objet de nombreuses rtalisations dans le cas ou p = 2. En effet B travers une reprksentation binaire, un code de Reed-Solomon correcteur de t symboles devient un code correcteur de t paquets de longueur en bit tgale B s. Le rendement d’un tel code est voisin de celui d’un code B.C.H. de mEme longueur corrigeant t bits. De plus la complexitC de rkalisation des dCcodeurs conventionnels est identique pour les deux codes. Ricemment de nouvelles rkalisations de codes de Reed-Solomon ont CtC proposkes utilisant la transformte de Fourier discrtte dans le codeur et dans le dicodeur. Au niveau du codeur la transformie grace B une interpolation de Lagrange se substitue B I’algorithme de division utilisC pour la dttermination de la redondance. Au niveau du dCcodeur I’utilisation de la transformte est possible h plusieurs Ctapes. Dans le cas oh I’on peut utiliser une transformte de Fourier rapide, la complexit6 du systeme se rtduit considtrablement. Le but de cet article est double. D’une part, il s’agit de faire une synthkse de I’utilisation de la transformte pour les codes de Reed-Solomon, d’autre part de montrer que I’utilisation de la transformie de Fourier rapide ou acctltrCe est possible pour certaines gtntralisations des codes de Reed-Solomon que nous definissons: pour elles un traitement e n paralltle est aussi possible. 2. Transformee de Fourier discrete

Soit q un nombre premier et n u n diviseur de q m- 1, oh m est un entier positif. Soit F p le corps de Galois d’ordre q”’ = n + 1 et p un ClCment primitif de ce corps. 327

S . Harari

32X

Soit A ={ar,,.... a , - ] } une suite de n CICrnents de F p . La transforme'e de Fourier discrtre de la suite A ; notCe 5 ( A ) est la suite 9 ( A ) = {bl,,. . . , b, I } dkfinie par "-1

6, = I

=n

aiPiJ, j = 0,1,.

.. ,n - 1.

(3.1)

La transform& de Fourier discre'te inuerse de la suite A notie $ - ' ( A ) est la suite F ' ( A ) = { b A , . . , bl (2.2) dCfinie par

oh N = n modq.

Soit A "[XI I'algibre des polynbmes a coefficients dans F p rCduite par I'idCal engendrC par ( x " - I). Les definitions de transformee de Fourier directe et inverse s'Ctendent de maniire tvidente. Soit a ( x ) = all + a l x

+ . ' . + an-lXn-l,

un polynbme de A " [ X ] ,

4 (( X~ )) = A ( X ) = A 11 + A I X

+

* * *

+ A,

~

I

x"

I

0u

A , = a ( a ' ) , i = 0 , 1 , ..., n - I .

Pour les algkbres de polynbmes now a w n s le risultat suivant.

z::,:

zy=l:

2.3. Theoreme. Soit u ( x ) = u , x ' et u ( x ) = u,x' d e u x polynbmes de l'algtbre A "[XI. Soit u * u ( x ) le polynbme dkfini par u

*u(x)=

c

n-l

I

c,x'

=lI

ou c, = u,v,, i = 0,1,.

. . , n - 1.

Nous auons les proprie'te's suiuantes:

F ( u ( x ) + u(x))= 9(u(x))+ F(U(X)),

9(0= )0, 9(1)= 1+x

+. . .+x"-I,

4 ( u ( x ) .u(x))= 9 ( u ( x ) ) * 9(u(x)), 9(u

* u(x))=

9(u(x)).9(u(x)),

9 - ' ( 9 ( u ( x ) ) )= u ( x ) .

Le dicodage rapide des codes de Reed-Solomon

3'9

Ce thCor6me, bien connu est dCmontrC dans [4] pour le cas discret. 3. Application de la transformke au codage des codes de Reed-Solomon

Un code de Reed-Solomon de longueur n = q m - 1, et de distance minimale d , est un code cyclique (de dimension k = n - d + 1). C'est en particulier un idCal de A " [ X ]de polyn6me gtnCrateur g ( x ) = x ( x - p i > .Ainsi tout mot du code vCrifie la relation

nfl:

c(x)E

c e c ( P ) = C ( P 2 ) = * . . = c ( p d - ' )= 0,

(3.1)

soit i ( x ) un polyn6me d'information de degrt < k . L'opCration de codage consiste A associer un mot de code c ( x ) i ( x ) . La maniche conventionnelle d'effecteur cette opCration revient a considCrer le reste de la division de x " - ' i ( x ) par g ( x ) , not6

Le polyn6me c(n) = x " - ' i ( x ) +

[

Xn-k

%I

appartient au code et est donc le mot de code recherchi. Pour aboutir a ce rCsultat on effectue une division d'un polynbme de degrC n par u n polyn6me de degrC d . Ceci nkcessite n 2 multiplications et n(n - 1) additions. Codage par transformie lire methode

Le codage systkmatique est possible avec une transformie de Fourier discrtte. Se I'on Ccrit la division euclidienne de I ( x ) = ~ " - ~ i ( par x ) g ( x ) nous avons I ( x ) =q(x)g(x)+ 4x1,

(3.3)

d ( R ( x 1) < d ( g ( x ) ) .

(3.4)

oh

Le mot de code associC B i ( x ) est alors c ( ~ ) = x " - ~ i ( x ) + r ( x ) .La determination de r ( x ) s'effectue par I'interpolation de Lagrange. En effet de la relation (3.3) nous tirons: r(cu') = I(a') pour j

= 0,1,.

. . ,d

- 1.

La condition (3.4) nous permet de dCterminer R ( x ) de manikre unique; grCce aux polyn6mes

330

S. Harari

En effet (3.5)

Le calcul des r ( a ’ ) se fait en considkrant les d - 1 premiers coefficients de la transformCe de Fourier discrkte de I ( x ) . Les E , ( x ) sont ( d - l)(n + d - 1) multiplications et ( d - l ) ( n + d - 1) additions. Les E i ( x ) sont pricalcults.

3.6. Exemple numerique. N = 255, d = 33. Le calcul des d - 1 coefficients de Fourier nbcessitent 852 multiplications et 1804 additions. La multiplication par les Ej nkcessite 960 multiplications et 960 additions, ce qui donne un bilan de 1812 multiplications et 2764 additions comparCes aux 7136 additions et 7136 multiplications du cas conventionnel.

3.7. Codage par transformee 26me methode. Un autre mCthode de codage non systkrnatique est possible par transformke. Soit i ( x ) le polynbme d’information; et Z(x) = x “ - ~ ~ ( x Le ) . mot F - ’ ( Z ( x ) ) appartient au code de Reed-Solomon de longueur n et de distance rninirnale d , car son image de Fourier A savoir Z(x) a ses d - 1 premiers coefficients nuls. Le cotit de cette opCration est une transformie de Fourier, donc n’ multiplications et n ( n - 1) additions.

4. Decodage des codes de Reed-Solomon

Le dCcodage des donnCes est la partie la plus complexe d’un systkrne de correction. La transformte permet ici de dtcoder les vecteurs r e p s et ceci a plusieurs niveaux. Rappelons le schCma d’un dicodeur. Celui-ci composC de plusieurs unites logiques: Le vecteur r e p est gardC dans un registre 2 decalage. Le syndrbme est catculk par une unite particulikre. Le syndrbme sert de donnCes A I’unitk qui calcule le polynbme localisateur et Cvaluateur d’erreurs. Une fois ce polynbme calculC la correction est effectuCe sur le mot r e p qui est alors dClivrC A I’utilisateur.

i-::-cb

Le dtcodage rapide des codes de Reed-Solomon

33 1

4.1. Schema du dCcodeur d’un code de Reed-Solomon canal

.-,gnaiaire

Calcul du polynome localisateur et kvaluateur d’erreur

syndrbme

4.2. l6re methode de decodage par transformee. Une lkre utilisation de la transformCe est utiliste au calcul du syndrdme. Le syndrdme vaut

OG R ( x ) est le mot requ et

le polynbme gentrateur du code. De la meme manikre qu’au codage, le reste de la division est connu si et seulement si le reste de la division par chaque facteur irrtductible est connu. Pour les codes de Reed-Solomon ces facteurs sont de degrC 1 ce qui rend la dktermination aisCe. Soit

s, = [ T x - ao ,],

i = l , ...,d - 1 .

On a trivialement que S, = r ( a ’ ) pour i = 1, . . . , d - 1 et S ( x ) = cy~,’r(a~)~ oh, (~x, ()x ) = ( n k , , ( x - , ~ ) ) / ( I 7 ~ ~ (, a k- a ‘ ) ) . 11 apparait alors que le calcul du syndrdme est une optration identique 2 celle du codage, en particulier que les S, ne sont autres que les coefficients de Fourier de r ( x ) , et que le gain en optrations a les mCmes valeurs. Pour la suite du dCcodage on prockde de manikre conventionnelle par I’algorithme de Berlekamp-Massey, en determinant un poiyn6me localisateur d’erreurs, ainsi qu’un polynbme kvaluateur.

4.3. 2eme mkthode de decodage par transformee. Une deuxikme mCthode d’utilisation de la transformCe a un schema difftrent de celui utilise dans la m6thode conventionnelle. En effet ici on ne calcule pas de syndrdmes ni de polyndmes Cvaluateur d’erreur. Soit c(x)=co+c,x

le mot de code Cmis,

+.*.+Cn-lX

n-I

332

S. Harari

r ( x ) = r o + rIx

+ - + ra-lx"-l *

*

le mot r e p . L'Cquation r ( x ) = c ( x ) + e ( x ) determine le polynbme d'erreur (inconnu) que I'on desire reconstituer. L'Cquation 9 ( r ) = 9 ( c ) + % ( e ) fournit une indication sur 9 ( e ) . En effet 9 ( r ) o = 9(e)", . . . , 9 ( r ) d - l = s ( e ) d - l car les termes correspondants de 9 ( c ) sont nuls par hypothkse. Soit

le polynbme localisateur d'erreur dans Ie plan de Fourier. Ce polynbme admet pour racines les positions des symboles errones. Par conskquent si on note A ( x ) = ho + Alx + + A,-lx"-' son image inverse de Fourier nous avons 1 'Cq ua t ion a

4

Aie,=O p o u r i = 0 , 1 ,..., n - 1 .

En prenant I'image de Fourier de cette Cquation nous obtenons I'Cquation de convolution

I

9(e).A(x)=O.

1

(4.5)

Dans cette Cquation 9 ( e ) est partiellement connu, et '4( x ) totalement inconnu. L'Cquation (4.5)se rCsout de la manikre suivante. Posons 9( e = Xil;E X I . L'tquation (4.5) se dkcompose en un systkme lintaire A,,e,,+ A

+ A2e,-, + . . . + Aren-,= 0, A,,e, + A ,el,+ A2e,- + . . + = 0, Aoe2+ A , e l + ,42el,+ . . + Aren-,+? = 0, I

'

Aoe.l+ A ]e2+ &el + . . * = 0, ,4,,e,-,

+ A ,en-:+ A l e n - l - l= 0.

I

J

Le systkme (4.6) est un systkme de n tquations, correspondant i la nullite de chacun des coefficients du polyn6me % ( e ) , A (x), n + t inconnues. A priori un tel systkme n'a pas de solution unique. Toutefois sa structure particulikre (incomplkte) permet de le risoudre avec unicite de la solution; dans certains cas. Parmi ces n equations ou les A, sont considCrCes comme inconnues, il y en a en t qui ne font intervenir que les coefficients eel, e l , . . . ,e2! I, qui sont eux connus.

Le dicodage rapide des codes de Reed-Solomon

333

Grace B ce sous systeme de t Cquations, on dktermine ies coefficients . t ,,, . . . , ,1. Pour retrouver les n - 2t valeurs encore inconnues de A (x) on substitue dans le systkme (4.6) les r valeurs de A , . On obtient alors un systkme de n - 2 t Cquations i n - 2 t inconnues E,. Une simple inversion de matrice fournit ces coefficients E , . Pour connaitre le vecteur d'erreur e ( x ) B partir de E ( x ) , une transformation de Fourier inverse est ndcessaire. La correction est alors achevke. 4.7. Remarque. Cet algorithme vaut aussi bien pour les codes BCH que pour les codes de Reed-Solomon h condition d'y ajouter des conditions supplCmentaires pour que les symboles soient dans le sous corps premier. 4.8. Remarque. L'Cquation (4.5) est analogue i celle que I'on trouve en utilisant la mCthode conventionnelle de dCcodage. L'algorithme de Berlekamp, utilisant un polyn6me auxiliaire peut aussi &tre utilisCe. 4.9. Remarque. Cet algorithme ne nkcessite pas la dktermination d'un polyn6me Cvaluateur d'erreur. Cette Ctape de la rCsolution est remplacCe par une transformke de Fourier inverse.

4.10. Rapidite de I'algorithme. Une transformke de Fourier discrete nCcessite O(n') opkrations. Une inversion de matrice O(n*)opCrations. Ainsi la solution de ce systkme est O(n*)opkrations.

5. Decodage au dela de la capacite

Un des avantages de la mCthode de dkcodage par transformke est que celle ci perrnet un dkcodage au deli de la capacitk de correction nominale du code, dans le cas d'une configuration corrigible. Soit A ( " ) ( x ) le polyn6me localisateur d'erreur. II s'obtient grace a un systkme rCcursif d'iquations, connu sous le nom d'algorithme Berlekamp-Massey: "-1

(1)

A, =

AY-"S,-,, j =O

(2)

A ("(x) = A ' ' - ' ) ( X ) - A,B('-')(x),

(3)

B"'(x)= (1 - S,)XB"-"(X) S,XA ; ' A ( ' - ' ) ( x ) .

+

r = 1,. . . ,2t, A " ( x ) = 1, B " ( x )= 1, S, = 1 si A,# 0 et degB"-"(x)> 1 et deg B". "(x) > max degACi'(x).6, = 0 sinon.

S . Harari

334

Tout polynhme :t ( x ) de degrC t polynbme localisateur d’erreurs si n - I

,2 ,I~s,-~= o

pour r = t

+v

ayant t

+v

racines distinctes est un

+ v + 1,. . . , 2 t .

=il

Le polynbme de plus bas degrC (s’il existe) correspond 2 la configuration d’erreur la plus probable. S’il est de degrt S t il est fourni par I’algorithme de Berlekamp-Massey. S’il y a plus de t erreurs, le mot requ peut aussi itre dtcodC de manikre unique. Supposons qu’il y ait t + 1 erreurs. Les deux polynbmes inconnus S2r+l,S2f+? seront suffisants, s’ils sont connus, pour dkterminer ,4 (x). Pour les diterminer, i I suffit d’ittrer I’algorithme de Berlekamp-Massey, deux fois en prenant S , , , , et Slrtlcomme inconnues:

, t ( 2 f + z ’ ( ~[ )I =- t i r f + l A 2 r + 2 A 2 ~ + l ~ ’ ] A ( ’ i ) ( x ) - [A2r+lX

+ (1 - ti?r+l)dLf+ZXZ]B(?r’(x),

oh &,+,E{O,I},

A 2 , + l ,A 2 r + l E F q met = O si =O. Les seules inconnues dans le membre de droite de I’Cquation sont tilril, et A 2 f + l . Pour les diterminer il suffit d’inverser par Fourier I’Cquation. On obtient A!”+”=

(1 - t i 2 r 1 1 4 2 , + : A : f 1 + I ~ ”]A!2f’ - [ A ? f + l ~ - ‘+ ( I - S 2 f + l ) 4 ~ f + ~ ~ ” ] b l ’ f ’ .

On doit determiner les inconnues, de manittre que la configuration d’erreurs ait au plus t + 1 composantes non nulles. Si deg A ” 5 t et le nombre de zCro de A‘:‘’ Cgale le degrC de A“”, alors le nombre d’erreur Cgale de degrC de ,4(2f’. S’il existe une solution unique pour A ( L r + Z ) avec t + 1 composantes nulles et une ,t ( 2 r + 2 ’ de degrC t + 1, une configuration d’erreur unique peut etre trouvCe. Soit J Z f = pkl et A 2 f + Z= pk:. Les cas a considerer sont A ! Z f + 2 1 = A j 2 0 - p k , C X -rb!:O, (1)

1

(4)

A j:f+l) = A ( 2 0 - p ‘>p‘LA I”]- p k , p -lb!2fl.

Chacun de ces cas doit 6tre explorC pour k l = 0,. . . ,q - 2 et k 2 = 0,. . . ,q - 2; en vue de A!Zft2’ = 0 pour exactement t + 1 valeurs de i. Pour ces valeurs des inconnues ,4 (zf’2)(x) est de degrC t + 1, A (”+” permet de dtcalculer SZf+,et S2f+2 et une transformee de Fourier inverse complkte de dCcodage.

Le dtcodage rapide des codes de Reed-Solomon

33s

6. Accklkration de la vitesse de la transformke

Une transformke de Fourier discrete sur n points nCcessite au plus n 2 additions et n' multiplications. D a m le cas ou n = 2" o n peut rCduire les multiplications B n logn et les additions B O ( n log n ) . Ceci [3] est un optimum dCterminC par Cooley-Tuckey. Seules les longueurs 17 et 257 permettent donc de dCfinir des codes de Reed et Solomon decodables par une transformCe de Fourier rapide. Le gain de vitesse offert par I'algorithme de Cooley-Tukey a conduit B deux gCnCrations. La premiere (Harari et Botrel [2]) porte sur les transformkes arithmktiques sur les entiers n de la forme n = q + 1 oh q = K . 2" + 1 , q premier et K impair. Elle repose sur le suivant. Lemme. Soit q un nombre premier de la forme q = K . 2" + 1, K impair soit a un e'le'ment primitif de F,. a est alors d'ordre 2" et a2' est d'ordre k . Demonstration. Tour avoir une transformke d'ordre n on transforme par k transformtes d'ordre 2", grlce B 1'CICment primitif a '. Ces transformations peuvent btre effectutes en parallde indkpendemment les unes des autres, grlce B un algorithme de transformCe de Fourier rapide. Le rksultat partiel est prolongC par 2" transformkes d'ordre k effectuCes avec I'CIkment a'" ; qui peuvent elles aussi btre impIantCes en parallele. Le bilan de cette transformCe est (pour un Cltment) k' additions et k' multiplications, et O ( m log n ) et b ( m log m ) additions et multiplications.

Les nombres premiers de la forme k . 2"

+ 1 sont frkquents:

13, 97, 197, 41, 641, 29, 113, 449, etc. La notion de transformee arithmktique rapide a donnk lieu B une autre genkralisation plus rCcente (Winograd, Reed et Miller). Ce groupe d'auteurs a obtenu le rCsultat suivant. Soit y E F L m une racine primitive m ikme de I'unitC ou d (2" - 1. La transformke en d-points sur F2m est donnCe par d-l

2

Aj = i =O aiyii, 0 6 d - 1 OC ai E F z m . Soit d = d , d z . . .d, avec ( d i , d j ) =1 pour i # j . En utilisant le thCor6me du reste chinois la transformie sur d points peut &tre dCcomposCe en une transformie multidimentionnelle sur FP .

336

S. Harari

Soit j un entier represent6 par (j,, . . .,j , ) oh j k = j mod dk (1 6 k zs r ) et yk une racine primitive &ibme de I’unitt; on a alors:

Conclusion Les deux derniers algorithmes prCsentCs pr6sentent une amelioration certaine par rapport B I’algorithme classique. Toutefois surtout sur les longueurs, il n’est pas dCmontr6 qu’ils sont optimaux. Pour les petites longueurs un ClCment important pour la complexit6 du systbme est le choix de 1’616ment primitif servant B effecteur la transform6e.

Bibliographie [ 11 Blahut, Transform techniques for error control codes, IEEE Internat. Symp. on Inf. Theory, Grignano, 1978. [2] J. Botrel and S. Harari, The efficiency of the decoding algorithms for Reed-Solomon codes and their generalisation, IEEE Internat. Symp. on Inf. Theory, Grignano, 1978. [3] D.E. Knuth, The Art of Computer Programming, Vol. 2 (Addison Wesley, Reading, MA, 1980). [4] McWilliams and Sloane, The Theory of Error-Correcting Codes (North-Holland, Amsterdam, 1977). 1.51 I.S. Reed, T.K. Truong, R.L. Miller and J.P. Huang, Fast transforms for decoding Reed-Solomon codes, IEEE Proc. 128 (1) (1981).


NOMBRE MAXIMUM D’ARCS D’UN GRAPHE ANTISYMETRIQUE SANS CHEMIN DE LONGUEUR 1 M. C. HEYDEMANN Uniuersitk de Paris-Sud. Centre Orsay, Laboratoire de Recherche en Informatique, Bititnetit 490. 91405 Orsay, France In this article we determine a number of arcs guaranteeing the existence of a path of length 1 in an oriented graph (or antisymmetric digraph) which is strong or not. The given condition is shown to be best possible except for a strong oriented graph with strictly more than 21 - 2 vertices, in which case we suggest a conjecture.

1. Introduction

I1 existe beaucoup de thiorkmes donnant des conditions sur les degrts des sommets d’un graphe orient6 pour que celui-ci contienne un chemin de longueur donnte (voir [l], (41, [9], [lo]). Par contre, le nombre minimal d’arcs suffisant pour qu’un graphe orient6, quelconque ou fortement connexe, contienne un chemin de longueur 1 n’est pas dttermint dans le cas gintral. Nous avons donne dans [3] des conjectures virifiies dans des cas particuliers. De mime que l’existence de chaines dans les graphes non orient& est rtsolue B partir de celle de cycles ([Ill, [12]), nos conjectures risultent par une construction simple, de conjectures sur I’existence de circuits en fonction du nombre d’arcs. Dans ce qui suit, nous allons considirer le cas des graphes orientis antisymttriques quelconques ou fortement connexes. Le Thiorkme 1 permet d’obtenir le nombre maximum d’arcs d’un graphe antisymitrique sans chemin de longueur 1. II peut se diduire facilement d’un thioreme d’existence de circuits (Thkorkme 2). Le Thiorbme 3 s’obtient aussi g r k e au Thiorkme 2, mais de manikre moins directe. I1 donne un nombre d’arcs suffisant pour qu’un graphe antisymitrique fortement connexe contienne un chemin de longueur 1. Nous montrons qu’il est le meilleur possible si n, nombre de sommets du graphe, vtrifie n S 21 - 2 et donnons une conjecture dans le cas n 2 21 - 1. 2. Definitions et notations

Nous suivons la terminologie de [2] mais rappelons ci-dessous quelques dtfinitions en precisant nos notations. 337

338

M. C. Heydemann

Tous les graphes considCrCs ci-aprks sont des 1-graphes sans boucles, c’est-adire des graphes orient& sans arcs multiples ni boucle. De plus ils sont antisymitriques, c’est-&dire sans circuit de longueur 2. Soit G un tel graphe. Nous notons: V(G): I’ensemble de ses sommets; E ( G ) : I’ensemble de ses arcs; ( x , y ) : I’arc d’origine x et extrCmitC y. U n chemin tle‘mentaire de longueur I est un l-graphe G tel que:

V(G) = { x , ,. . . ,x ~ + ~et} E ( G )= {(xi,xi+,),1 G i s I } . Comme nous ne considCrons que des chemins CICmentaires, nous omettrons dans ce qui suit I’adjectif Cltmentaire. A ( G ) dCsigne la longueur maximum d’un chemin de G. Si A est un sous-ensemble de V(G), G - A dCsigne le sous-graphe de G engendrC par V(G) - A. Si A est riduit 5 un sommet x, on note G - A = G - x. Si H est un sous-graphe de G, G - H dCsigne le sous-graphe de G engendrC par V ( G ) - V ( H ) . Pour A et B sous-ensembles de V(G), on note:

E ( A , B ) = ( ( x , y ) E E ( G ) , x E A ,y E B ) U { ( x , y ) E E ( G ) , x E B Y, E A } . Si H et K sont des sous-graphes de G on utilise la notation abrCgCe E ( H , K ) pour E ( V ( H ) ,VW)). Pour x E V ( G ) ,on note f ( x ) I’ensemble des voisins de x et si H est un sous graphe de G, f ( H ) = U . , , , H ) T ( x ) .

Theoreme 1. Soit G un graphe antisymttrique avec n 3 1 + 1 sommets et ve‘rifiant ] E ( G )> ( (1/21) [ n 2 ( f- 1)- r(I - r ) ] avec n = hl - r, 0 s r < 1. AIors G contient un chemin de longueur 1. Ce thCor2me peut itre obtenu comme consCquence directe du thCor&me de Turan (voir [5, ThCorkmes 7-9, p. 1101) sachant qu’un graphe oriente complet est chemin hamiltonien, ou bien, par une construction simple (voir [6, p. 70]), comme corollaire du thCorkme suivant dCmontrC dans [8].

Theoreme 2. Soit G un graphe antisymtrrique fortement connexe d’ordre n virifiant : l P ( G ) l > m [n’(k - 3 ) + 2 n - k

+ 1- r ( k - 2

- r ) ] = @(n,k ) ,

Nombre maximum d’arcs

avec n = h (k - 2) - r + 1 , 0 G r < k - 2, k entier, k circuit de longueur 3 k.

339

2 3.

Alors G contient un

Remarquons que le ThCorttme 1 est le meilleur possible, ce qui signifie encore que (1/21)[n2(1 - 1)- r ( 1 - r ) ] est le nombre maximum d’arcs d’un graphe antisymttrique sans chemin de longueur 1, au vu de l’exemple suivant. Exemple. Graphe D l ( n ,I ) . Soit D l ( n ,1) le graphe ayant n sommets, avec n = hl - r, 0 s r < I, ces sommets se partitionnant en 1 stables X i , avec IX, 1 = h - 1 pour 1 S i S r, I Xi I = h pour r + 1 S i s 1 et dont les arcs sont tous les arcs (x, y ) pour x E X , , y E Xi,1 s i 6 j s 1. Ce graphe a (1/21)[n2(1 - 1 ) - r ( 1 - r ) ] arcs et ne contient aucun chemin de longueur 1. Dans le cas des graphes antisymitriques fortement connexes, nous avons le thtorkme suivant annonct dans [7]. Theoreme 3. Soit G un graphe antisyme‘trique, fortement connexe d’ordre n, et soitl unentierue‘rifiant 1 s n - 1 . Si G ue‘rifie I E ( G ) l > i n ( n - 1)-3(n - 1 ) alors G contient un chemin de longueur 1. Demonstration. Elle s’effectue par rtcurrence sur n. Posons t(n, I ) = h ( n - 1 ) - 3(n - I ) et montrons que le thtorkme est vrai pour n=l+l. Soit G un graphe antisymttrique fortement connexe d’ordre n verifiant ( E ( G ) l > t(n,n - 1 ) = h ( n - 1 ) - 3. Alors, d’aprtts le Thtorttme 2, G contient un circuit de longueur 3 n - 2. I1 est facile de voir que si G est un graphe antisymetrique fortement connexe dont la longueur maximum d’un circuit C est c ( G ) on a A(G)>min{n - l,c(G)} et si G - C n’est pas stable, A ( G ) 3 min{n - 1, c ( G ) + 1). I1 suffit donc de montrer que si c(G) = n -2, G - C n’est pas stable. Or, d’aprks le lemme suivant. Lemme 1 ([6], [S]). Soit G un graphe antisyme‘trique fortement connexe et soit C un circuit de G de longueur maximum I. Alors pour tout sommet y E V(C) on a ~€(y,C)IGl-l. Nous avons ( E ( G - C, C)l s 2(n - 3). Si G - C est stable, il vient:

I€(G)lG?(n - 2 ) ( n - 3 ) + 2 ( n - 3 ) = t ( n , n - I ) , ce qui contredit I’hypoth2se. Par constquent A ( G ) = n - 1.

M. C. Heydemann

340

Puisque le theortime est vrai pour n = 1 + 1, nous allons faire une recurrence sur n. Nous supposons n L 1 + 2 , le thCortime vrai pour tout n’ tel que 1 + 1 6 n’ < n. Soit G un graphe antisymetrique, fortement connexe, d’ordre n et soit c la longueur maximum d’un circuit de G. Si on a c 2 inf(l, n - 2), alors G contient un chemin de longueur 1. Nous supposons G sans chemin de longueur I, donc c s inf(1- 1, n - 3), et cherchons A montrer 1 E ( G ) (S t(n, I ) . Soit C un circuit de G de longueur c dont nous preciserons le choix ultkrieurement. Nous allons considkrer deux cas. Cas 1. I1 existe une composante fortement connexe A de G - C et un sommet a de A fels que € ( A ,G - A ) = E(a, C). Afin de majorer I E ( G ) ( nous allons distinguer plusieurs sous-cas suivant la valeur de 1 V ( A ) (. Cas 1.1. A est riduit ir un sommet a. Le graphe G - a &ant fortement connexe, nous pouvons lui appliquer I’hypothtse de rkcurrence: ( E ( G - a ) / G t ( n-l,l)=$(n -l)(n-2)-3(n-l-l).

Par ailleurs, on a ( E ( a ,C ) (s c

- 1 (Lemme

( E ( G ) (S i ( n - l ) ( n -2)-3(n

l), d’oG

- 1-l ) + c

- 1,

lE(G)ISt(n,l)+c-(n-3).

De c s n - 3, on deduit IE(G)lS t(n, I ) . Cas 1.2. A a au plus n - 1 sommets, et n’est pas re‘duit d un sommet. La forte connexite de A implique alors que A contient au moins 3 sommets. Nous posons a = ( V ( A ) ( , d Y o G 3 S a G n - 1 . Le graphe G - A + a est fortement connexe, a au moins 1 + 1 sommets, et au plus n - 2 sommets. On peut donc lui appliquer I’hypothkse de recurrence. D’ou ( E ( G- A + a ) / s t ( n - a

+ l)(n -a)-3(n

-a

+ 1- I )

en majorant J E ( A ) ( S t a ( a- l ) ,

on en dCduit puisque E ( A , G - A ) = E ( a , G - A ) , ( E ( G ) I s f ( n- a + l ) ( n - a ) - 3 ( n - a + l - l ) + & ~ ( a l),

soit E ( G ) l G i n 2 - t n ( 2 a - l ) + a ( a -1)-3(n

ou

-I)+3a -3,

Nombre maximum d’arcs

21I

Posons A = ( a - 1 ) ( a - n +3). De 3 < a < n - 1 S n - 3, on deduit A S 0, d’ou lE(G)lSt(n,O. Cas 1.3. A a au moins n - 1 + 1 sommets. Du fait que, pour tout x E V(A - a ) , on a E(x, G - A ) = 0, on dCduit

I E ( G ) I < t n ( n- 1 ) - ( a - l ) ( n - a ) , soit

l E ( G ) I s t ( n , l ) + 3 ( n- / ) - ( a - l ) ( n -a). PosonsA=3(n-l)-(a-l)(n-a).

Or,onaa-l*n-l

et n - a a A a 3 ,

d’ou

(a-l)(n-a)a3(n-l)

et

AGO.

On en dCduit IE(G)I < r(n,1). Cas 2. Supposons que 1’on ne se trouue pas dans le Cas 1. Soit H le graphe obtenu en contractant le circuit C en un de ses sommets x et en ne conservant pour tout sommet y E T ( C ) - C qu’un seul arc entre y et x. On peut montrer comme dans [8] que, puisque I’on ne se trouve pas dans le cas de 1, le choix du sens de I’arc conservC peut etre fait de telle sorte que le graphe H soit fortement connexe. Le graphe H est alors un graphe antisymtitrique, fortement connexe, d’ordre n - c + 1, vCrifiant ( E ( H ) I3 I E ( G - P ) l , si P dCsigne le chemin C - x. De plus, on peut voir que si H contient un circuit de longueur c ‘ alors G contient un chemin de longueur c + c’ - 2. Puisque nous avons supposC A ( G) s 1 - 1, ceci implique c + c ’ - 2 < 1 - 1 soit c ’ s 1 - c + 1 . D’aprks le Theoreme 2, on a donc

I E ( G - P ) I < 1 E ( H ) I s @ ( n- c + 1, I - c + 2 ) . Nous allons alors prCciser le choix de C et x par le lemme suivant, dtmontre dans [6]. Lemme 2. Soit G un graphe antisyme‘triquefortement connexe, don1 la longueur maximum d ’un circuit est c. Alors il existe un circuit C dans G de longueur c, et un sommet x de C tels que pour tout y e V ( C )on a J E ( y , C - x ) l s c -2.

Demonstration. Choisissons donc un circuit C de longueur c et x E V(C) tels que pour tout V(C) on ait

ye

J E ( y , P ) l S c- 2 .

M. C. Heydemann

342

On a alors IE(P, G - P ) l S ( c - 2 ) ( n - c

+ 1)+ 1.

D’oh en majorant IE(P)I par ;(A - 1)(A -2),

I E ( G ) ( s @ ( n- c

+ 1, I - c

+2)+(c -2)(n - c + I )

+f(c-I)(c -2)+1. Pour achever la demonstration, il suffit donc de montrer

@ ( n- c

+ 1,1 - c

+ 2 ) + ( c -2)(n

-c

+ 1)+4(c - l)(c -2)+

1s t(n,l),

ce qui rksulte d’un calcul assez facile, et achkve la demonstration du theoreme. Nous allons montrer que le thkorkme precedent est le meilleur possible pour n c 21 - 2, ou encore. Proposition. Le nombre maximum d ’arcs d ’un graphe d ’ordre n antisyme‘trique fortement connexe, sans chemin de longueur 1, atlec n ~ 2 1 - 2 , est $ n ( n- 1)-3(n - I ) . Demonstration. Nous definissons le graphe T(n,I) comme suit, suivant la parite de n - 1. Si ( n - I ) est pair et n c 21 - 1, nous posons s = 1 - ( n - 1)/2 (Fig. 1). L’ensemble des sommets de G = T ( n ,I ) se partitionne en V ( G )= X,, avec IX,1 = 4 pour 1 c i s ( n - l ) / 2 , et X,= {x,}pour ( n - 1)/2< i s s. Nous posons

u:=,

E(G)={(x,,x,),x,EX,,X,EX,,I~~ @ ( n ,q + 2).

Bibliographie [ I ] J. Ayel, Longest paths in bipartite graphs. [2] C. Berge, Graphes et Hypergraphes (Dunod, Paris, 1970). 131 J.C. Bermond, A. Germa, M.C. Heydemann et D. Sotteau, Chemins et circuits dans les graphes orientis, in: Proc. CofI. Montreal, IY7Y. Ann. Discrete Math. 8 (1980) 2Y3-3W. [4] J.C. Bermond, A. Germa, M.C. Heydemann et D. Sotteau, Longest paths in digraphs. Combinatorica 1 (4) (1981) 337-341. [S] J.A. Bondy and U.S.Murty, Graph Theory with Applications (The McMillan Press, Ltd., 1976). 161 M.C. Heydemann, Existence de chemins et circuits d a m les graphes, T h h e d'Etat. UniversitC Paris-Sud, Orsay, lY80.

346

M. C. Heydemann

M.C. Heydemann, On cycles and paths in digraphs, Discrete Math. 31 (1980) 217-219. M.C. Heydemann, Cycles in strong oriented graphs, Discrete Math. 38 (1982) 185-190. M.C. Heydemann, Degrees and cycles in digraphs, Discrete Math., a paraftre. B. Jackson, Long paths and cycles in oriented graphs, J. Graph Theory 5 (1981). D.R. Woodall. Sufficient conditions for circuits in graphs, Proc. London Math. SOC.(3) 24 (1972) 739-755. 1121 D.R. Woodwall, Maximal circuits of graphs I, Acta Math. Acad. Sci. Hungar. 28 (1976) 77-80.

[7] [8] [9] [lo] [l I ]

Annals of Discrete Mathematics 17 (1983) 347-359 Company

0 North-Holland Publishing

TREES AND LENGTH FUNCTIONS ON GROUPS Wilfried IMRICH and Gabriele SCHWARZ Instirut fuer Mathematik und Angewandte Geometrie, Montanuniuersitaet Leoben. A -S700 Leoben. Austria

Length functions on groups were introduced by Lyndon to axiomatize cancellation arguments used in the proofs of subgroup theorems for free groups and free products. To every group with a length function we associate a metric which can be imbedded into a tree on which the group acts as a group of automorphisms. The original length function can then be recovered from the action of the group on the tree. With the aid of results of Tits on automorphism groups of trees it is possible to give simpler proofs of many results o n length functions. Further, we characterize groups with length functions whose set of non-Archimedean elements is an unbounded group, thereby extending results of Wilkens and Hoare [14]. [ h ]on length functions and normal subgroups. We also characterize the case in which the set of Archimedean elements is a group.

1. Introduction

In an attempt to axiomatize certain cancellation arguments used in Nielsen’s proof of the subgroup theorem for free groups Lyndon [8]developed a set of axioms for a ‘length function’ from a group into the set of natural numbers. He used them to derive theorems which contain the subgroup theorem of Nielsen and the theorems of Kurosh and of Grushko-Neumann. Subsequently integerand real-valued length functions have been investigated by Harrison [ 3 ] , Chiswell [2] and Hoare and Wilkens [4],[ 5 ] ,[6], [ 141. Chiswell also established a connection with Serre’s theory of groups acting on trees [ 101 by constructing a pathwise connected contractible metric space T to every group G with a length function, such that G acts on T as a group of isometries. (For integer-valued length functions T is a tree in the graph theoretic sense.) By making use of several concepts developed for the embedding of metrics into trees Chiswell’s arguments were simplified in [7], where it was also observed that some results of Tits [12] on groups acting on trees generalize to actions of groups on so-called tree-like metric spaces T. Such generalized trees were also introduced by Tits [13] and further investigated by Tignol [ I l l . In this paper these concepts are applied to the investigation of groups with length functions. In particular, results of Wilkens and Hoare [14], [6] on length functions and normal subgroups are extended and the proofs are simplified. 347

W . Imrich, G . Schwarz

348

In the next three sections we shall summarize several results about length functions, viz., realizations of metrics by trees and automorphisms of trees. 2. Length functions and metrics x

A length function on a group G is a function from G into R, which we write as Ix I , satisfying the following axioms:

w

Al. IlI=O. A2. ( x I = I x - ' I . A4. c ( x , y ) < c ( x , z ) implies (I x I + I Y I - I XY -'IW

that

c ( y , z ) = c ( x , y),

where

c(x, y ) =

The numbering of the axioms is that of Lyndon [8]. We note that, in a free group, c ( x , y ) can be interpreted as the length of the largest common terminal segment of the reduced words representing x and y. Axiom A4 states that of the three numbers c ( x , y ) , c(x, z ) , c ( y , z ) , two are equal with the third no smaller. Since c ( x , 1) = c ( y , 1) = 0 by A l ' and A2 we thus immediately obtain Lyndon's axiom A3. c ( x , y ) a O , and subsequently I x I = c ( x , x ) 3 0. This suggests the definition of a distance d ( x , y ) = I x - l y I in G. For, d is nonnegative, symmetric, d ( x , x ) = 0 and d satisfies the triangle inequality A3, because c(x-', z - l y ) = ( J x - l y I

+ I z - ' y I - J x - I 2 1)/2 2 0

and hence d(x, z )

d ( x , y ) + d ( y , 2).

However, there may be elements x # y in G with d ( x , y ) = 0. We thus define a relation by

x

-y

if and only if

d ( x , y ) = 0.

Clearly this is an equivalence relation, the equivalence classes being the left cosets of the subgroup M of G consisting of all elements of zero length. As x z implies d ( x , y ) = d ( z , y ) by

-

d ( x , y ) zs d ( x , 2 ) + d ( 2 , y ) = d ( z , y )

-

d ( z , x ) + d ( x , y ) = d ( X , y ),

-

d is a well-defined metric on GI . It is easy to see that G acts on G I by left multiplication, since left multiplication leaves d invariant and is compatible with - .

349

Trees and length functions on groups

We have used A4 up to now only to derive A3. Thus the construction of the metric space G / on which G acts by left multiplication depends only on the axioms Al’, A2 and A3. We shall see in the next section how A4 allows us to embed G I - into a (generalized) tree on which G acts.

-

3. Embedding of metrics in trees

Let T be a (weighted) tree and the distance p between any two vertices of T be defined as usual. Then p induces a metric (V, p ) on the vertex set V of T. On the other hand, if ( X ,d ) is a metric space and if there is an isometry a of ( X , d ) into (V, p ) we say T realizes the given metric. If aX is not contained in any proper subtree of T we speak of an optimal realization. It has been shown by Pereira [9] that a finite metric ( X , d ) can be embedded into a tree if and only if every submetric on four points can be realized as a tree. Equivalently, Buneman [ l ] showed that this is the case exactly if ( X , d ) satisfies the so-called four-poinf condition : For any four elements x, y , u, u of X two of the three sums

d(x,y)+d(u,u),

d(x,v)+d(y,u),

d(x,u)+d(u,y)

are equal and not smaller than the third. Needless to say A4 implies the four-point condition on the metric ( G / ,d ) . This is not hard to show and can be found in [7]. (G/ ,d ) will be infinite in general though, and not necessarily integer-valued. We have to generalize the concept of a tree in order that Pereira’s theorem still holds.

-

-

Definition ([ll], [13]). A free T is a non-empty, metric space which has no subspace homeomorphic to a circle and in which any two points are the endpoints of an interval, i.e., for any two points x, y there exists an isometry a of an interval [0, d ] into T with a ( 0 )= x and a(d)= y .

In [7] an equivalent definition was given and it was shown that the completion of a tree is again a tree. It is thus convenient to require all trees to be complete metric spaces, as is the case in [Ill. We also slightly alter the concept of realizing a metric. Definition. A tree Trealizes a metric space X if there is an isometry a of X into T. The realization is optimal if a X is not contained in any proper subtree of T. Theorem 1 (Imrich [7]). A metric is realizable by a free if and only if it satisfies the four-point condition. Any two optimal realizations of a metric by a free are isomorphic.

350


This implies that any isometry of a metric satisfying the four-point condition uniquely extends to the optimal realization by a tree. Thus one obtains the following generalization of a result of Chiswell [2]. Theorem 2 (Imrich [7]). Let G be a group with a length function satisfying Al', A2 and A4. Then the metric ( G / ,d ) has a unique optimal realization by a tree T on which G acts as a group of isometries.

-

If T realizes (G/ isometry maps

- ,d ) there

G / -= { x M

IX

is an isometry cp of (G/

- ,d ) into

T. This

EG},

I

where M = { y I y 1 = 0}, into T. By abuse of language we will simply write x instead of q ( x M ) . Further we note that G acts on G / by left multiplication and that the action of G on T is defined by

-

g : c p ( x W w cP(gxM), i.e., g : x H gx. For any x E G we can thus consider x E T as the image of 1 E T under the action of the group element x on T. In symbols

x =xl Since y is an isometry, d ( x , y ) = I x - ' y I is the distance between cp(xM) and q ( y M ) in 7'.This means that ( x I is the distance between the points 1 and x in T. For integer-valued length functions T is a tree in the usual sense and one can apply Serre's theory of groups acting on trees for the investigation of groups with length functions, as has been done by Chiswell [2]. In the general case, however, it seems appropriate to apply results of Tits [12], [13] on automorphisms of trees to the study of length functions on groups. Finally we remark that every group G acting on a tree T as a group of isometries can be given a length function by choosing a point u in T and by defining the length of an element x in G as the distance between u and xu. By Theorem 2 every length function on a group can be defined in this manner.

4. Automorphisms of trees

An automorphism of a tree T is an isometry x of T onto itself. We say an automorphism x of T stabilizes a subset P of T if xP = P. If x stabilizes every point of P we say x fixes P. The set of points fixed by x will be denoted by T".If T' is not empty it is a subtree of T.


35 1

We will in particular be interested in automorphisms fixing and stabilizing intervals, half-lines, lines and ends of T. An interval (resp. half-line, resp. line) of a tree is the image of a closed interval [0, d ] of the real line (resp. the real half-line [0, m), resp. the real line) under an isometry a. Two half-lines L, L' are called equivalent, if L i l L' is also a half-line. The equivalence classes with respect to this relation are called ends of T. We say a subtree A of T contains an end b, if A contains some half-line in b. It is not hard to see that to any two distinct ends b, b' there exists exactly one line containing b and b'. One says this line joins b with b'. Furthermore, an automorphism x of Tstabilizes (resp. fixes) an end b if xb = b (resp. if T" contains b ) .

Theorem 3 (Tits [13]). Let x be an automorphism of a tree T. If T" is nonempty, then every subtree of T stabilized by x has a nonempty intersection with T" and T" contains every end stabilized by x. If T" is empty, then x stabilizes exactly two ends 6, 6' and every subtree stabilized by x contains the line L joining b with b'. In the latter case we say that x induces a nontrivial translation on L. This is justified by the following observation: Let a be an isometry of the real line onto L. Then a - ' x ( ~is an isometry of the real line without fixed points, and therefore a translation. We shall further find it convenient to make use of the following results.

Lemma 1 (Tignol [ll]). Let x be an automorphism of a tree T and v E T. If T" # 0, the interval [ v , x v ] contains exactly one point of T'. This point is the midpoint of the interval. Lemma 2 (Tignol [ 111). Let G be a group of automorphisms of T. If every element of G fixes at least one point of T but if G fixes no point of T, then G fixes exactly one end of T, 5. Archimedean and non-Archimedean lengths Let G be a group with a length function. An element x of G is Archimedean if lxzl > Ix I and non-Archimedean if l x z l S I x I . Further, let N denote the set of non-Archimedean elements of G. A length function is called Archimedean if N = (1) and non-Archimedean if N = G. The next lemma characterizes Archimedean elements by their action o n the unique optimal realization T of (G/ , d ) .

-

Lemma 3. A n element x of G is Archimedean if and only if T' = O . It is non-Archimedean if and only if it has a fixed point.

352

W. Imrich, G. Schwarz

Proof. Suppose T' # 0. Then, by Lemma 1, the midpoint u of [l,x I] is the only point of this interval in T'. Let n be an integer. As the length of x " is the distance between 1 and x " l it is at most equal to the distance between 1 and u plus the distance between u and X " 1, the latter being the same as that between 1 and u because x " [ l , u ] = [ x " l , u ] . Thus I x " Is 1x1

for every x with T" # 0. If T' = 0, let L be the line on which x induces a nontrividl translation. Further let u be the projection of 1 onto L, i.e., the unique point on L for which [ I , u ] n L = { u } . Clearly x [ l , u] n L = xu and x z [ l , u ] n L = x'u. Thus IX2l>IXI~

and 1 x21- I x 1 is the distance by which every point of L is moved under the action of x (Fig. 1).

Fig. I .

From this figure it is also clear that the following relations hold for any x P N and any positive n :

I = l x l + ( n - N I x 2 l - b I), 2 ( x " J - I x L n1 = 2 ( x I - l x 2 1 . Further it is clear that Ix'I - ( x I is invariant IX"

under conjugation (Harrison [ 3 ,

Theorem 2.8]), i.e.,

- I x 1 = 1 y x y 1 - 1 yxy-' 1,

JXZJ

because if x shunts L by a certain distance then yxy-' shunts y L by the same distance. We also note that T x is nonempty if and only if T y x Y is.- 'Thus the set N of non-Archimedean elements is normal in G (Wilkens [ 14, Proposition 3.41). We also easily obtain the following strengthened version of a result of Harrison [3, Theorem 3.101. Theorem 4. Let G be an abelian group with a length function. If N is a proper subgroup of G there exists a unique line L in T stabilized by G. If N is trivial, then G is a subgroup of the additive group of real numbers.

Trees and lengrh functions on groups

353

Proof. Let x E G \ N and y E G. By Lemma 3 there exists a unique line L stabilized by x in the optimal realization T of ( G / , d ) . Since G is abelian,

-

x ( y L ) = y x y - ' ( y l ) = yL. By the uniqueness of the line stabilized by x we thus infer yL = L, which proves the first assertion of the theorem. If N is trivial every nonidentity element of G induces a translation on L. The observation that L is an isometric imbedding of R into T concludes the proof of the theorem.

6. Bounded lengths on N

If the elements of N are bounded, the subtree S of T spanned by the images of 1, N I ={xi

IX

EN),

under the action of N is bounded. Clearly S is invariant under the action of N on T. Since every bounded (closed) tree S has a unique center s (Imrich [7, Proposition 41) every element of N fixes s. Thus N consists of all elements of G fixing s and is a (normal) subgroup. This case is characterized by Wilkens [ 141 with the concept of an extension of a length function. A length function x I+ Ix 1 on G is an extension of a length function x I+ lx 1, on K , a normal subgroup of G, by a length function x I+ Ix on H = G / K if

1

1x1,

1x1

=

if x E K,

If(x)12 if xtif K ,

where f : G -+ H

=

G / K is the projection homomorphism.

Theorem 5 (Wilkens [14, Theorem 5.31). If G is a group with a length function and N # G then the lengths of the elements of N are bounded if and only if one of the following equivalent conditions is satisfied : (i) l a x l = ( x l f o r a l l a E N , x E G \ N ; (ii) N is a normal subgroup of G and the length function on G is an extension of a non-Archimedean length function on N by a n Archimedean length function on H = GIN. Proof. Clearly (ii) implies (i). For let f : G + H = G / N be the projection homomorphism. Then f ( a x ) = f ( x ) for any a E N, x E G \ N and

W . Imrich, G. Schwarz

354

I 4 = 1x1 by the definition of the extension of a length function. Suppose (i) is satisfied and let a E N, x E G \ N be arbitrarily chosen. Since N is normal there is an a ' E N with a'x = x u . Thus ~ x l = / a ' x l = ~ x a By l . the triangle inequality, d ( x , x u ) S d(1, x ) + d(1, xu), i.e., l a I = 4 xI

As x was arbitrarily chosen this means that the lengths of the elements of N are bounded. Let N be bounded. Then N is a normal subgroup. Choose a E N , x E G \ N and let L be the line stabilized by x in the unique optimal realization T of ( G / - , d ) . Clearly axa-' acts as a translation on aL. We wish to show that aL = L and that a fixes every point of L. Suppose first that aL and L are disjoint or have at most a finite interval in common. Let u be the projection of 1 onto L. Then the distance between x " u E L and axnu E aL can be made arbitrarily large by increasing n. As d ( x " , x " u ) = d ( a x " , ax"u) = d(1, u ) the distance between X " and ax" can also be made arbitrarily large. However, since N is normal there exists an a' with ax" = xna' and

I a'1=

d(1, a ' ) = d ( x ",x " a ' ) = d ( x " , ax"),

in contradiction to the boundedness of N. Thus a L and L have at least a half-line in common. Let w E L . Then the interval [ w , a w l is in L U aL and the midpoint u,which is fixed by a, is in L and aL. Let L , and Lr be the two half-lines into which u separates L. If a is not the identity on L, then it cannot be the identity on L , and Lz. Suppose a L , # L , . Then a L , f l L I is at most of finite length. Let u be defined as above. Then it is clear that either d ( x " u , a x n u ) or d ( x - " u , a x - " u ) can be made arbitrarily large, contrary to the boundedness of N. This proves that a fixes every point of L . By the way, this implies that I a I < I x 1, not only I a I S 2 1 x I as previously shown, because

I

l a = d ( 1 , a ) S d(1, u ) + d ( u , a ) = 2d(l, u ) < I x I.

The identity 1 ax I = I x 1 follows from

d ( l , x ) = d(l,xu)+d(xu,x), and

d ( l , a x ) = d ( l , a x u ) + d ( a x u , a x ) =d ( l , x u ) + d ( x u , x ) . It remains to be shown that the given length function on G is an extension of a non-Archimedean length function on N by an Archimedean one on GIN.


355

To see this we note that TIN, i.e., the space obtained from T by identifying any two points equivalent under elements of N, is again a tree. Further, since every line L stabilized by an element x E G \ N is fixed pointwise by every a f N there is an isometric injection of these lines into TIN. Finally, if v is the projection of 1 onto L, it is clear by d ( 1 , v ) = d (a,av ) = d (a, v )

that no two points of the interval [l,v ] are equivalent under any element of N. ( T I N is a subtree of T.) Since the points of T I N are in bijective relation to the orbits of N on T we can define the action of GIN on T I N by

( x N ) ( N w )= Nxw. By the above,

dTIN(N,XN)= d , ( l , x ) for any x E G \ N and

l N / = O together with l x N l = l x l for x E G \ N define an Archimedean length function on GIN. Of course, this can also be verified directly. 7. Abelian groups We show first that N is a group when N is abelian and then that N is bounded when G is abelian and N f G. Lemma 4. Let G be a group with a length function. If N is abelian, then N is a group.

Proof. Let a, b E N and let v be a fixed point of a in the unique optimal realization T of ( G / ,d ) . Since

-

abv = bav = bv,

a fixes bv and thus also the interval [ v , b v ] .By Lemma 1 we infer that the midpoint w of this interval is also fixed by b. Thus abw = w and ab is in N since it has a fixed point.

Theorem 6 (Wilkens [14]). If G is an abelian group with a length function and if N is a proper subgroup of G, then N is a bounded subgroup of G. Thus the length function x + l x I on G is the extension of a non-Archimedean length function

W, Imrich, G. Schwarz

356

x + Ix 1, on N by an Archimedean length function x +Ix on H = GIN. Moreover, there exists a c 2 0 and an embedding g :H +R such that c is an upper bound for the lengths of the elements of N and Ix 12 = c + l&)l for x E H, x # 1. Proof. By Theorem 4 every element of G stabilizes one and the same line L in T. Let a E N . If a is not the identity on L it must induce a reflexion on L, but then a and x cannot commute because x induces a translation on L. Thus every element in N fixes L and N is a bounded subgroup of G. By Theorem 5 the given length function on G is the extension of a non-Archimedean length function x -+ Ix 1, on N by an Archimedean length function x - - , ~ x /on ~ H = GIN. We note that H acts on L as a group of translations. Further, since L is an isometric embedding of R into T there is an induced action of H on R as a group of translations, which yields the claimed embedding g : H R. Finally, inspection of Fig. 1 shows that 1 g(x)l = Ix.'l - I x 1 for x E G \ N and that -+

Ix I2 = c + lg(x)l, where c/2 is the distance between 1 and L in T.

8. Unbounded lengths on N

The next theorem characterizes the case when N is an unbounded subgroup. Theorem 7. Let G be a group with length function and N a proper subset of G. Then N is an unbounded subgroup of G if and only if N has no fixed point, but a fixed end b. (This end is uniquely determined and is the only end stabilized by G.)

Proof. Let N be a proper subset of G without fixed point but with a fixed end b. Then the product xy of any two elements x, y E N also fixes b and thus has a fixed point. Hence xy E N and N is a group. Since N has no fixed point it is unbounded. If N is an unbounded subgroup it fixes exactly one end b of T by Lemma 2, because every element of N fixes a point, but N does not fix a point. Let x E G \ N and a E N . Then there is an a ' € N with ax = xu'. By

a(xb)=x(a'b)=xb, a stabilizes the end xb. Since elements in N have fixed points they can only fix


357

ends and since b is the only end of T fixed by every a in N we must have xb = b. Thus G stabilizes 6. Clearly there can be no other end c of T stabilized by G, for then it would have to be fixed by all a in N . To obtain an example of a length function on a group G such that the subgroup N of all non-Archimedean elements is unbounded it thus suffices to pick an end b of a reasonably homogeneous tree T, to let G be the group of all isornetries of T stabilizing b and to set N equal to the subgroup fixing 6. A corresponding length function is then obtained by the choice of an arbitrary point u and by defining the length 1x1 of an element of G as the distance between u and xu. As a more concrete example we further list one due to Hoare and Wilkens [6]. Let G be the group defined by

I

G = (u,g, i E B , u-'g,u = g,+,).

G is a so-called HNN-extension of a free group N on the free generators g,, i E Z, with a single stable letter u. It is known that every element of G has a unique representation of the form au r ,

where a E N and r E h. If a = gy,'g:: * * . g& in reduced form, define

rn(a) = 2max{i,, .. . , i k } , and the length of x = au' by

1 au '

= max{rn ( a )

+ r, r, - r }.

Hoare and Wilkens [6, Theorem 21 show that i..is is a length function on G such that the subgroup N has unbounded lengths, and consists of all the nonArchimedean elements of G. For details we refer to [6]. We should like to remark though that the unique end b fixed by the action of N on the optimal realization T of ( G / - d ) is defined by the half-line containing the points 1, u-',u - 2 , . . . and that a similar construction works if N is the free abelian group on the generators g,, i E Z . 9. Archimedean subgroups

Let G be the group of isornetries of the real line. We obtain a length function on G by choosing a point u on this line and by defining the length of every element x in G as the distance between u and xu. Clearly the Archimedean elements of G form a subgroup isomorphic to the additive group of real numbers and the index of this subgroup in G is two. The non-Archimedean elements, on the other hand, form a set of unbounded lengths.


358

Theorem 8. Let G be a group with a length function. If the set of Archimedean elements together with 1 is a proper subgroup A of G, then A is isomorphic to a subgroup of the additive group of real numbers. Proof. By Theorem 4 it suffices to show that A is abelian. Let x E A \{l},L the line on which x induces a translation, and let a be arbitrarily chosen in G \ A . We show first that aL = L. To see this let u be a fixed point of a and u the projection of u onto L. Then x " u is the projection of x "U onto L. We further note that x "a must be in N since A is a group and that xnau = X " U . By Lemma 1 this implies that the midpoint w of the interval [ U , X " U U ] is fixed by X " Q . We note that w is also the midpoint of the interval [ u , x " u ] . Hence w is on L. From x"aw = w we infer aw = x-"w and thus also a w # w. As d ( u , w ) = d ( u , a w ) it is clear that aw is the reflexion of w in u on L (Fig. 2). By the uniqueness of paths in T the element a thus maps [ u , w ] into [u, a w l . Since x " u has arbitrarily large distance from u it follows that a reflects L in u, and thus aL = L. Further, let y be an arbitrary Archimedean element. Then ay is in N since A is a group and thus ( a y ) L = L. But then yL = a - ' L = L and x and y commute.

aw=xnw

7-7" u=au

x"u

Fig. 2.

References [I]P. Buneman, A note on the metric properties of trees, J. Comb. Theory, Ser. B 17 (1974) 48-50, [2] 1.M. Chiswell, Abstract length functions in groups, Proc. Cambridge Philos. Soc. 80 (1976) 451-463. [3] N. Harrison, Real length functions in groups, Trans. Amer. Math. Soc. 174 (1972) 77-106. [4] A.H.M. Hoare, On length functions and Nielsen methods in free groups, J. London Math. SOC. 14 (1976) 188-192. [S] A.H.M. Hoare, An embedding for groups with length functions, Mathematika 26 (197Y) YY-102. (61 A.H.M. Hoare and D.L.Wilkens, On groups with unbounded non-Archimedean elements, manuscript. 171 W. Imrich, On metric properties of tree-like spaces, Beitrage zur Graphentheorie und deren Anwendungen, Oberhof (DDR), (1977). 129-156; edited by Sektion MAROK der Technischen Hochschule Ilmenau.


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[8] R.C. Lyndon, Length functions in groups. Math. Scand. 12 (1963) 209-234. 191 J.M.S. Simbes-Pereira, A note on the tree realizability of a distance matrix, J . Comb. Theory, 6 (1969) 303-310. [ 101 J.-P. Serre, Groupes discrets, Extrait de I’Annuaire du College de France 1969-1970. I I I ] J.-P. Tignol, Remarque sur le groupe des automorphismes d’un arbre, Ann. SOC.Sc. Bruxelles 93 (1979) 196-202. [ I ? ] J . Tits, Sur le groupe des autornorphismes d’un arbre, in: Essays on Topology and Related Topics (Memoires dkdies 5 G. De Rham) (Springer, Berlin-Heidelberg-New York, 1970) pp. 188-211. [I31 J . Tits, A “theorem of Lie-Kolchin” for trees, in: Contributions to Algebra: A Collection of Papers Dedicated to Ellis Kolchin (Academic Press, New York-San Francisco-London, 1977) pp. 377-388. [I31 D.L. Wilkens. Length functions and normal subgroups. J. London Math. SOC.22 (1980) 339348.



MAXIMAL CYCLES IN BIPARTITE GRAPHS Bill JACKSON Mathematics Department, Goldsmiths' College, London, England

The purpose of this note is to describe several sufficient conditions for long cycles in bipartite graphs, and to mention some related problems. In the following, G will denote a simple bipartite graph with bipartition V(G) = A U B where ( A ( = a and IBI=b. Suppose each vertex of A has degree at least k. It can easily be seen that G need not contain cycles of length greater than 2 min{a, k } . The following theorem determines the maximum function f ( a , k ) such that if b S f (a, k ) , then G must contain a cycle of length at least 2min{a, k } . Theorem 1 111. Suppose G is such that each vertex of A has degree at least k 2 2. (i) Zf a s k and b S 2k - 2, then G contains a cycle of length 2a. (ii) If a a k and b S [ a / ( k - 1)1 ( k - l), then G contains a cycle of length at least 2k. The fact that the function f(a, k ) given in Theorem 1 is maximum follows by considering a separable graph, with a single cut vertex belonging to B, and each block containing k vertices of B. Let g ( a , k ) be the maximum function such that if b s g(a, k ) , then G must contain a cycle of length 2i for all i, 2 s i s min(a, k ) . It follows from Theorem l(i), by deleting all but i vertices of A from G, that g(a, k ) a 2 k - 2 . Moreover, the example above shows that g(a, k ) = 2k - 2 for a S k. It remains an open problem, however, to determine g(a. k ) for a > k . Our only bound follows from the fact that if b 3 k d / a then G need not contain a 4-cycle. Since, however, G must contain a 4-cycle if b S d a k ( k - I), we are tempted to conjecture that g ( a , k ) > d a k ( k - 1). Another open problem would be to increase the bounds on b given in Theorem 1, under the added hypothesis that G is '-connected.

Conjecture 1. Suppose G is 2-connected and each vertex of A has degree at least k. (i) If a s k and b c 3(k - 2 ) + 1, then G contains a cycle of length 2a. (ii) Zf a 2 k and 36 1

B. Jackson

367

where &k is equal to one if k is even, and zero if k is odd, then G contains a cycle of length at least 2k.

In a slightly different vein, Voss and Zuluaga [3]have shown the following. Theorem 2 [3]. Suppose G is 2-connected and has minimum degree 1. Then G contains a cycle of length at least 2 min{a, b, 21 - 2}. Let Kk.l be the complete bipartite graph with k vertices in one set, and I vertices in the other, and u and u be two vertices of the 1-set of Kk.I.By considering the bipartite graph obtained by taking several disjoint copies o f Kk., for k I, and associating each of the vertices u and u to two vertices u * and u * respectively, one can see that the bound on the length of a maximal cycle, obtained in Theorem 2, is, in a sense, best possible. We have been able to improve the bound, however, for bipartite graphs in which the largest set o f the bipartition has large minimum degree. Theorem 3 [2]. Suppose G is 2-connected, and that each vertex of A and B has degree at least k and 1 respectively. If a 2 b, then G contains a cycle of length at least 2 min(b, k + 1 - 1,2k - 2). Let a, b, k, and I be integers such that a b 3 k 2 1 2 2. By considering the graph obtained from two disjoint complete bipartite graphs K,.,, and K l . b - kby , joining each vertex in the k-set of KO-,.,to every vertex in the 1-set o f K h one may deduce that the bound on the length o f a longest cycle obtained in Theorem 3 cannot be improved, even for bipartite graphs of connectivity greater than two. One could still improve the statement of the theorem, however, by relaxing the condition that a 3 b. Using Theorems 1 and 3, we have recently determined the minimum number of edges required in a bipartite graph to insure the existence of a cycle of length at Least 2m. Theorem 4 [2]. Let G be a bipartite graph and m be an integer such that 2 s m s a a b . If

[

a

,E(G)I >

+ (b - l ) ( m- 1 )

(a+b-2m+3)(m-l)

then G contains a cycle of length at least 2m.

for a

S

2m -2,

foraa2m-2,

Maximal cycles in bipartite graphs

36.3

Let G, be the graph obtained from K , , - , . b by adjoining a set of ( a - rn + I ) vertices and joining each new vertex to a single vertex in the b-set of K,, -,.,,. Let G, be the separable graph with two blocks K,-,,+,.,,,-, and K,,-I.h-m+.' where the cut vertex lies in the (rn - 1)-set of the first block and the ( b - rn +')-set of the second. Then the graphs G I and GZdemonstrate that Theorem 4 is, indeed, best possible. Again, however, it remains an open problem to improve the bounds on I E ( G ) l under the added hypothesis that G is 2-connected. We close by giving two further conjectures concerning cycles in bipartite graphs which are perhaps more attractive than those mentioned above.

Barnette's Conjecture. Every 3-regular, 3-connected, bipartite planar graph is hamiltonian. Haggkvist's Conjecture. Every k -regular, 2-connected bipartite graph on at most 6 k vertices is hamiltonian.

References 111 B. Jackson, Cycles in bipartite graphs, J. Comb. Theory, Ser. B 30 (1981) 332-342. (21 B. Jackson, Long cycles in bipartite graphs, submitted. [3] H.J. Voss and C. Zuluaga, Maximale gerade and ungerade kreise in Graphen I. Wisz. 2 . Techn. Hochsch. Ilmenau 23 (4) (1977) 57-70.



EXTENSION OF TURAN’S AND BROOKS’ THEOREMS AND NEW NOTIONS OF STABILITY AND COLORING IN DIGRAPHS H. JACOB C1M.S. S4 bd. Raspail. 7‘i0#6 Paris. France

H. MEYNIEL 4.7 rue Mazarine. 7.5006 Paris, France We define notions of quasi stability and quasi coloration in directed graphs and we extend Turin’s and Brooks’ theorems.

1. Introduction

Let G = (V, E ) be a digraph without loops or multiple edges. We shall call quasi stable a subset S of V such that the subgraph G s of G generated by S does not contain any circuit. We shall call quasi coloring a partition of G in quasi stable sets. Note that if G is a symmetric graph the notions of stability and coloring are identical with the notions of stability and coloring in the corresponding nonoriented graphs. Our purpose in this paper is to give some generalizations for digraphs using the above definitions of classical theorems related to nonoriented graphs, in particular, Turin’s Theorem [7], Brooks’ Theorem [ 2 ] , Lovasz’s Perfect Graph Theorem [4],and also a conjecture which is in fact equivalent to Hadwiger’s Conjecture [ 3 ] . We shall denote by a ‘ ( G )the maximal cardinality of a quasi stable and by y ’ ( G ) the minimal cardinality of a quasi coloring of G. Definitions and notations are those of Berge [ 11. 2. Turan’s Theorem

Let us consider two integers n, k with n 3 k > O and note that 9 = [n/k]*= [(n - I ) / k ] + 1 and hence n = k ( q - I ) + r, O < r S k . Let us consider the graph Gn.kformed with k distinct cliques, r of them having 9 vertices, the ( k - r ) others having 9 - 1 vertices. (Cliques are subsets of vertices joined in any possible manner.) We have the following theorem. 365

H. Jacob, H . Meyniel

366

Theorem 2.1. If G is a graph with n ( n 3 k ) vertices with a t ( G ) n VA C V ; (2') a ( G i ) = 8(GA) VA C V; (3') -y(GA)= w ( G ; ) VA C V. Thus we obtain the above equivalence in G.

References [ I ] C. Berge, Graphes et Hypergraphes (Dunod, Paris, 1973). [2] R.L. Brooks. On coloring the nodes of a network. Proc. Cambridge Phil. Soc. 37 (1941) 103-197. [ 3 ] H. Hadwiger. Uher eine Klassifikation der Streckencomplexe, Vierte Naturforsh Ges. Zurich SS (IY33) 133-132. [ 3 ] L. Lovasz, Normal hypergraphs and the perfect graph conjecture. Discrete Math. 2 (1972) 253-257. 151 E . A . Nordhaus and J.W. Gaddum. On complementary graphs. Ann. Math. Monthly h3 (lCJ5h) 175-1 77. [hl I . Tomescu. Sur le probltme du coloriage des graphcs geniralists. Note in C.R. Acad. Sci. Paris

267 (1968) 250-252. 171 P. T u r i n . An extremal problem in graph theory, Mat. Fiz. Lapok 48 (1941) 436-452.


SYMMETRIC REPRESENTATIONS OF BINARY MATROIDS F. JAEGER I.M.A.G.. B.P. 53X, 38041 Grenoble Cedex, France To every symmetric matrix A with coefficients in GF(2) is associated on the one hand the binary matroid M ( A ) (matroid of the linear independence of columns, of which A is a representation), and on the other hand the simple graph G ( A ) , the vertex-to-vertex adjacency matrix of which has the same non-diagonal elements as A. W e characterize some classes of binary matroids (bipartite, bicyclic, without bicycles, cographic, planar without hicycles and some others) by the existence of a symmetric representation A with certain properties (most of them involving properties of the graph G ( A ) ) .

I . Definitions

I . I . Binary matrices, spaces and matroids Let A be an (n x m)-matrix over GF(2). Let E be the set of columns of A we denote by A ' the corresponding column-vector of A, which is an element of [GF(2)]".The subsets I of E such that (A ', e E I ) is a linearly independent family of vectors form the independent sets of a binary matroid on E which we shall denote by M ( A ) ; A is a representation o f M ( A ) (over GF(2)) ([ll]). Let E = (el,. . . ,e,?,}.We identify each subset F of E with its representative vector cf,, . . . ,f,") E [GF(2)]", where fi = 1 iff e, E F. Then the elements of the space KerA are identified with the cycles of M(A). The circuits of M ( A ) are then the non-empty cycles minimal by inclusion. The cocycles of M ( A ) are the elements of the space (KerA)'; this space is generated by the rows of A. In the sequel all matrices have coefficients in GF(2).

(I E I = m); for every e E E

1 . 2 Symmetric matrices and graphs

For definitions on graphs the reader should refer to [l]. Let A be an (n x n ) symmetric matrix. We denote by G ( A ) the simple graph on n vertices u , , . . . , un,where ui is adjacent to u, (i# j ) iff the element of A on the ith row and jth column is equal to 1. G ( A ) is the graph of A. 37 1

F. Jaeger

372

2. Representation theorems

2.1. Universality of symmetric representations We shall characterize some classes of binary matroids by the existence of symmetric representations of a given kind. The first and simplest result of this type is the following.

Proposition 1. A matroid is binary if it has a symmetric representation over GF(2). Proof. Let M be a binary matroid on n elements of rank r ; M has a representation over GF(2) of the form

R

= [I, :C ] ,

where I, is the ( r X r)-identity matrix and C is a ( r x ( n - r))-matrix. Then consider

Clearly R is a symmetric ( n X n)-matrix. Moreover R ' has the same row space as R since ['C : 'CC] = 'C [ I ,: C]. Hence R' is a symmetric representation of M. From now on all matroids are assumed to be binary. 2.2. 1-diagonal symmetric matrices and bipartite matroids A square matrix is said to be a-diagonal (0E GF(2)) whenever all its diagonal elements are equal to LY. A matroid is said to be bipartite whenever all its cycles (equivalently: all its circuits) are of even cardinality (see [lo]).

Proposition 2. A matroid is bipartite if it has a I-diagonal symmetric representation. Sketch of Proof. (a) If M is a bipartite matroid, the proof of Proposition 1 gives a symmetric representation R ' of M which is always 1-diagonal; indeed it is easy to show that every column of C has an odd number of 1's and thus 'CC is I-diagonal. (b) Conversely, let A be a 1-diagonal symmetric matrix. Let X be a cycle of M ( A ) ;we shall consider X as a set of columns and also as a set of rows. The

Symmetric represenrations o/ binary matroids

373

corresponding square symmetric submatrix of A (row set X , column set X ) has an even number of 1's (the sum of the columns is the zero-vector), hence 1 X I is even.

2.3. Bicyclic matroids and bipartite graphs Let M be a binary matroid. A bicycle of M is a cycle of M which is also a cocycle of M. The bicycles of M form a space B ( M ) ; the rnatroid with cycle space B ( M ) is called the bicycle matroid of M . A matroid will be called bicyclic if it is the bicycle matroid of some matroid. Clearly a bicyclic matroid is bipartite. Proposition 3. A matroid is bicyclic iff it has a 1-diagonal symmetric representa tion, the graph of which is bipartite. The proof uses a representation of binary matroids by bipartite graphs introduced by de Fraysseix in [4].

2.4. Matroids without bicycles A matroid is without bicycles if its only bicycle is the empty set (the zero-vector). Such matroids have nice properties (see [4], [7], [9]).

Proposition 4. A matroid is without bicycles iff it has an idempotent symmerric representation.

Proof. Let A be a symmetric ( n X n ) matrix which is idempotent, i.e.. A' = A . This can be written A ( A + I ) = 0. (I is the unit ( n x n ) matrix.) Then if % and .I[ are the row spaces (equivalently column spaces) of A and A + I respectively, we have % C Yt'. On the other hand, V i E { I , . . . , n } the sum of the ith columns of A and A + I has only one non-zero coordinate at i t h position. It follow5 that 'f2 + X has dimension n. Now,

n = dim(%' + Yt) = dim % + dim Yt - dim(%'n X) = dim %'

+ ( n - dim X') - dim(%' n X).

Hence dim(V n X)+ (dim X '- dim %')= 0. Each term of this sum being non-negative must be zero. Hence % = 3t' and %' n 5Y = (0). The matroid with cocycle space 72 is without bicycles (% n %' ' = {0}) and ha? representation A.

374

F. Jaeger

The converse part of this proof is easy using the concept of interlacement graph of a matroid without bicycles introduced in [8] by Rosenstiehl.

2.5. Cographic matroids and chord graphs A matroid is cographic if it is isomorphic to the cocycle matroid of some graph (see [ 1 11, Section 2.4). A chord graph is a simple graph isomorphic to the intersection graph of a family of chords of a circle (see 131 for an introduction to chord graphs). Proposition 5. A matroid is cographic iff it has a symmetric representation, the graph of which is a chord graph.

The proof of this result is given in [5]. One can also prove the following. Proposition 6. A matroid is cographic and bipartite iff it has a 1-diagonal symmetric representation, the graph of which is a chord graph.

(One half of the proof is given by Propositions 2 and 5 . ) Using a result of de Fraysseix ([4],Proposition 6), it is easy to show the following. Proposition 7. A matroid is the bicycle matroid of the cycle matroid of a planar graph iff it has a I-diagonal symmetric representation, the graph of which is a bipartite chord graph.

From Propositions 5 and 7 it follows that the bicycle matroid of the cycle matroid of a planar graph is cographic. Proposition 8. A matroid is the cocycle matroid of a planar graph and is withour bicycles iff it has an idempotent symmetric representation, the graph of which is c1 chord graph. Proof. Let A be an idempotent symmetric matrix, the graph of which is a chord graph. A and A + I represent two dual matroids (see the proof of Proposition 1) without bicycles, and both are cographic by Proposition 5. By Whitney’s Duality Theorem, these matroids correspond to dual planar graphs. The converse part of the proof follows from a result by Rosenstiehl ([8], Theorem 12).

We conclude with a result related to topological graph theory.

Symmerric represenrarions of binary marroids

375

Proposition 9. A matroid is the cocycle matroid of a graph which can be 2-cell imbedded in an orientable surface with exactly one face iff it has a 0-diagonal symmetric representation, the graph of which is a chord graph. Remark. The graphs which can be 2-cell imbedded in an orientable surface with exactly one or two faces are called upper-embeddable; the study of these graphs has led to many interesting problems and results; see [2], [ti], [12], [13]. 2.6. Graphic matroids A matroid is graphic if it is isomorphic to the cycle matroid of some graph (see [ I I , Section 1.101). Let G be a graph. The algebraic line-graph of G is a simple graph, the vertex-set of which is the edge-set of G, two edges being adjacent if they meet exactly once (i.e., loops become isolated vertices and two parallel edges are not adjacent). We have the following partial result.

Proposition 10. A matroid is graphic with even rank iff it has a 0-diagonal symmetric representation, the graph of which is an algebraic line -graph. Sketch of Proof. A matroid is graphic with even rank iff its cocycles are the cocycles of a connected graph with an odd number of vertices. On the other hand, the cocycles of a connected graph G with an odd number of vertices are generated by the cocycles associated to the boundaries of edges of G ; these cocycles are precisely the rows of the vertex-to-vertex adjacency matrix of the algebraic line graph of G . 2.7. The 0-diagonal case

It follows from Euler’s formula that a graph 2-cell embedded in an orientable surface with exactly one face has even Betti number, i.e., its cocycle matroid has even rank. Hence the 0-diagonal symmetric matrix, the existence of which is asserted by Proposition 9, has even rank. Proposition 10 presents another kind of 0-diagonal symmetric matrix with even rank. In fact we have the following general result. Proposition 11. Every 0-diagonal symmetric matrix has even rank. The proof will appear elsewhere.

376

F. Jaeger

References [l] C. Berge, Graphes et Hypergraphes (Dunod, Paris, 1974). [2] 0. Chevalier, F. Jaeger, C. Payan and N.H. Xuong, Odd rooted orientations and upperembeddable graphs, in: C. Berge et al., eds., Combinatorial Mathematics, Proc. Internat. Coll. on Graph Theory and Combinatorics, Annals of Discrete Mathematics 17 (North-Holland, Amsterdam, 1983) pp. 177-181 (this volume). [3] J.C. Fournier, Graphes de cordes, hypergraphes de chaines d’un arbre et matroi’des graphiques, in: C. Benzaken, ed., Actes du Colloque “Algkbre Appliqute et Combinatoire”, Grenoble (1978) 164-171. 141 H. de Fraysseix, Local complementation and interlacement graphs, Discrete Math. 33 (1981) 29-35. [ 5 ] F. Jaeger, Graphes de cordes et espaces graphiques, to appear. [6] R.D. Ringeisen, Survey of results on the maximum genus of a graph, J. Graph Theory 3 (1979) 1-13. [7] P. Rosenstiehl, Mot et vecteur d’un graphe mile, in: J.C. Bermond and Cori, eds., Journees de Combinatoire et Informatique (1975) 317-328. [8] P. Rosenstiehl, Les graphes d’entrelacement d’un graphe, Coll. Inter. CNRS no 260, Problemes Combinatoires et Thtorie des Graphes, Orsay (Editions du CNRS, 1976) 359-362. [9] P. Rosentiehl and R.C. Read, On the principal edge tripartition of a graph, Ann. Discrete Math. 3 (1978) 195-226. 101 D.J.A. Welsh, Euler and bipartite matroids, J. Comb. Theory 6 (1969) 375-377. 111 D.J.A. Welsh, Matroid Theory (Academic Press, London, 1976). 121 N.H. Xuong, Upper-embeddable graphs and related topics, J. Comb. Theory 26 (1979) 226-232. 131 N.H. Xuong and C. Payan, Sur un thtorkme min-max en thtorie des graphes (d’aprts L. Nebesky) in: C. Berge et al., eds., Combinatorial Mathematics, Proc. Internat. Coll. on Graph Theory and Combinatorics, Annals of Discrete Mathematics (North-Holland, Amsterdam, 1983) pp. 527-534 (this volume).

Annals of Discrete Mathematics 17 (1983) 377-382 Publishing Company

0 North-Holland

SUMS OF VECTORS AND TURAN’S GRAPH PROBLEM G.O.H. KATONA Mathematical Institute, Reriltanoda 1-3-15, 1053 Budapest, Hungary

1 . Introduction Let X be a Hilbert space and 0 s rn Q n integers. If a , , . . . , a, E X , then N , ( a l , . . . , a,) denotes the number of sets A = { i l , . . . , i K } , [ A 1 = g such that

Finally, let us consider the minimum N , ( X . n , m ) = minN,(al ...., a,), where the minimum is taken over all the 5 quences al, . . . , ( 1 d i 4 m ) , [ ) a ,I[< 1 ( m < i Q n); n and rn are fixed. The aim of this paper is to investigate the values N ? ( X ,n, m ) and N , ( X , n, m ) . I t is. following the author’s talk in Marseille, a version of [6], with less proofs, but giving a wider view of the subject,

2. Vectors and graphs Theorem 1 ( [ 3 ] ) . If X is a Hilbert space of at least two dimensions then

Proof. I t is easy to see that for any three vectors al, a:, a 3 satisfying IIaII(,Ila,ll, there is a pair i f j with 11 a, + a, [I 2 1. I t is enough to verify this for 3 dimensions. Then it follows for any Hilbert space as the 3 vectors span a %dimensional subspace. Suppose al,. . . , a, satisfy the condition [la,11 3 1 (1 S i S m).Define a graph G = ( X , E ) o n the vertex-set X = { u , ,. . . , a,”}. The unoriented edge { a , ,a,} is in E iff ) / a , a, 1) 3 1. By the above remark, any 3 vertices of G span at least one

11 a l l [2 I

+

311

G.O.H. Katona

378

edge, that is, G contains no spanned empty triangle. So we can use the following special case of Turan's theorem [12]: If a graph on m vertices contains no spanned empty triangle then the number of edges is at least [((m- 1)/2)']. Therefore, N,(a,, . . . ,

that is,

The following construction proves the equality. X is not one-dimensional. consequently there are two orthogonal vectors e and f in X with lengths Ile 11 = 0.99, (1 f 11 = 0.2. Put a l = * = akm121 = e -f , = * . . = a, = - e -f and a,,,+l= * = a, = f . Here, 112f 11 = II(e - f ) + ( - e - f ) / l = 0.4, Il(e-f)+fl/=/l(-e-f)+fll=0.99 hold, hence /la,+a,II>1 occurs iff either 1 =S i, j S [ m / 2 J or [m/21 < i, j 6 m. The number of these pairs is really [(m- 1)/2]'. The proof is complete.

-

+

It is interesting that the one-dimensional X needs somewhat more complex tools.

Lemma 1 [4]. If thegraph G = ( X l U X 2 , E ) ( X , n X 2 = 0I X , 1l=nl,IXzI=nz) contains no empty triangle spanned by at least two vertices from X 2 , then

Unlike in the higher-dimensional case, here the sums ai + ai, 11 ai 11 5 1,II a, 11 < 1 should also be considered. In the optimal case they are not all smaller than 1. The above lemma makes us able to use these sums, too. The following theorem follows by a proof similar to that of Theorem 1.

Theorem 2 ([7]).I f X is the real line then

if n 2 2 m .

Sums of vectors and Turan's graph problem

379

There is a more general question considered in the literature. The following notation is needed for posing it:

N, ( X , c, n, m ) = min No ( a ,, . . . , a, ), where 0 s c < m, 0 6 m S n and the minimum is taken over all the sequences a ,, . . . , a,, I)a,Il>c ( I s i s m ) , IIa,I( x holds for 1 S i S m. Therefore

The inequality 2N2(X,n, m ) + m t(m - 2 m ) + m - I 3 - 2 ;)2 n2 n’

(

follows from Theorem 1. However, m / n = P(II 6

I(

2

(7)

x). By (6) and (7) we proved

PI P( 11 5 + 77 (1 2 x ) 3 4P’(l( 5 II 3 x ).

(8)

However, if 5 and 77 are (independent and identically distributed) arbitrary random vectors then we need a ‘continuous’ variant of Theorem 1, that is, of Turan’s theorem, where ‘number of’ (vertices, edges) is substituted by ‘measure of’. This is worked out in [5]and independently in [ 2 ] .Consequently ( 8 ) can be proved for such general random vectors too [4]. Theorem 2 similarly implies

I

!--(I

P(I 5 + 77

I2 x) 2

P’(I

- P ( / ~ l ~ x ) ) if’ P()51>x)>!,

5 I 2 x)

otherwise.

It is interesting to mention that these inequalities are improvable. Simple constructions give equality in them. Lower estimations of P((l5 + 77 11 2 x ) by use of P(ll51(2 c x ) are worked out in [7] and [lo].

G.O.H. Katona

382

The probabilistic version of Theorem 3 is P(II 51 + (2 + 5311 3 x ) 3 iP*(II 51 II 3 x ) ( l - P(ll51II 3

while our efforts concerning N3(X,n, n ) can be formulated in the form

P(IISI + 5*+ 5311 3 x, IISlII 2 x, 115211 3 x, IIS.lll3 x ) 3 CP'(II 51II 3 x), where c = 5/14 is proved for any X , c = 419 is proved for the real X,and c = 519 is conjectured. Interested readers can find more about the geometrical tools in [ll]. I also would like to draw attention to the forthcoming paper (or papers) of Sidorenko

POI. References [ I ] G. Bereznai and A. Varecza, On the limit of a sequence, Publ. Math. Debrecen, to appear. (21 B. Bollobis, Measure graphs, J. London Math. SOC.21 (1980) 401-407. [3] G.O.H. Katona, Graphs, vectors and probabilistic inequalities, Mat. Lapok 20 (1969) 123-127 (in Hungarian). [4] G.O.H. Katona, Inequalities for the distribution of the length of sums of random vectors, Teor. Verojatnost. i Primenen. 15 (1977) 466-481 (in Russian). [5] G.O.H. Katona, Continuous versions of some extremal hypergraph problems, Coll. Math. SOC. J . Bolyai 18, Cornbinatorics, Bolyai SOC.Budapest, 1978. [6] G.O.H. Katona, Sums of vectors and Turin's problem for 3-graphs, Europ. J. Combin. 2 (1981) 145-1 54. [7] G.O.H. Katona, 'Best' estimations on the distribution of the length of sums of two random vectors, Z. Wahrsch. Verw. Gebiete, submitted. [8] G. Katona, T. Nemetz and M. Simonovits, On a graph-problem of Turin, Mat. Lapok 15 (1964) 228-238. (in Hungarian). [9] G.O.H. Katona and B.S. Stechkin, Cornbinatorial numbers, geometrical constants and probabilistic inequalities, Dokl. Akad. Nauk SSSR 251 (1980) 1293-1296. [lo] A.F. Sidorenko, manuscript without title. [ll] A.F. Sidorenko and B.S. Stechkin, Extremal geometrical constants, Mat. Zamet. 29 (1981) 691-709 (in Russian). (121 P. Turin. On an extremal problem of graph theory, Mat. Fiz. Lapok 48 (1941) 436452 (in Hungarian).


L’HYPERCUBE INFINI ET LA CONNEXITE DANS LES GRAPHES INFINIS Jean-Marie LABORDE C N R S , M A G BP S3X, 38041 Grenoble, Cedex, Ftance

In this paper some properties and characterizations of the infinite hypercube are presented. This is used in particular as an illustration for the graph-theoretical concept of connectivity in the infinite case.

1. Definition des hypercubes infinis

Definition. Quelque soit I’ensemble a, nous dkfinissons le cube Q, comme le graphe simple dont I’ensemble des sommets est 9 ( a )= 2“ et ob deux sommets x et y sont adjacents si et seulement si leur diffkrence symktrique x A y est de cardinal un. Ce graphe peut donc aussi &tre considCrC comme I’ensemble des familles indexkes par a 06 xi E (0,l}, deux de ces familles Ctant reliCes par une arCte ssi elles ne different exactement que pour un seul i E a. Exemple. a = 3 = (0, 1,2}, V (Q,) = 2’ = (000,001, . . . ,111) (Fig. 1).

Fig. 1.

ProprietC 1. a = /3 ( a et P sont kquipofenzs) @ Q, = Qs (Q, et Qp sont isomorphes en tunt que gruphes). Demonstration. ‘ 3 ’ Si f est une bijection de a sur 0, considkrons l’application f ’ de V(Q,) dans V ( 0 , ) dCfinie par f ’ ( a = { f ( x))IEoco

>

f ’ est trivialement injective et surjective; 383

J. M . Laborde

384

f ’ comme son inverse respectent la structure de graphe sur 2“

1

( ( aA b = 1 @ I / ’ ( a ) A f ’ ( b ) l = 1).

‘+’ Soit a

un sommet de Q, ( a C a ) et b son image dans Qo par un isomorphisme de Q, sur Qp.On remarque que l’ensemble des voisins de a, N ( a ) est Cquipotent a a d’oh a = N ( a ) = N ( b )= p.

Ceci nous autorise A ne consid6rer dorknavant que des cubes Q, OG a est un cardinal. (Puisque modulo l’axiome du choix, tout ensemble est 6quipotent B un cardinal.) 2. Caracterisation de I’hypercube Q, pour a infini

Pour a C a,on note af l’ensemble des parties a distance finie de a, c’est-a-dire:

af={xCcu,laAxIEN}.

On peut Ccrire

u

2” =

Xf

x €20

et, comme xf r lyf # 0 implique x f = yf d’une part et d’autre part g r k e B l’axiome du choix, on peut h i r e aussi sous forme d’une partition:

2” =

u

Xf.

%€ID=

EnonGons alors la propri6tC suivante.

PropriOte 2. Tous les sous-graphes de Q, engendrks par diffe‘rents a, sont isomorphes (donc en particulier d celui engendre‘ par gf). Demonstration, En effet, pour af et bf donnCs, consid6rons l’application 4 ar-* b f ,

4 :x

~

(A a x)Ab

(= Q A b A x ) .

On a

b A ( a A x A b ) = a A x AO= a A x , ou encore pour x E af

( bA ( a A x A b ) /= ( a A x 1 E N ; d’ob 4 est bien une application de af dans bt. D’autre part, si x et x ’ ClCments de Qu sont dans ar et vCrifient Ix A x ‘ / = I , alors,

14(x)A

+ ( X I ) (

=

1 a A x A b A a A X ’A b I = I X

A x’l= 1,

L'hypercube injini

385

et { $ ( x ) , $ ( x ' ) } est bien une ar&te du graphe engendrC par b,. Enfin 4 est bijective, car involutive, et de mtme, pour { y , y ' } arCte de b r , { + - ' ( y ) , ~ $ - ' ( y ' ) } est une arete de a f ce qui finit d'Ctablir la propriCtC. Supposons maintenant a infini; comme le cardinal de l'ensemble des parties finies de a est a, I doit satisfaire 1'Cquation fournie par (1): 2"

=I

x

ff.

De la Z = 2" et on a le thCorkme suivant. Theoreme. Pour a in,fini le 0-cube Q, est isomorphe au graphe (non connexe) constituk de 2" exemplaires du graphe 0" (connexe) des parties finies de a, relikes par une arite ssi leur difkrence symktrique est de cardinal 1.

D'aprks le thCorkme prCcCdent, le problkme de la caractCrisation de Q, est ramen6 B celle de Qa le sous-graphe de Q, engendrC par les parties. finies de a. Montrons pour cela le lemme suivant. Lemme. G connexe est isomorphe a Q, ssi - tout sous-graphe fini de G est inclus dans un hypercube fini, sous-graphe de

G; -

l'un des points de G est de degrk a.

Demonstration. Les conditions sont Cvidemment nkcessaires, reste 5 montrer qu'elles sont suffisantes. Pour cela, construisons un monomorphisme de G dans 0"de la faGon suivante: Tout d'abord si x est I'un des points de G de degr6 a,on associe h x la partie vide de a,de code {O,}iEo et B chacun des a voisins de x un singleton de a, de code { x , } ~ ~tel , que 3j E a,V i Z j, x, = 0 et x, = 1. G Ctant connexe, il est alors possible Ctant donnC un point y quelconque de G, de considCrer le plus petit hypercube contenant un chemin de longueur finie reliant x 5 y et d'associer B y le code, somme des codes des voisins de x dans cet hypercube. La premikre condition assure bien siir la compatibilitC de toutes ces affectations entre elles ainsi qu'avec la structure de graphe de G et 0,ce qui achkve la dkmonstration. Rappelons la caractCrisation suivante [l] des hypercubes finis: G connexe est un hypercube fini ssi - deux aretes incidentes et distinctes de G appartiennent 5 un et un seul quadrilatkre (4-cycle) - le degrC minimum 6 de G est fini et G possbde 2' sommets. L'explicitation de la premikre condition du lemme, grrice B cette dernikre caractkrisation conduit alors au suivant.

J. M. Laborde

386

Theoreme. G est un hypercube infini, isomorphe ri Q, s s i : (i) G posskde 2” composantes connexes G, ; (ii) chaque Gisatisfait: - l’un de ses sommets est de degri a, - deux de ses ar6tes incidentes et distinctes uppartiennent ri un et un seul [quadrilatkre 1, - chaque sous -graphe connexe fini de Giposskde au plus 2’ sommets (oh A en est le degre‘ maximal). 3. Remarques sur la connexite dans les graphes infinis

Bien que manipulant des graphes infinis, nous avons utilisk ici la connexiti selon sa dkfinition habituelle. On dit qu’un graphe est connexe ssi Ctant donne deux de ses points x et y quelconques, il existe un chemin de longueur (finie): xI,=x,Xl)

..., x , = y .

Selon cette dkfinition un objet aussi ‘ramassk’ que Ie cube sur N n’est pas connexe. Divers arguments de thkorie Clkmentaire des ensembles montrent sans doute que le concept de chemin de longueur infinie: xo=

x , . . . , x , , . . . . ,x, = y

reliant ii travers une famille infinie d’intermediaires conskcutifs est inconsistant; cependant, une faGon simple de s’en persuader est de considirer la situation Cventuelle suivante. Supposons que nous ayons pu gknkraliser la difinition de la connexitk de fagon Q rendre QN connexe. Deux exemplaires indipendants de QN devraient cependant consituer un graphe non connexe; or, compte-tenu du thCorkme, ce graphe apparait aussi comme QN hi-mCme, connexe par hypothbse. Ainsi apparait le caractkre illusoire d’une tentative de modification dans le sens indiqui du concept de connexiti. Notons pour finir que dans le cas infini trbs peu de thkorkmes subsistent. Par exemple, I’hypothkse pour un graphe, que son degrt minimum 6 soit supirieur ii In/ZJ - 1 qui devient ici 6 = n ne suffit pas i en assurer la connexitk: K , U KN, par exemple, n’est pas connexe.

Bibliographie [ 11 J.M.Laborde and S.P.Rao Hebbare, Another characterization of hypercubes, Discrete Math. 39 (1982) 161-166.


THE AUTOMORPHISM GROUP OF THE SMALLEST NON-AFFINE HALL TRIPLE SYSTEM J. LACAZE and L. BENETEAU Universiti Paul Sabatier, U.E.R. Mathirnatique, 118 Route de Narbonne, 31077 Toulouse, Cedex, France Every HTS is associated with an M,-loop, just as every affine space over IF, is associated with an elementary abelian 3-group. If E is an MJoop, denoie by X ( E ) the corresponding HTS. Then 1 Aut K(E)I = IE 1.1 Aut E 1. When E is free among the M,-loops which are nilpotent of class 2, the groups Aut E and Aut X ( E ) may be described. In particular, the automorphism group of the non-affine HTS of order 3' turns out to be a semi-direct product of GL(3,F,) by the exponent 3-group B, on 3 generators whose order is 3'.

1. Introduction

The Hall triple systems (HTSs) are the Steiner triple systems in which any three non-collinear points generate an affine plane or, equivalently, in which any symmetry is an automorphism (a symmetry a, with fixed point a is defined by a, ( x ) = y whenever { x , a, y } is a line). Such an HTS is said to be abelian when it arises from some affine space over IF3 = GF(3). Any HTS has order a power of 3, even in the non-abelian case. Moreover any two minimal generator sets of such a system have the same cardinal number n + 1; we call n the dimension of E (see [l]). The smallest non-abelian example of HTS contains 81 = 34 elements (see 141). It is denoted by %'(L3),and it is the only non-abelian 3-dimensional HTS. Our purpose here is to describe the automorphism group of X ( L 3 ) . 2. Fischer groups 2.1

Definition. A Fischer group G is a group in which the subset E of the order 2 elements satisfies: (i) E is a generator set of G ; (ii) Vx, y E E, ( ~ y=) 1.~ The derived subgroup G' = D ( G ) of such a Fischer group is a 3-group and satisfies [G : G'] = 2. 387

J . Lacaze, L . Beneteau

388

The following properties will show why the study of the HTSs (and particularly of their automorphism groups) is connected with the Fischer groups. 2.2 Let E be an HTS, and G the pcrmutation group of E generated by the symmetries. Then G is a Fischer group such that Z ( G )= (1).

Conversely, if G is a Fischer group, let E be the subset of the order 2 elements. If we define the ‘lines’ of E as the 3-subsets of, the form { x , y , z } with z = xyx = yxy. then E turns out to be an HTS.

2.3 In fact, we have by 2.2 and 2.3 a one-to-one correspondence between the HTSs and the Fischer groups in which Z ( G )= (1) (see [7]).

3. M1-loops or exponent 3 commutative Moufang loops

Definition. An M3-100p is a set E provided with a commutative binary law x , y + x y admitting a unit 1 and satisfying V x , y , z E E, X~ ( x * y ) = y and

-

(x * y ) * ( x * z ) = x ’ . ( y

*

2).

In particular, x 3 = x z . x = l . Loosely speaking, the MJoops are to the HTSs what the elementary abelian 3-groups are to the affine spaces over F3. More precisely we have the following. 3.2 If (E, is an M3-loop, E may be seen as an HTS in which the ‘lines’ are the 3-subsets of the form { x , y, ( x * y)’}. a )

3.3 Conversely, let E be an HTS, and u E E. If x , y E E, the set { u , x , y } is contained in a subsystem isomorphic with AG(2,3). Define x . y = z , where z is Y

The automorphism group of the smallest non-afine Hall triple system

389

the 4th point of the parallelogram { u , x , z , y } . Then (E,;) is an M3-loop. Moreover, if u E E, we have ( E , ; ) ( E , ; ) . 5

3.4 By 3.2 and 3.3 we have a one-to-one correspondence between HTSs and M,-loops (see [5]). In particular, we shall denote by L , the M3-loop associated with the HTS X ( L ) of order 34.

3.5. Nilpotency of MJoops Let us give three definitions in an M3-loop ( E ; ) : (i) if x, y , z E E, the associafor is (x, y , 2) = (x * ( y * z))-'

*

((x * Y ) * z ) ;

(ii) the derived subloop D ( E ) is the subloop generated by all the associators of E ; (iii) the upper central series : Z o ( E )= {l},

1

Z I ( E )= {z E E Vx, y E E, (x, y , z ) = 11, Z , + , ( E ) = { z E EI V x , y E E , ( x , y , z ) E Z ~ ( E ) } k, a l . Proposition. E , D ( Eis) an k-vector space of order 3" ( n is called the dimension of E 1. Remark. Any minimal generator set of the n-dimensional MJoop E involves n elements. Theorem (Bruck-Slaby [3]). If n ( n Z , - i ( E ) = E.

2 ) i s the dimension of the M1-loop E, then

Definition. An M,-loop E # (1) is said to be centrally nilpotent of class k if and only if Z k - l ( E ) E s = Z, (E). The theorem of Bruck-Slaby says that an n-dimensional M3-loopis centrally nilpotent of class k s sup{n - 1,l). 4. Description of L , and of its automorphism group

L , is the only non-associative M3-loopof order 34 = 81; its dimension is 3

J. Lacaze, L. Benefeau

390

4. I . Description of L ,

Let 5' be a 4-dimensional vector space over IF, and { e l ,e 2 ,e l , a } a basis of 5'. For any two elements

X = x l e l+ x2e2+ x3e3+ acr and

Y = y l e l+ y2e2+ y3e3+ ba, define

x - Y = x + Y + (x1y2 - x 2 y , ) ( x , -

y3)a

Then (T;) is an M3-loop of order 34. 4.2 Let f be the canonical mapping from Aut L3onto Aut(L3,D(L,)). We know that L3,Dcr,, is a 3-dimensional vector space over IF,. Then Aut L1may be described as follows. 4.3 Theorem [ 2 ] . The following exact sequence splits : f

Kerf-AutL3-GL(3,F,)-(l).

(1)

Moreover Kerf = Z : , so that Aut L , is the semi-direct product of GL(3, F,) by Z: Q Aut L3.

5. Automorphism group of HTSs 5.1 Let E be an M,-loop and X ( E ) thc corresponding HTS. Any f of Aut X ( E ) may be uniquely expressed as a product of an element g of Aut(E) by a symmetry, say f = a, og. Therefore [Aut X ( E ) l = 1 E I./Aut E 1, and Aut X ( E ) is I-rransitive. But Aut X ( E ) is non 2-transitive in the non-abelian case. Recall that 2?(LJ) is the only non-abelian HTS of order 3'. and B, the exponent 3-group on 3 generators whose order is 3'.

The autornorphism group of the smallest non-ufine Hull triple system

39 I

5.2 Remark (see [6]). The centerless Fischer group G associated with %(L,) has order 2 x 3' and its derived subgroup is G' = B,.

5.3 Theorem. There exists a splitting exact sequence of the form:

(~)-,B~-,Au~(~(L~))-,GL(~,IF~)-,(~). Aut(X(L,)) is the semi-direct product of GL(3, IF,) by BJa Aut(X(L,)). 5.4 Remark. These results may be generalised. Any n-dimensional M,-loop of class 2 is of order S 3'. where 2, = n + (;), and there exists only one such loop of order 3'", say L(R2).Then Aut L(n,2) is the semi-direct product of GL(n,IF,) by Z;(;)a Aut L,,,,.,. Then Aut(%'(L,,.2,)) is the semi-direct product of a normal subgroup B, by GL(n,FJ). Moreover, B, is the exponent 3-group on n generators whose order is 3'. where s, = n + (1) + (Y). The corresponding centerless Fischer group is of order 2.3'n. A more detailed account of these facts is to be found in [6]. The reader who is only interested in the description of thc automorphism groups of M3-loops may restrict himself to [2].

References [ l ] L. Beneteau, Topics about Moufang loops and Hall triple systems, Simon Stevin 54 (IY80) 107- 124. [2] L. Beneteau et J. Lacaze, Groupes d'autoiiiorphismes des boucles de Moufang commutatives. Europ. J. Combin. 1 (1980) 299-309. [3] R.H. Bruck. A Survey of Binary System\ (Springer, Berlin-Giittingen-Heidelberg. 1958). [4] M. Hall Jr., Automorphisms of Steiner triple systems, IBM J. Res. Develop. (1960) 460-472, MR 23 A # 1282. [5] M. Hall Jr., Group theory and block designs, in: Proc. Internat. Conf. Theory of Groups (Australia Nat. Univ., Canberra, 1965) (Gordon and Breach, N.Y., 1967) pp. 115-144. [6] J. Lacaze, Thtse de 3kme cycle, Universitt P. Sabatier, to appear. [7] Y.I. Manin, Cubic Forms: Algebra, Geometry, Arithmetic (North-Holland, AmsterdamLondon-New-York, 1974).



UN ALGORITHME LINEAIRE POUR L’ENUMERATION DES CRENEAUX D’UNE SUITE Abkelkader LAHRICHI I.N.S.E.A., Rabat, Marocco

Jean Luc SERET Elecrricirt! de France, Clamarr, France

In a sequence of integers, a ‘hole’ covers a subsequence of consecutive terms the maximum of which is smaller than both boundary terms. This concept leads to various problems. Among them, the main problem is to enumerate all the holes of a finite sequence. By defining the notion of ‘support of a hole’ and by using a ‘heap of supports’ with decreasing heights, we introduce here a linear algorithm to solve the problem mentioned above.

1. Description du probleme

On considkre des suites de n + 2 entiers, tels que u(0)= u(n autres termes u(j) ayant des valeurs finies,

+ 1) = w,

les

1.1. Dtfinition d ’ u n cre‘neau

O n appelle cre‘neau un triplet d’entiers, not6 [d, h,f[,tel que (1) d E [ l , n ] et f E [ d + l , n + l ] , (2) h = max u ( i ) pour i E [d,f - I ] et h < min{u(d - l), uv)}. d, h, f sont, respectivement, le dCbut ( i n c h ) , la hauteur et la fin (exclue) du crineau [d, h,f[. 1.2 Exemple. Soient n = 10 et u = (m,3,3,1,0,2,3,2,3,4,1,~).Le vecteur u est schCmatisC sur la Fig. 1.

394

A. Lahrichi, J. L. Serer

1.3. Support d’un cre‘neau On appelle support du crCneau [ d ,h , f [ , tout indice i E [d,f - 11 vCrifiant u ( 1 ) = h. On dit aussi que i supporte [d, h , f [ . Dans I’exemple de la Fig. 1, [1,4,11[ a un support unique, i.e., 9. Par contre, [1,3,9[ en a quatre, i.e., 1, 2, 6, 8.

Fig. 1

1.4. Le problime EnumCrer I’ensemble des ‘crCneaux de u.

Utilite. Le problkme consistant i trouver tous les crCneaux d’un vecteur donnC se pose friquemment pour rCsoudre certains problhmes d’ordonnancement [ 11, (21, [41 et PI. 2. L’algorithme 2.1. Algorithme cre‘num

(donnees: n, tableau u [0, n + 11: ENTERS; resultats: nc, tableaux d[l, n ] , h [ l , n], f [ l , n]: variables: i, pile p: ENTIERS)

it O nc + O empiler 0

ENTIERS;

Un algorirhme liniaire

305

Tant que i S n + 1 repeter tant que u ( i ) > u (dernierb)) rCpCter {dans d, h, f on mCmorise tous les crCneaux de fin i } nc + n c + 1 f(nc)+i h ( n c )+ u (dernier(p)) dCpiler d ( n c )+ 1 + dernierb) si u ( i ) = u (dernierb)) alors dbpiler { i et dernier(p) supportent le mime crCneau. Tous les crCneaux qui dCbutent en dernier(p) ont CtC gCnCrCs. On conserve i qui posskde encore la facult6 d’ktre dCbut de crCneaux}. empiler i i+i+l

2.2. Indication sur la validite‘ et la complexite‘ Durant chaque phase i, on gCnkre I’ensemble des crCneaux de fin i. Si nc est le nombre de crCneaux de u, on passe ne fois dans la boucle interne. Or nc S n, d’ob le rbsultat: L’algorithme cre‘num est en o(n).

Remerciements

Nous tenons ii remercier M. Gondran pour les conseils prCcieux qu’il nous a prodiguC lors de cette Ctude.

Bibliographie [ I ] J. Erschler, G. Fontan et F. Roubellat, Potentiel sur un graphe non conjonctif et analyse d’un problime d’ordonnoncement i moyens limitis, RAIRO Rech. Opir. 13 (1979). (21 M. Gondran et A. Lahrichi, Implications et heuristiques en combinatoire, Contribution Nationale Francaise au Congrts I Fors 81, Hambourg. [3] M. Gondran et M. Minoux, Graphes et algorithmes, Eyrolles 79 dans la collection des Etudes et Recherches EDF. [4] R.L. Graham, E.L. Lawler, J.K. Lenstra and A.H.G. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling: A survey, Ann. Discrete Math. 5 (1979) 287-326. [ 5 ] A. Lahrichi, Ordonnancements: la notion de ‘partie obligatoire’ et son application aux probltmes cumulatifs, Thise de 3” cycle. I, Programmation, Universite de Paris VI, 1979. [6] A. Lahrichi, La notion de partie obligatoire et son application i la recherche des crineaux. RAIRO Inform. Comput. Sci. 15 (1981) 13-27.


Annals of Discrete Mathematics 17 (1983) 397-41 1 @ North-Holland Publishing Company

LE POLYNOME DE MARTIN D’UN GRAPHE EULERIEN Michel LAS VERGNAS Centre National de la Recherche Scientifique, Universiti Pierre et Marie Curie, U.E.R. 48. 4 place Jussieu, 75005 Paris, France

We show that given an Eulerian directed graph G there is a polynomial m ( G ; l ) in one variable with non-negative integer coefficients satisfying certain inductive relations, such that m ( G ;1) is the number of Eulerian circuits of G. This solves a conjecture due to Martin. Similar results hold for undirected Eulerian graphs and 4-regular graphs drawn o n surfaces. Using properties of these polynomials we derive bounds for the number of Eulerian orientations of an undirected graph with even degrees.

1. Introduction

Dans sa these [3] Martin a introduit pour diffkrentes familles de graphes (graphes non-orientis 4- et 6-rCguliers, graphes orient& pseudo-symttriques 4-rCguliers) un polyndme m(G)en une variable, liC 2 I’CnumCration des cycles (circuits) eulkriens de G.Dans chaque cas le polyndme m(G)est dCfini par des relations de ricurrence faisant intervenir des graphes obtenus B partir du graphe G par des optrations de riduction et il est montrC que les diverses facons de proceder aux riductions conduisent au m&me polyndme. Une conjecture gCnCrale est posCe dans (3, p. 1261 (voir plus bas) concernant l’existence de tels polyn6mes pour des graphes quelconques. Dans le present article nous dkfinissons sous le nom de polyndme de Martin un polyndme CnumCrateur des k -partitions eulkriennes d’un graphe G.N o u s montrons que ce polyndme vCrifie les relations de rCcurrence posCes par Martin, Ctablissant ainsi sa conjecture. La dCmonstration fonctionne dans diffkrents contextes: graphes eulCriens orient& (Section 3), non-orient& (Section 4), graphes 4-rCguliers traces sur des surfaces (Section 6). L’utilisation des propriCtCs de ces polyndmes nous permet en Section 5 d’exprimer le nombre d’orientations eulkriennes d’un graphe G en fonction de I’Cvaluation m (G;3), d’ou en corollaire des bornes pour ce nombre d’orientations. Les rCsultats de cet article avaient CtC partiellement annoncks dans [ 2 ] . Signalons Cgalement [ 11 oh I’on trouvera en particulier des gCnCralisations dans les cas de la sphere (ou du plan), du plan projectif et du tore de thkoremes de Martin [3], [4] et de Rosenstiehl et Read [5] sur les partitions eulCriennes croisantes et non-croisantes et le polyndme de Tutte de graphes planaires. 397

398

M . Las Vergnas

2. Terminologie et notations

Les graphes considiris dans le present article sont finis avec Cventuellement des boucles et des arttes multiples. Un graphe G est dCfini par un ensemble S ( G )de sommets et un ensemble E ( G )d’arttes. A chaque arkte e E E ( G )sont associks 2 sommets (confondus dans le cas d’une boucle): les extre‘mite‘s de e. On distingue une extrCmitC initiale et une extrCmitt terminale lorsque le graphe G est orientt (une artte est alors plus communCment appelte un arc). Le degre‘ d’un sommet x E S ( G ) , not6 dG( x ) , est le nombre d’arCtes ayant x pour extrtmitt, comptCes avec la multiplicitC 2 dans le cas d’une boucle. Dans le cas orientC, le demi-degre‘ exte‘rieur, not6 dA(x), est le nombre d’arcs ayant x pour extrCmitC initiale. On dCfinit de mtme d ; ( x ) , le demi-degre‘ inte‘rieur. On sait depuis Euler en 1736 que dans un graphe non-orient6 G il existe un parcours fermt utilisant exactement une fois chaque ar&te- appelC maintenant un cycle eule‘rien - si et seulement si G est connexe et de degrt pair en chaque sommet. Nous dirons qu’un tel graphe est eule‘rien. Plus gCntralement nous appellerons k-partition eule‘rienne de G une partition en k 3 1 cycles de l’ensemble E ( G ) des arktes de G. Un cycle est ici un parcours fermt, i.e. une suite ( x 0 , e l ,x I ,e z ,x2,. . . ,x f - l ,e f ,x f = xo) de sommets x, et d’arttes e, de G telle que pour i = 1,2,. . . , I, x , - ~et x , soient les extrCmitCs de e, - utilisant au plus une fois chaque artte. L’entier 1 est la longueur du cycle. Notre etude portant essentiellement sur des denombrements il convient de souligner que nous ne distinguons pas les cycles correspondant A deux suites se dtduisant I’une de I’autre par permutation circulaire ou par inversion du sens de lecture. Par contre la prksence de boucles peut apporter des multiplicitts (voir au dCbut de la Section 4). L’exemple de la Fig. 3 illustre ces dtfinitions. On a des definitions analogues dans le cas orientt. Un graphe orient6 G est eule‘rien si et seulement si il est connexe et d A ( x ) = d ; ( x ) pour tout x E S ( G ) . Une k-partition eule‘rienne (orientke) de G est une partition en k 3 1 circuits de l’ensemble E ( G ) des arcs de G. Un circuit est un parcours fermC dtcrivant les arcs dans le sens de leur orientation et les utilisant au plus une fois; il est dCfini une permutation circulaire prks par une suite (x,,, e l ,xI,e2,x?,. . . ,x f I , e l ,xl = x,)) telle que pour i = 1,2,. . . , I, e, ait x , - pour ~ extrCmitC initiale et x, pour extrCmitC terminale. Un anticircuit dans un graphe orientt est un parcours fermt dtcrivant les arcs alternativement dans le sens de leur orientation et en sens contraire, et les utilisant au plus une fois chacun. Dans les problkmes de dinombrement on ne distingue pas deux suites dtfinissant un anticircuit qui se dCduisent I’une de l’autre par permutation circulaire ou par inversion du sens de lecture.

Le polyndme de Martin d'un graphe eule'rien

399

3. Le polynome de Martin d'un graphe eul6rien orient6

Soit G un graphe eultrien orientC. Nous dCsignons par fk ( G ) le nombre de partitions euleriennes de G comportant k circuits. Nous appelons p o l y n h e de Martin de G, noti m ( G ) ,le polynBme tnumtrateur en une variable dCfini par

Exemple (Voir Fig. 1).

m ( G ; 4') = 2 f 2 + 44'.

Fig. 1.

Rappelons la difinition de l'optration de reduction introduite par Martin dans [3]. Soit G un graphe eulCrien orient6 et x un sommet de G de demi-degrk d & ( x )= d G ( x ) = d, comportant b boucles incidentes h x, b 6 d - 1. Soient e : resp. e, i = 1,2,. . . ,d - b les arcs de G non-boucles d'extrCmitCs terminales resp. initiales x. Pour toute permutation u de {1,2,. . . ,d - b } , u E Lh, soit G" le graphe obtenu ii partir de G en enlevant le sommet x et les arcs incidents et en ajoutant pour chaque indice i = 1,2,. . . ,d - b un arc ayant pour extrCmitC initiale celle de e : et pour extrCmitC terminale celle de err(,)(Fig. 2). Proposition 3.1. Soit G un graphe eule'rien orient6 et x un sommet de demi-degrk d 2 1 comportant b =sd - 1 boucles. O n a

La Proposition 3.1 exprirne que m(G) vCrifie la contrepartie orientke de la conjecture CnoncCe par Martin dans [3, p. 1261. En particulier dans le cas d'un

M. Las Vergnas

400

Fig. 2

graphe eultrien orient6 4-rCgulier nous identifions ainsi m ( G ) le polynbme CtudiC dans [3, Chap. IV]. Observons cependant que 1’hypothl.se faite dans [3], x non-sommet d’articulation de G, n’est pas nkcessaire. La relation de la Proposition 3.1 est valable dans tous les cas A condition d’Ctendre les definitions de f k ( G )et de m ( G ) B des graphes G ndn-necessairement connexes, vtrifiant d L ( x )= dG(x) en tout sommet x. Si G = G I+ G 2+ * . + G,o i ~G I ,Gz, . . . , G,, p 3 1, sont des graphes eulkriens 2 B 2 disjoints on a

~(e)

Lemme 3.1.1. Soit G un graphe eulkrien orient4 et x un sommet de G de demi-degre‘d 3 2. Soit G’ le graphe obtenu a partir de G en enlevant b S d - 1 des boucles incidentes ri x. On a

m ( G ; [ ) =(5 + d

- b - 1)(5

+d -b).

*

.((

+ d -2)m(G‘;g).

Demonstration. Considtrons d’abord le cas b = 1. Les k-partitions euliriennes de G se ripartissent en deux classes: (1) Celles ou la boucle considCrke e est un circuit de la partition. En enlevant e on Ctablit une bijection entre les k-partitions eulkriennes de G de ce type et les (k - 1)-partitions eukiennes de G’. (2) Celles oh e n’est pas un circuit. E n enlevant e on obtient une k-partition eulkrienne de G‘ et clairement il existe d - 1 k-partitions eulkriennes de G de ce type correspondant h une k-partition eulkrienne de G’ donnCe.

Le polyndrne de Martin d'un graphe eulirien

40 1

On a donc pour tout k 2 1 (avec la convention f,,(G')=O) f k ( G ) = fi I ( G ' ) +( d - I ) f k (G'). II vient

Le cas gtneral s'obtient aistment par ricurrence sur le nornbre de boucles 6. Demonstration de la Proposition 3.1. Dans le cas ou aucune boucle n'est incidente a x, b = 0, on a clairement fk ( G ) = c,GS,fk (G"). et par suite m ( G ) = C,,.,,m(G"). On obtient la Proposition 3.1 pour b quelconque en utilisant le Lemme 3.1.1. Soit Bd,d 2 1, le graphe eulirrien orient6 constitue par d boucles incidentes a un meme sommet. On a rn (GI) = I . Pour d 3 2, en appliquant le Lernrne 3.1 . I avec b = d - 1 on obtient

m (B, ;f ) = f ( f + 1)(5+ 2). . . ( 5 + d

- 2).

Observons que d'apres I'une des dirfinitions de c ( d , k ) nombre de Stirling sans signe de premiere espkce, on a

fm (8, ; 5 + 1 ) = f ( 5

+ I)({ + 2 ) . . . ( 5 + d - 1) = 2 c ( d , k ) C k . k all

D'O" fk (B,) = c ( d , k ) rirsultat classique, les k-partitions eulirriennes de B, etant ividemment en bijection avec les permutations d'ordre d comportant k cycles.

Theoreme 3.2. Soit G un graphe eule'rien oriente' de degre' maximal 2d 3 4. Alors m ( G ) est divisible par m ( B d ) et le quotient est un polynbme a coefficients nun -ne'gatifs.

En particulier m ( G ) est un polynbme a coefficients non-ntgatifs sans terrne constant. Dhonstration. Nous faisons la dirrnonstration par recurrence sur le nombre n de sornmets de G. Pour n = 1 le thCor6me est trivialement vrai. Supposons

402

M. Las Vergnas

n 3 2. Soit xo un sommet de G de degrC 2d. Tout graphe connexe avec au moins deux sommets contenant au moins deux sommets non d’articulation, il existe un sommet x de G, x # xo,tel que le sous-graphe G \ { x } soit connexe. Pour ce choix de x tous les graphes G “ de la reduction utiliske dans la Proposition 3.1 sont eulCriens, et de degrC maximal 2d puisque xo conserve son degrC. Le ThCorkme 3.2 est alors une consCquence de la Proposition 3.1 et de I’hypothkse de rkcurrence. Proposition 3.3. Soient G I ,G2,. . . ,G,, p 5 2 , des graphes eule‘riens orientis 2 a 2 disjoints, et pour i = 1 , 2 , . . . ,p, xi un sommet de Gi. Soit G le graphe obtenu a partir de la reunion des Gi en identifiant les sommets xi entre eux i = 1,2, . . . ,p. O n

a

oit di est le demi-degri de x, d a m G,. Demonstration. I1 suffit de considtrer le cas p = 2 , le cas gCnCral s’en dtduisant facilement par rtcurrence sur p, Calculons inductivement m ( G ) au moyen de la Proposition 3.1 en effectuant d’abord des reductions de G en tous les sommets de G2 diffkrents de x? = xI (dans un certain ordre). Clairement les diverses rCductions produisent toutes finalement le m&megraphe G’ obtenu B partir de G Ien attachant d z boucles en xI. D’apr8.s la Proposition 3.1 on a m ( G ) = P. m(G’) otI P est un certain polyn6me en 5. D’aprks le Lemme 3.1.1 on a

m(G‘;5) = (5 + d l - 1)(5 + d l ) . . (5 + dl + d 2- 2)m(GI;5), soit

I1 vient

Effectuons maintenant la mime succession de reductions sur Gz. D’aprks la Proposition 3.1 on a m(G2)= Pm(Bd2).D’otI la proposition. La Proposition 3.3 gCnCralise le Lemme 3.1.1. Elle gCnCralise egalement les relations d’induction relatives aux points d’articulation donnCes dans [3, pp. 70-731 dans le cas des graphes eulCriens orient& 4-reguliers.

Le polyndme de Martin d'un gtaphe eulirien

403

La dCfinition de m ( G )implique que son degrC est Cgal au nombre maximal de circuits 2 i 2 disjoints de G moins un (ces circuits partitionnent alors I'ensemble des arcs de G). Si G est de degrC maximal 26 3 4 on a m ( G ; - ( d - 2 ) ) = m ( G ; - ( d - 3)) = * * = m ( G ; O )= 0 d'aprks le ThCorkme 3.2. D'autre part, par difinition de m ( G )on a m ( G ;1) = f r ( G )le , nombre de circuits eulCriens de G. Proposition 3.4. Soit G un graphe eule'rien oriente'. O n a

ou S ( G ) est l'ensemble des sornrnets de G.

Demonstration. La Proposition 3.4 peut se dCmontrer par rkcurrence sur I S(G)l en utilisant la Proposition 3.1. En effet on a (avec les notations de la Proposition 3.1)

m (G ;2 ) = ( d - b + l)(d - b

+2 ) - d

m (G";2).

4

CtSd-a

D'ou la proposition comme

par I'hypothkse de rCcurrence et IS,-, 1 = ( d - b ) ! . On peut aussi remarquer que par difinition de m ( G ) , m ( G ;2 ) = Ck,,f k ( G) est le nornbre de partitions euliriennes de G. Or une partition eulkrienne de G est dCterminCe par le choix en chaque sommet d'une bijection entre les arcs entrants et les arcs sortants (y compris les boucles qui sont i la fois arc entrant et arc sortant). I1 existe clairement d L ( x ) ! telles bijections en chaque sornmet d'ou la proposition. Nous rappelons pour memoire deux propriCtCs remarquables, Ctablies dans [ 3 ] , du polynbme de Martin d'un graphe eulCrien orient6 G 4-rCgulier: (1) m ( G ;- 1) = ( - I)"(G'( -2)"'c'-' oh n ( G ) est le nombre de sornrnets de G et a ( G ) le nombre d'anticircuits de G [3, p. 801. Dans le cas o i ~G est obtenu i partir d'un graphe planaire 4-rCgulier non-orientk en orientant les ar&tesde faqon que Ies frontikres des faces soient des circuits, cette propriCtC se rattache ii 1'Ctude du polynbme de Tutte du graphe des faces blanches (ou des faces noires) de G (voir [ 3 , pp. 51-81J, [4], [SJ, et Cgalement [l], [2], oh ces propriCtCs sont renforckes et gentralistes dans les cas de la sphkre (ou du plan), du plan projectif et du tore). ( 2 ) Le coefficient du terme en 3 de m ( G ;6) est non nu1 si et seulement si G est 4-arCte-connexe [3, p. 911.

M. Las Vergnas

404

4. Le polyn6me de Martin d’un graphe eulCrien non-oriente Soit G un graphe eulCrien non-orient&. A toute k-partition eulkrienne de G comportant k , cycles de longueur 1 nous associons une multiplicit6 Cgale i 2b‘G’-kl,oij b ( G ) dksigne le nombre de boucles de G. Le choix de cette multiplicit6 revient B distinguer deux sens de parcours d’une boucle dans un graphe non-orient6. Pour. k 3 1 nous dksignons par fk(G) le nombre de k-partitions euleriennes de G compt6es chacune avec sa multiplicit6. Nous dCfinissons le polynbme de Martin de G par

Exemple (voir Fig. 3). = 96, = 96, f,= 30, f. = 3. ft

f2

G

m ( G ; 5) = 3L(L + 2Y.

Fig. 3.

Les propriCtCs du polynbme de Martin d’un graphe eulCrien non-orient6 sont semblables i celles du polynbme de Martin d’un graphe eulCrien orient6 6tablies dans la section pr6cCdente. Nous CnonGons les principaux rCsultats. Les dimonstrations, analogues i celles de la Section 3, sont 1aissCe.s au lecteur. Soit G un graphe eulCrien non-orient6 et x un sommet de G de degrC 2d comportant b boucles. Soit 9 I’ensemble des partitions en paires non-ordonnkes de I’ensemble des 2d - 2 b aretes non-boucles incidentes i x. On a

1 9 1 = (2d - 26

- l)!! = (2d

-26 - 1)(2d - 2 b - 3 ) . * * 3 . 1 .

Pour tout P E 9 soit G ” le graphe obtenu en ajoutant i G \ { x } pour chaque classe de T une arCte joignant les extr6mitCs diff6rentes de x des deux aretes de cette classe. Proposition 4.1. Avec les notations pr&ct!dentes on a

La Proposition 4.1 exprime que m ( G ) v6rifie la conjecture CnoncCe dam [3, p. 1261. Dans [3] Martin avait montrC I’existence d’un polynbme satisfaisant i

Le polynime de Martin d'un graphe eulkrien

105

cette conjecture dans le cas des graphes 4-riguliers et 6-r6guliers, en virifiant la compatibiliti des relations. En particulier nous identifions ce polyn6me A m (G). Soit Bd,d 3 1, le graphe non-orient6 consistant en d boucles incidentes a u n mime sornmet. On a f k ( B d ) = 2 d - k c ( d , k )pour 1 S k S d, m ( B , ) = 1 et pour d 3 2 , m(Bd;5))=[(5)+2)...(l+2d-4) .

Theoreme 4.2. Le polyndme de Martin d ' u n graphe eule'rien non -0riente' de degre' maximal 2d 4 est divisible par m ( B d ) el le quotient est un polyndme a coefficients entiers non -ne'gatifs.

En particulier m (G) est un polyndme a coefficients entiers non-nigatifs sans terme constant. Nous laissons au lecteur le soin d'inoncer (et de dtmontrer) la contrepartie non-orientie de la Proposition 3.3 (il su%t de changer m ( & ) en m(Bd)). Proposition 4.3. Soit G un graphe eule'rien non-oriente'. O n a m(G;3)=

n

(dG(x)-l)!!

XES(G1

5. Orientations euleriennes Le polyn6me de Martin d'un graphe eulirien non-orient6 se ramkne B des polyn6mes de Martin de graphes orientts au rnoyen de la proposition suivante.

Proposition 5.1. Soit G un graphe eule'rien non-oriente'. O n a

la sommation porte sur l'ensemble des orientations euliriennes (? de G et b ( G ) est le nombre de boucles de G.

OM

Demonstration. A toute k -partition eulkrienne de G faisons correspondre les k -partitions euliriennes orienties obtenues pour les diffirents choix de sens de parcours des cycles de la partition. Si k l cycles de la partition considkrte sont rtduits a des boucles, celle-ci intervient avec la multiplicitt 2b'G'-kl dans I'Cvaluation de f k (G). D'autre part le nombre de k-partitions euliriennes orienties associies est alors 2k-k1. Donc dans I'tvaluation de 2'fk (G) I'aide de cette correspondance les k-partitions eultriennes orienties assocites 2 une k-partition eulirienne donnte sont obtenues avec une multipliciti de 2""'. D'ou puisque pour les difftrentes k-partitions euliriennes de G on obtient toutes les

M. Las Vergnas

406

k-partitions eulCriennes des diffCrentes orientations euliriennes 6 de G , la relation 2kf, (G) = 2 b ' G ' C A f k (6)ou la sommation porte sur toutes les orientstions eulCriennes 6 de G . I1 vient

En particulier m (Bd ;5 ) = 2 d - ' m(fi, ;5 / 2 ) et le ThCorkme 4.2 est un corollaire immCdiat du ThCorkme 3.2 et de la Proposition 5.1. De meme la contrepartie non-orientCe de la Proposition 3.3 (IaissCe au lecteur) s'obtient immidiatement 2 partir de cette proposition. La Proposition 4.1 peut Cgalement etre obtenue a partir de la Proposition 3.1. Nous laissons les details au lecteur. La Proposition 5.1 peut ktre utilisie pour le dknornbrement des orientations eulkriennes d'un graphe non-orientC. Theoreme 5.2. Le nombre d'orientations eule'riennes d 'un graphe eule'rien non oriente' G est e'gal 13 (auec les notations pre'ctdentes):

Demonstration. D'aprks la Proposition 5.1 on a

m ( G ;4 ) = 2b'")-'

m A

(e;2 )

ou la sornmation porte sur I'ensemble des orientations eulkriennes D'autre part on a

d'aprks la Proposition 3.3, d'oh le thCorkme.

6

de G .

Le polynbme de Martin d'un graphe eulirien

407

En particulier si G est rtgulier de degrt 2d avec n sornmets, sans boucles, le nombre d'orientations euleriennes est 2(d !)-"m(G ;4). Pour un graphe 4rkgulier sans boucles ce nombre est 2-'"-''m(G ;4). Exemple (Voir Fig. 4).

f,= 44, = 32, f,=5. f2

m ( G ; 4") = 512 + 124".

G G a 16 orientations eultriennes: 8 sont isomorphes 5 d , et 8 sont isomorphes A

G,

d,

e 2

m ( G , ; 5) = 34"+25.

m(G,;l)=2C2+4l.

Fig. 4.

En corollaire des ThCorbmes 3.2 et 5.2 on obtient une borne infCrieure du nombre d'orientations eultriennes d'un graphe non-orientC. Pour la simplicit6 nous donnons cette Cvaluation dans le cas d'un graphe rtgulier de degrC 2d sans boucles. La proposition s'ttend clairement au cas general.

Proposition 5.3. Le nombre d 'orientations eule'riennes d 'un graphe non -0riente' rigulier de degre' 2d de n sommets, connexe, sans boucles, est au moins &a1 a 2'((2d - l ) ! ! / d ! ) " .

Demonstration. D'aprbs le T h C o r h e 5.2 le nornbre d'orientations eultriennes

M. Las Vergnas

408

d’un graphe non-orienti G rCgulier de degrC 2d de n sommets, sans boucles, est Cgal A 1 2(d !)” (G; 4)’ D’aprbs le ThCorkme 4.2, m ( G ) est divisible par m(Bd)et le quotient est 6 coefficients positifs, donc

a

m B d;3) ; 4 m(G;3). m(G;4)3 m(Bd O r m ( G ; 3) = ((2d - l)!!)” d’aprks la Proposition 4.3, d’ou la proposition. Dans le cas d’un graphe 4-rCgulier la borne de la Proposition 5.3 vaut 4 ( 3 - ’ . On obtient facilement une borne supkrieure en utilisant la Proposition 4.1. Parmi les graphes connexes 4-rkguliers de n sommets, n 3 4, le graphe de la Fig. 5 a le plus grand nombre d’orientations eulkriennes, soit 9.2”-3 (sur un nombre total d’orientations de 22n) .

*-a

. . .... . . .

Fig. 5.

Dans le cas d’un graphe 4-rkgulier la Proposition 5.1 combinCe avec I’un des rksultats citCs en fin de Section 3 fournit une interprbtation de m ( G ; - 2 ) en termes d’anticircuits. On a en corollaire la proposition suivante.

Proposition 5.4. Soit G un graphe non -oriente‘ connexe 4-re‘gulier. Le nombre d’orientations eule‘riennes de G ayant un anticircuit eulkrien (i.e. relies que a (C?) = 1) est divisible par 4.

z:

Demonstration. D’aprbs la Proposition 5.1 et le rCsultat citk de Martin

2 m ( G ;- 2 ) =

cc m ( G ;

- 1)= ( -

1)”

(-2)”(’)-’

c

ou n est le nombre de sommets de G, et a , ( G ) le nombre d’orientations eulCriennes 5: de G telles que a((?) = k. Remarquons que ai.(G) est pair pour tout k puisque I’inversion de tous les arcs de G ne change pas a ( e ) . Donc

Le polyndme de Martin d'un graphe eulirien

409

comme m ( G ;- 2 ) est divisible par 2 d'aprks le T h C o r h e 4.2, I'CgalitC prCcCdente entraine que a , ( G )est divisible par 4.

6. Le polynome de Martin d'un graphe 4-rkgulier track sur une surface

Une dCmarche analogue h celles des sections 3 et 4 peut ttre suivie pour les

k -partitions eulkriennes non-croisantes d'un graphe non-orient6 4-rCgulier track sur une surface. Nous en donnons une brkve description. Dans ce cas nous considCrons les cycles de la partition cornme des parcours fermCs sur la surface. Nous disons qu'une partition eulCrienne est non -croisante si pour tout somrnet x E S ( G ) du graphe G les deux occurrences de x dans les cycles de la partition sont de la forme ou bien (.. . , e l , x , e z ,..) . et (. . . , e3,x, e4,. ..) ou bien (. . . ,el,x, e4,. ..) et (. . .,e2, x, e,, . . .) (voir Fig. 6). Cette difinition est ambigue dans le cas d'une boucle incidente en x. La succession des sommets et des arites du graphe (abstrait) n'est pas suffisante pour dtcrire le parcours sur la surface. Nous laissons au lecteur le soin de prCciser la difinition dans ce cas (par exemple en remplaGant une boucle par deux 'demi-boucles', voir [l, 2, 31).

Fig. 6.

Nous dtsignons par f i ( G ) le nombre de k-partitions euliriennes non croisantes de G et nous dCfinissons le polynsme de Martin de G trace' sur la surface par

O n dCmontre comme plus haut les propriCt6s suivantes du polyn6me rn "(G): (1) Si aucune boucle n'est incidente en x ES(G) on a

m " ( G ) = rn"(GJ+ rnx(G2), ou G I et G fsont les graphes obtenus A partir de G comme il est montrC sur la Fig. 7.

M . Las Vergnas

4 10 X

G

G2

G1

Fig. 7.

(2) Si une boucle est incidente en x E S ( G ) (et l S ( G ) I 3 2 ) on a (Fig. 8)

n

m " ( G ) =5m"(G').

G

G'

Fig. 8 .

O n a mX(B2;5) = 5. Le polyn6me m " ( G ) est un polyn6me h coefficients entiers non-nkgatifs sans terme constant. (3) Si x E S ( G ) est un sommet d'articulation de G alors (Fig. 9)

m " ( G ) =5m"(G')mx(G'').

G'

G

Fig. 9.

Lorsque x est de degrC > 4 les propriktCs (2) et (3) peuvent &re en dCfaut. Le polyn6me m " ( G )peut avoir des coefficients nkgatifs lorsque G a des sommets de degrk>4.

Le polynhe de Martin d'un graphe eule'rien

31 1

Dans le cas ah G est track sur la sphkre (ou le plan) le polyn8me rnx(G)est liC aux polyn8mes de Tutte de graphes associCs A G [3, 4, 51. Ces propriCtCs se gCnCralisent au plan projectif et au tore [l, 21.

References [ 1 1 M. Las Vergnas, Eulerian circuits of 4-valent graphs imbedded in surfaces, Coll. Math. SOC.J. Bolyai 25 (Algebraic Methods in Graph Theory, Szeged, Hungary, 1978) (North-Holland, Amsterdam, 1981) 451-477. [2] M. Las Vergnas, On Eulerian partitions of graphs, in: R.J.Wilson ed., Graph Theory and Combinatorics, Res. Notes in Math. No. 34 (Pitman, San FrancisceLondon-Melbourne, 1979) pp. 62-75. [3] P. Martin, EnumCrations eulkriennes dans les multigraphes et invariants de TutteGrothendieck, Thtse, Grenoble, 1977. [4] P. Martin, Remarkable valuation of the dichromatic polynomial of planar multigraphs, J. Comb. Theory, Ser. B 24 (1978) 318-324. [5] P. Rosenstiehl and R.C. Read, On the principal edge tripartition of a graph, Ann. Discrete Math. 3 (1978) 195-226. '



T-CRITICAL

HYPERGRAPHS AND THE HELLY PROPERTY

J. LEHEL Computer and Automarion Institute of the Hungarian Academy of Sciences, H-1502 Budapest, Kende u-13-17, Hungary

1 . Introduction

An early result of Gilmore can be formulated as a sufficient (and necessary) condition on the ‘local’ structure of a hypergraph having the Helly property. Berge and Duchet [2] have extended Gilmore’s condition to hypergraphs having the Helly property of ‘higher dimension’. By emphasizing the strong relationship between such conditions and the concept of multitransversal number of 7-critical hypergraphs, we are able to prove some further conditions which have the following general form (with I, t, k and w as integer parameters): Let H be a hypergraph with at least k edges and such that any k edges of it have a nonempty intersection. If for every A C V ( H ) , ( A 1 = 1, the partial hypergraph H‘ with edge set { e E E ( H ) : le fl A 13 w ) satisfies T ( H ’ ) s t, then T ( H ) ?. The vertex set and the edge set of a (simple) hypergraph H is denoted by V ( H ) and E ( H ) , respectively. H‘ is called a partial hypergraph of H if E ( H ’ ) C E ( H ) . The transversal number of a hypergraph H is defined as min{ 1 TI: T c v ( H ) , ( e n T J3 1 for every e E E ( H ) ) . A hypergraph is said to be 7-critical if the removal of any edge reduces the transversal number. If the transversal number of a 7-critical hypergraph H is known, say T ( H )= t + 1, then H will be called ( t + 1)-critical. A hypergraph is called k -intersecting if it contains at least k edges and any k of its edges have a .f“ nonempty intersection. As an extension of the well-known Helly property (see [I]) we define the ( k , t)-property as follows. Definition. A hypergraph H verifies the ( k , t)-property if T ( H ‘ ) St holds for every k-intersecting partial hypergraph H’ of H. 413

J . Lehel

414

The (k, I)-property for hypergraphs was investigated first in [ 2 ] under the name of Helly property of order k. An early result of Gilmore says: The hypergraph H has the (2,l)-property (or Helly property) iff for every A C V ( H ) ,/ A I = 3, the partial hypergraph H’ with edge set {e E E ( H ) : le n A 1 2 2 ) satisfies T ( H ’ ) =1. We will show that analogous characterizations of hypergraphs having the (k, t)-property follow from the solution of extremal problems posed on k intersecting (t + 1)-critical hypergraphs.

2. The (k, [)-property

First we introduce the notion of multitransuersals. Let H be a hypergraph with lower rank at least w, i.e., { ( H )= min{ I e I: e E E ( H ) }2,w. A set A C V ( H ) is called a w-transversal of H if le fl A

I

w for every e E E ( H ) . We denote by T, ( H ) the w-transversal number of H : T, ( H )= min{ [ A

I: A C V ( H ) , 1 e n A 12 w

for e E E ( H ) } .

The existence of w -transversals in k-intersecting hypergraphs is guaranteed by the following observation.

Lemma. If t, k and w are positiue integers such that w s t ( k - 1 ) + 1, and H is a k-intersecting hypergraph with T ( H ) * t + 1, then l ( H ) > w. Proof. Suppose that for some e,, E E ( H ) , { ( H )= leal, and consider a partition of e,, into t-element sets: ‘I

e , , = (J T,, T,nT,=0 ( l s i < j s q ) , ,=I

IT,J=t ( l s i s q - 1 )

and

0 < 1 T, 1 c t. None of the T’s is a transversal of H, therefore there exist edges of H such that e, n T, = 0 (1 s i G 4). Since e , , n e , ) = 0 and H is k-intersecting, we have k iq and therefore

(m=,

7-critical hypergraphs and the Helly property 4

I

I ( H )= ieoI = , = I T, 1 = (q - 1)t + 1 T,

11s

I 3 t ( k - 1 ) + 1 3 w.

It is proved in (41 that T,,, (H)S ( t + 1)” for every ( t + 1)-critical hypergraph H with [(Id)>w. Let l(t, k, w) denote the smallest integer 1 such that T~ ( K ) G 1 for every k-intersecting ( t + 1)-critical hypergraph K with [ ( K ) 3 w. Theorem. Let H be a hypergraph, t, k and w be positive integers, and w s t(k - 1) + 1 . Then the next three statements are equivalent: (i) H has the (k, r)-property ; (ii) H has no k-intersecting ( t + 1)-critical partial hypergraph; (iii) For every A C V ( H ) ,1 A I S l ( t , k, w), the partial hypergraph H’ with edge set { e E E ( H ): 1 e n A 1 3 w } verifies the (k, t )-property. Proof. It is easy to see that (i) and (ii) are equivalent and (i) implies (iii). To prove (iii) .$ (ii) suppose indirectly that K is a k-intersecting ( t + 1)-critical partial hypergraph of H. By the lemma, K has w-transversals and by the definition of l(t, k, w), ( K ) S I(?, k, w). Let A be a w-transversal of K such that ] A I s l(r, k, w). The hypergraph H‘ defined by this set A in (iii) contains K as partial hypergraph. H’ has the (k, ?)-property, therefore T ( K ) Ct, which is a contradiction. 0 An immediate consequence of the theorem is the following.

Corollary. Let H be a k-intersecting hypergraph, t, k, w and 1 be positive integers such that w s t(k - 1)+ 1 and 1 3 I(?, k, w). If for every A C V ( H ) ,( A I = I, the partial hypergraph H’ with edge set {e E E ( H ) : I e n A 1 S w} satisfies T ( H ’ )s t, then T ( H ) s t. 3. On the l(r, k, w) function

Determining the values l(t, k, w) seems to be rather hard in general. We give the solution of this extremal problem in some special cases. If t = 1 and w = k then l(1, k, k ) = k + 1 is a corollary of the following result. Proposition 1. If K is a k -intersecting 2-critical hypergraph then T k ( K )= k

+ 1.

Proof. Let E ( K ) = { e , ,e l , . . . , em},m 3 k + 1, and xI,xz, . . . ,x, E V ( K )be vertices such that x,E e, iff i# 1. It is clear that {x,,xz,. . . ,&+I}is a k-transversal set Of K, SO T k ( K ) s k + l . On the other hand any set B C V ( K ) ,I B 1 = k, is not a k-transversal, since for

J. Lehel

416

a vertex x E B there exists an edge of K which does not contain x, so ?-k(K)*k+l. 0 We draw attention to the fact that the structure of a 2-critical hypergraph K is entirely described by Proposition 1 , since K is ( I E ( K ) (- 1)-intersecting.

Corollary (Berge and Duchet [2]). The hypergraph H has the Helly property of order k if and only if for every A C V(H), 1 A I = k + 1 , the partial hypergraph H’ with edge set { e E E ( H ) : I e n A I k } satisfies 7 ( H ’ )= 1. Proof. By Proposition 1 , 1 ( f , k, w ) = k + 1. Observing that H‘ is a k-intersecting hypergraph, the statement follows easily from the main theorem (with t = 1, k=w). 0 Proposition 2. 1(2,2,3) = 9. Proof. Let H be a 2-intersecting 3-critical hypergraph with edge set { e l ,e 2 , .. . ,e m } ,For every 1 s i S m we choose a 2-element set J of V(H) such that f, n e, # 0 iff i # j and we denote the graph of the f,’s by G. It is clear that ?-(G)2 2. We consider three cases: Case A: T ( G )2 4 . Let G‘ be a 4-critical partial graph of G. By a theorem of Erdos and Gallai [3] which says that ?--critical graphs have at most 27 vertices, wehave)V(G’)lc8.Since)e,r l V(G’)l>3for 1 S i S m , i n t h i s c a s e ? - ~ ( H ) c 8 . Case B: ?-(G)= 3. Let G’ be a 3-critical partial graph of G and suppose that E(G’)=(f,,fi,. . . , f , } . Then le, n V ( G ‘ ) ( a 2 f o r1 G i G p and le, V ( G ’ ) l s 3 for p < j G m. G‘ is equivalent to one of the graphs 3K2, K 2+ K,, C,and Kq. I n each case V ( G ’ )can be completed by adding new vertices x , E e, or x,, E e, (7 el, as shown on Fig. 1, to obtain a 3-transversal of at most 9 elements. In the case of G ’ = C,,the vertex x,, should be replaced by x , or x, if le, fl V(G’)/ = 3 or 1 e, f l V (G’) 1 = 3, respectively.

0

0

2‘

0

Iw8

0

0

x, x&

0

0

x3

x9

0 ’f,

0

0

‘5 ‘34

0

xf,

0

0

x36

Fig. 1.

Case C : T ( G ) = 2. We may suppose that m 3 6 . Let {p, q } be a transversal of G and let F be a maximal forest of G such that (p, 4 ) # ! E ( F ) and I dF( p )d F ( q ) lis minimal. Here d F ( x )is the degree of vertex x in F. It is easy to see that d F @ ) + d F ( q ) 3 4 .If F has a partial graph F’ isomorphic to 29, or Psthen a 3

417

r-critical hypergraphs and the Helly property

transversal containing at most 8 elements can be constructed starting from V ( F ’ ) as shown in Fig. 2. We note that lei f l V ( F ’ ) 3 ) 3 for 4 < i S m follows from I e, f l { p , q } )S 1, and instead of a point xi,E ei n e, other points such as xi E ei or xj E ej may be chosen if it is sufficient to complete the 3-transversal.

Fig. 2.

Finally, if dF(p) = 1 then F has a partial graph F’ isomorphic to K 1+ K , , and since I d F @ ) - d F ( q ) l is minimal, every edge from E ( G ) \ E ( F ’ ) contains q. In this case V ( F ‘ )plus two points from el is an 8-element 3-transversal of H. The case d F ( q )= 1 is the same. Thus 1(2,2,3)G9 is proved. Fig. 3 shows the edge-vertex matrix of a 2-intersecting 3-critical hypergraph with 3-transversal number equal to 9. 0 V

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

E 1

2 3 4 5 6 7

8 9

1 0 0 0 0 0 1 0 0

0 1 0 0 0 0 1 0 0

0 0 1 0 0 0 0 1 0

0 0 0 1 0 0 0 1 0

0 0 0 0 1 0 0 0 1

0 0 0 0 0 1 0 0 1

1 0 0 0 0 0 0 1 1

0 1 0 0 0 0 0 1 1

0 0 1 0 0 0 1 0 1

0 0 0

0 0 0

1

0 1

0 0 1 0 1

0 1 1 0

0 0 0 0 0 1

1 1 0 0

1

0 1

1 0

1 1 1 1

0 0

1

0

1

1

0 0

1

0

0 0

0

1

1 1

1

Fig. 3.

Proposition 3. 1(2,2,2) = 5.

Proof. Let H be a 2-intersecting 3-critical hypergraph with T ~ ( H2)5. There exist such hypergraphs, e.g., five lines given in general position in the plane generate a 4-uniform 2-intersecting 3-critical hypergraph on the ten points of intersection whose 2-transveral number is equal to 5 ; the 6-uniform hypergraph with 17 vertices and 7 edges shown in Fig. 4 has the same property. Let E ( H ) = {ei}?=l,where m 3 5 may be supposed. Let G be the same graph as in the previous proposition, i.e., E ( G ) = {fi}?=l and fi n e,# 0 iff i# j . It is easy to

418

J. Lehel

Fig. 4.

show that G must have two independent edges, say f , and f 2 . Then f l U f.. U {x,,} is a 5-element 2-transversal ( x l rE e l f l e 2 ) . 0 The next two Gilmore-like conditions immediately follow from Proposition 3, Proposition 2 and the main theorem.

Condition 1. If H is a 2-intersecting hypergraph and if for every A C V ( H ) , ( A I = 5 , the partial hypergraph H’ with edge set { e E E ( H ): 1 e n A 13 2 ) safkfies T ( H ’ )s 2, then T ( H )zs 2. Condition 2. If H is a 2-intersecting hypergraph and if for every A C V ( H ) , I A 1 = 9, the partial hypergraph H ’ with edge set { e E E ( H ) : I e n A I 3 3} satisfies r ( H ’ )6 2, then T ( H )S 2. Some further special result concerning the function I ( f , k , w ) may be found in [41.

References [ 11 C. Berge, Graphs and Hypergraphs (North-Holland, Amsterdam, 1973). 121 C. Berge and P. Duchet, A generalization of Gilrnore’s theorem, Recent advances in graph theory, Proc. 2nd Czechoslowak Symp., Prague, Academia Prague (1975) 49-55, [3] P. Erdos and T. Gallai. On the maximal number of vertices representing the edge5 of a graph. MTA Mat. Kut. Int. KOzI. 6 (1961) 181-203. [4] J. Lehel. Multitransversals in r-critical hypergraphs, to appear.


ON CERTAIN FAMILIES OF DISJOINT PERFECT MATCHINGS IN Kz, Andri LENTIN Universift?Rent? Descarres, U.E.R. de Marhimatiques, Logique formelle er Informarique, 12 rue Cujas, 7S005 Paris, France How can one obtain 2n - 1 disjoint perfect matchings in the complete graph K , , hy "turning" a given perfect matching round a fixed vertex? How many ways are there to do it? Such problems lead us to investigate, for Z/2n - 1, some properties of functions obtained hy taking the odd part (in the functional sense) of permutations.

I . Introduction 1.1

A family of disjoint perfect matchings in the complete simple graph K 2 , will be called complete if it contains all the edges. Clearly, a complete family of disjoint perfect matchings (CFDPM) includes 2n - 1 such matchings. CFDPM's raise a series of combinatorial problems, the first of which we shall state here in a geometric manner. Let {a,a ' , b, b', .. . ,1, l'} be the set of vertices of Kz,. Imagine a ' as the center of a circle regularly graduated by the integers 0, 1 , 2 , . . . ,n - 1, - n + 1,. . ., - 2, - 1 mod 2n - 1 or, equivalently, by the elements of Z/2n - 1 . Problem. Is it possible to dispose a, b, b', . . . , I, I' on the circle (i.e., to assign them abscissae) in such a way that the matching M = aa', bb', . . . generates a CFDPM by turning round a'? , I l l

This problem and analogous ones have been studied by the author ([I], ['I, (31). I n order to formulate a necessary and sufficient condition, we first define the pseudo-distance S ( x , y ) between two vertices: For x # a ' # y, S(x, y ) is the absolute value of the difference of their abscissae, whereas S(a',x ) = S(a',y ) = 01. Now the required condition is that { & ( a ,a ' ) ,S(b, b'), . . . , S ( I , 1 ' ) ) = {'=, 1 , 2 , . . . , n - 1).

It is worth drawing attention to the (obvious) fact that this condition is equivalent to 319

A. Lentin

-120

{2S(a,a'),2S(b,b'),. . . ,2S(I,l')} = {m, 1,2,. ..,n - I}. When this condition is satisfied, the circle containing a, b, b', . . . defines a permutation q which contains a fixed point a' and a cycle of length 2n - 1. This permutation q generates a CFDPM v ( M ) by acting on the matching M . An example should serve to clarify the various definitions and properties above. Figs. 1 and 2 show such a generating permutation q = (a')(afbb'def'd'ce'c')and its action.

Fig. 2. Second matching.

Fig. I . Initial matching.

The generated CFDPM q ( M ) is as follows: M,,= M = M , = M" =

...

...

aa' bb' cc' fa' b'd e'a

d d ' ee' ff' ec f'c' bd'

... . . . ... ... ...

M , ,= M s ' " = c ' a ' f b

d'e' b'f' dc

I

.

.

ae

One can easily define the M-type of a generating permutation q, a notion we shall present though examples: g.p. ( a ' )( a f b b' d e f ' d ' c e' c ' ) (= 5 11' 3 4 5' 3' 2 4' 2') its type g.p. ( a ' ) ( ad ' f ' e c g' c ' e' b b' f d g ) (m 3' 5' 4 2 6' 2' 4' I 1' 5 3 6 ) its type

The M-type of q is a circular word in a suitable alphabet. Notice that for every integer n there exists a particular type which we shall call the symmetric type (Fig. 3). 1.2. Types from an algebraic point of view

We use again a graduated circle. Without loss of generality we shall assign to the symbol 03 the abscissa 0. Given a type 7,there exists a mapping T of Z/2n - 1 onto itself such that T( - i ) and T ( i ) are, respectively, the abscissa of the origin

Certain families of disjoint perfect mafchings in

Fig. 3. Symmetric type 1'3'5'. . ,642

a

K2,

1 21

2'4'6'. . ,531.

and of the extremity of the arc of length 2i, 2i E {1,2,. . . , n - 1). Clearly, T is a permutation of b / 2 n - 1. So a type can be algebraically defined as a permutation T such that T(i)-

T( -

2

i)- . -

1.

I n other words, given an integer n, the family (e2"= % of types can be identified with the family of permutations of Z/2n - 1, the odd part of which coincides with the identity permutation I. (The type T = I is the symmetric type.) Remark. The tth power of a generating permutation is a generating one iff the integer t is relatively prime with 2n - 1. This fact allows us to define classes of equivalent types. The following table gives the number of types and classes for 2 G n S 7:

/%:"I I'&/

1 1

1

I

1 1

3

z

9

3

25 5

133 14

The determination of the exact value of (e2" 1 (the cardinality of % ? " ) in general seems to be a very difficult and maybe hopeless problem. The aim of the present paper is to connect this problem with other ones of the same kind. This gives a better understanding of the nature of the problem and allows one to formulate a (hopefully) reasonable conjecture concerning a bound for 1 (e2" 1. It seems (at least to the author) that the small amount of material available for this purpose deserves attention.

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A. Lenrin

2. Permutations and symmetric partitions

2.1. Taking the odd part Let R be a ring and f : R + R a function. By taking the odd part (henceforth TTOP) of f we get an odd function f defined for every x by

f(x)=f(

x) - f ( - x

2

6

Using this notation we have, for example, G ( x ) = sh(x), or = sh. A well-known and useful property of l T O P is the 'right exit property':

An example of this is exposin = sh osin. The ring we shall be interested in is H/2n + 1. Let cp be a permutation which belongs to the symmetric group acting on it. For every element a , the function defined by cpo ( x ) = cp(x)+ a is also a permutation and we have (pa = (p. So we can only consider a permutation cp satisfying cp(0) = 0. Throughout the paper we shall put A:, = A = Z/(2n + 1)-(0) and conveniently write cp E Sym(A). Let O(A) be the set of all odd functions g : A + A . TTOP determines a mapping from Sym(A) into b ( A ) . Since we have IB(A)I = n'", (Sym(A)l= ( 2 n ) ! and n2" > ( 2 n ) ! for n 3 3 , we conclude that the mapping above is not surjective. Surjectivity however will appear below in connection with an analogous mapping from Sym(A) into another range. 2.2. Symmetric partitions

For x, y E A, put x [ ( p ] y iff @ ( x ) = @ ( y ) .This equivalence relation determines a partition P(cp). As @ is odd, x [ @ ] y implies - x [ @ ]- y. So the blocks of P(cp) can be paired in such a manner that each block of a pair is symmetric to the other by changing plus to minus and conversely. On these grounds, P(cp) will be called a symmetric partition. Let ItsAbe the set of all symmetric partitions of A. The automorphism group Aut(fisA)is the subgroup of Sym(A ) the elements of which transform every symmetric partition into an isomorphic one. Clearly, function ally odd permutations 8 (i.e. = 8) belong to this group. Hence, if we denote by 0 the group formed by all such permutations, we have 0 S Aut(llsA).What is more, as can be easily proved, equality holds. By a sorted symmetric partition, we mean a partition every block of which is either a subset of A + = { 1 , 2 , . . . ,n } or of A - = { - 1, - 2 , . . . , - n } . Notice that every class 6 of isomorphic symmetric partitions contains sorted partitions. Now, there exists a natural bijection

Certain families of disjoint perfect matchings in K,,

323

between sorted partitions of A and ordinary partitions of A +. So the number of isomorphism classes in USA is p ( n ) , the number of partitions of the integer n. TTOP naturally defines a second mapping P :Sym(A)+ USA.To state an adapted terminology: a permutation q induces (p and determines a partition P ( q ) by means of Cp, the induced function. As seen in the Introduction, P is not injective. On the other hand, we shall prove below that it is surjective. The first step will be proving the following theorem. Theorem 1. Given any isomorphism class 7i of symmetric partitions, one can determine a sorted partition T E 7i by means of a function . i , = j,where both T and J, E Sym(A) are canonically associated with 6. Proof. Part A. Again, we use the graduated circle (here 2n + 1 points, 0,1,2,. . . , n, - n, . . . , - 1). 7i is characterized by a partition of n :

7 i : n= m , + . . . + m , + m , + , + . . . + m , , m , s m , + , . By a representation of 7i we mean a set of arcs - or conveniently chords - such that (a) for every m , , there exist m, chords of the same length 1, ; (b) for i # j , I, # I,. The construction being straightforward, an example will suffice to make it clear to the reader (see Fig. 4). Of course a unique 7i may admit different representations, so we define a regular representation by adding the following conditions. (i) Every arc has one extremity in the upper half-circle and its other extremity in the lower one. (ii) If the representation contains some arc, it contains also the arc symmetric to the first one. (iii) Upper extremities of equal arcs are contiguous (i.e., non-separated by foreign points).

-7

--f

Fig. 4. A representation of 7i : 7 = 1 + 1 t 2 + 3.

A. Lenrin

424

We now show how to construct the canonical representation of the class .li characterized by the partition n = rn,+ * + rn,, m,< m,, We assign to rn, the shortest regular chords on the side opposite to 0 (the left side). We assign to ni2 the shortest remaining regular chords on the side of 0 (the right side). We continue like this, left, right, left . . . . It is easy to prove that the lengths of the arcs increase strictly when passing from rn, to m,+,. As an example, the class 7i considered above admits the canonical representation given by Fig. 5.

-5

Fig. 5 . Canonical representation of Si : I + 1 + Z + 3

Part B. From the canonical representation of 6,we can obtain a canonical sorted partition 7~ E 6 and a canonical permutation J,. We have already considered the particular case where 7i is the class 6 which includes the only partition 0,the blocks of which are the singletons. Of course, the canonical element of 6 is 0 and J,, = I. Inspection of this case suggests to take in general J, as follows. First take J ( i ) = i for i E {1,2,. ..,n} and take J ( - i) equal to the abscissa of the second extremity of the arc whose first extremity has abscissa equal to i. The sorted partition 7~ appears clearly, and each of its blocks contains consecutive integers. In the case of 7~ considered above, we have

The rule of formation is simple and we could get, if necessary, an explicit formula for J ( - i). This proves Theorem 1. 2.3

After we have obtained for 7i a canonical representation, it remains t o show how each member of 7i can also be obtained. First let us study in a more detailed

Certain families of disjoint perfect matchings in K,,

425

manner the action of 0 on USA.For 0 E 0, cp E Sym(A) and writing cp = (cpO)O-', we get by (*) (pe 0 - ' = 4. Hence, the equality Q ( x )= Q ( y ) holds iff

g(O-'(y)).

cpO(O-'(x)) = It follows that cp0 determines the partition O-'(n), where 7r = P(cp). Now let 0, be the subgroup of 0 the elements of which fix 7~ in the whole. The set of permutations which determine 7r is closed by right multiplication by 0,. On these grounds, we can make the following definition.

Definition. A family 9 of permutations which determine a symmetric partition 7r will be called @,,-free (or simply, free) if 9 n $0;= 0, where 0':= 0,- { I } . We can now formulate our second theorem.

u:=,

Theorem 2. Let 0 = 0,0, be the decomposition of 0 in left cosets modulo 0, and let 3,= 93 = {PI,. . . ,& } be a maximal free family of permutations which determine 7r = P ( J ) . Then all the permutations belonging to the same set 930,0j determine the same and unique partition 0, (r).Partitions determined by different sets are different.

Proof. The proof follows directly from the foregoing. 93 is nonempty and since Aut(Usn)= 0, every member of 7~ is obtained. 0 Theorem 2 can be refined by taking into consideration induced functions by means of which partitions in are determined. In the same way, one can choose for 9,a 'quasi-canonical' family. Let 0 ; be the subgroup whose elements fix each block of 7r individually. Clearly, 0:C 0,. Moreover, let us consider the set 0,, the elements of which are permutations 6 such that 1 It is also a group, since j implies J6 = J; hence = J K ' . (i) By virtue of property (*), equality and conversely. (ii) If 2 = 37 then J = J6' = 5 6' = J(66') = J88'.Now by the very definition, j = JO e ( t l x ) J ( x )= J ( O - ' ( x ) ) ,hence 0, is the group which fixes each block of 7r = P ( J ) , and we have 0, = 0:. (For 8 , E 0, \ 0: we should have TOl # 1) On these grounds, we consider the coset decomposition 0, = 0:O"'. From the foregoing we can formulate as a conclusion the following theorem.

+

z=

z=

u:=,

Theorem 3. Every maximal free family B, of Permutations which induce (where J = J , is canonically associated with +) can be taken as a family 93,in the sense of Theorem 2. All classes BJ0B,8"'determine the same 7~ but by means of different functions JO'i'. Two distinct classes induce distinct functions. Corollary. The cardinality of the set of permutations which determine all the partitions of an isomorphism class 7i is 13,12"n!, where 93, is a maximal family

426

A . Lentin

which induces = Jmand J = J,, is the permutation canonically associated with the sorted partition r.

(Recall that 2"n ! = I 0 1 .) Example 1. As previously, the class 0 includes the only partition 0 formed by the singletons. In this case r = 0,0:=( I ) and 0, = 0. The family of all permutations which induce f = I, i.e., V, is free. Example 2. 7i is the class which contains r = { 1 , 2 , .. . , n } { - 1, - 2 , . . . , - n } as the only sorted partition. All arcs in the representation have the same length. On the first block J ( i ) = i and on the second one J ( - i ) = - n - 1 + i. 0 ;= Sym[n]. J O ; I = n ! ;0, = 0 l x ( Z / 2 ) "= 0 and hence /%,I= 1.

2.5. A conjecture It follows from the foregoing that

the sum running over the different classes 7i. Since there are p(n) such classes, the mean value of the cardinalities of the families is ( 2 n ) ! / p ( n ) 2 " n ! We . conjecture that the cardinality of V is always less than the mean value:

2.6. Final remark

M-types and others that we have investigated (but not dealt with here) are more or less analogous to Skolem sequences (cf., Amar, Fournier and Germa, this volume, pp. 7-10 and pp. 11-17) but of course they are circular and not ordinary words. References [ I ] A . Lentin. Sur certains ensembles constituirs dans 5,. par 2n - 1 involutions sans point fixe et discordantes, Publications de 1'U.E.R. de MathCmatiques de I'Universite, Rene-Descartes, Septembre 1Y79. 121 A. Lentin, Sur certains ensembles constituis dans Gj2"par 2n - 1 involutions sans point fixc et discordantes, Etalantes, types, genres, C.R.A.S. de Paris, SCrie A 289 (1979) 559-561. [3] A . Lentin, Sur certains ensembles constitues dans 2,. par 2n - I involutions sans point fixe et discordantes, Groupe d'automorphismes. UnicitC du genre, C.R.A.S. de Paris, SCrie A 289 (1979) 591-593.


DECOMPOSITIONS EN CHAINES D’UN GRAPHE COMPLET D’ORDRE IMPAIR R. LOPEZ BRACHO* Universire‘ du Maine, 72017 le Mans, Cedex, France We study the decomposition of the complete graphs of odd order into trails of given lengths. A decomposition of graph G into trails is a set of its trails such that each edge belongs to one and only one of these trails. For a graph G of orders, we shall say that a decomposition of G into trails C , ,C,, . . . , C, is a factorization of G into trails if we can direct the edges of G in such a way that every trail becomes a directed walk and every point of G becomes the origin of one directed walk and the extremity of one directed walk. We shall say that G is universally factorizable into trails if for every sequence of non-negative integers I , , I?, . ..,Is (s = I V ( G ) l )such as I , + I, f . . . + I, = E ( G ) , there is a factorization of G into trails of lengths equal to I , , I,, . . . .1,. It is proved that the complete graphs of odd order are universally factorizable into trails. This result is used in another work to study the achromatic number of trees with at most one vertex of degree higher than 2; these trees are called stars.

1. Introduction Nous dCfinissons les factorisations en chaines et Ctudions les conditions necessaires et suffisantes pour qu’un graphe complet d’ordre impair soit factorisable en chaines. Les rCsultats ainsi obtenus sont utilisCs en [4] pour dkterminer les Ctoiles n-minimales, une Ctoile Ctant un arbre dont tous les degrCs sauf eventuellement un sont infCrieurs ou Cgaux i 2. Les factorisations en chaines peuvent &tre CtudiCs pour d’autres graphes. Cela est fait pour les graphes cubiques et rkguliers de degrk 4 dans un article de Bouchet et Fouquet [2].

2. Definitions et proprietes elementaires

On considkre des graphes finis et non orient&, qui Cventuellement possCdent des boucles et arCtes multiples. En fait nous nous intkressons essentiellement aux graphes complets. Si G est un graphe, V ( G )est I’ensemble des sommets de G et E ( G ) est l’ensemble des arCtes de G.

* Etudiant boursier du Consejo Nacional de Ciencia y Technologia, Mexique. 427

428

R. Lbpez Bracho

Une ddcomposition de G en chaines est un ensemble de chaines C , ,C?, . . . ,C, tel que chaque ar6te de G appartient Q une et une seule de ces chaines. Faisons remarquer que ces chaines ne sont pas ndcessairement CICmentaires, si bien que un m6me sommet peut-&tre rencontri plusieurs fois. On considerera aussi les chaines de longueur nulle constitu6es pour un seul sommet (X,,), ce sommet Ctant A la fois extremitC initiale et extrCmitC finale de la chaine. Construisons alors le graphe G’ tel que V(G‘)= V(G). E ( G ) = { e l ,e 2 , .. . ,e s } ,les extrCmitCs de chaque ar&teei &ant celles de la chaine C,. On dira que la dCcomposition en chaines C,,C2,. . . , C, est une factorisation en chaines si G’ est un 2-facteur. En d’autres termes, cela veut dire qu’on peut orienter les chaines C,,C2,.. . , C, de telle faGon que chaque sommet de G soit extrCmitC initiale, et extrCmitt finale, d’une et une seule de ces chaines.

Remarque. S’il existe une factorisation en chaines C,,C2,.. . ,C, d’un graphe G, alors (i) 1 v ( G ) l = s , (ii) tous les sommets de G ont degrC pair. Soit m le nombre d’ar6tes de G, un partage de m en s parties designera une representation de m telle que m = 1 , + I , + * * + I, 06 chaque 1, est un entier supCrieur ou Cgal Q 0. On dira qu’un partage m = 1, + I z + * + I, est rCaIisable par une factorisation de G en chaines C1,C2,.. . ,C, si les longueurs de ces chaines sont respectivement l , , I,, . . ., I,. Enfin on dira que G est universellement factorisable en chaines si chaque partage de m en n parties ( n = I V(G)l) est rkalisable par une factorisation de G en chaines.

-

-

3. Factorisation en chaines du graphe K2s+l

Un graphe universellement factorisable doit avoir tous ses degrCs pairs; nous allons voir que cette propriCtC est suffisante pour les graphes complets.

Theoreme. Soit s a 1 un entier, le graphe K2,+,est universellement factorisable en chaines.

On commence par Ctablir deux lemmes, nkcessaires pour la dbmonstration du thCori5me. Lemme 1. Soit un entier s 3 1 ef un partage de M = (2s + 3 ) ( s + 1) en (2s + 3 ) parties. I1 est possible d’ordonner les fermes du partage en M = I , + l2 + . * + llecl,

Dkornposirions en chaines d’un graphe cornplet d’ordre impair

429

lorsque ces termes ne sont pas tous egaux h 2, de telle facon qu ’il existe une suite de naturels A , , Azr . . . ,A2r+3 ve‘rifiant les conditions suiuantes: (i) A , = I,, A, = 1? et A , + Az est pair; (ii) A 7 S I , et A , + h2+ A, = 3 (mod 4); (iii) V i > 3 : A, G I, et Ai est pair; (iv) A , + h2+ * + h2r+3 = 4s + 3. +

+

Preuve. On considere les 3 plus petits termes de la suite. Deux parmi eux ont la mCme paritC; on rtordonne alors la suite de telle faGon que ces termes soient 1, + 1,. Ainsi, en posant A , = 1, et A 2 = 12, A , + A 2 sera pair, et la condition (i) sera ainsi satisfaite. De plus, nous allons Ctablir I’inCgalitC: 1, + I ,

(1)

4s.

Supposons I , + I , > 4s. On a alors I , (ou I?)> 2s et par suite 14,

1%. . .,L t 3 > 2s,

on a

1, +

* *

+

lZs+3>

2 ~ ( 2 ~ 4)S>z + 1.

Mais comme

I , + I,

+ + + 13

14

* * *

+

Lr+3

= (2s

+ 3)(s + 1)

et 1, + 1, > 4s. on a

l4+ * - . + 12s+3 < (2s + 3)(s + 1) - 4s s 2s’

+s +2.

InCgalitC irrtalisable avec s 2 1. On a ainsi vCrifiC, I’inCgalitC 1, + l2 s 4s pour tout s 5 1. Nous faissons maintenant la suite de la preuve pour le cas s 3 2. Pour dtmontrer qu’on peut dCfinir un entier A, z s I., tel que A , + A z + A3 = 3 (mod 4), on va montrer qu’il existe dans la suite L,14,. . . , l,r+3au moins un terme de valeur suptrieure ou Cgale ii 3. S’il n’existe pas de tel terme, les entiers 1 3 , 1 J , . . . r l l r + 3sont tous inftrieurs ou tgaux 2 2, ce qui nous donne:

I , + 1 2 +..* . + l , s + 3 s I , + I ,

+ 2(2s + l),

1’CgalitC

c 1,

2s+7

= (2s

+ 3)(s + 1)

,=I

implique alors (2s + 3)(s

+ 1)

I , + 1 2 + 2(2s + 11,

R. Lbper Brucho

430

compte tenu de I’intgalitt (l), il vient: (2s + 3)(s + 1) - (4s + 2) S 4s,

soit 2 ~ ’ - 3+~1 SO,

intgalitt qui est vCrifite seulement pour s E [1/2,1]. Donc, si s 2 2, il existe un terme au moins Cgal ti 3 dans la suite I,, I,, . . . , I , , + l ; on peut rtordonner de telle faGon que ce terme soit l,. Pour montrer qu’on peut satisfaire aussi aux conditions (iii) et (iv), on prockde comme suit. Soit A I’entier maximum tel que A $ d I , et A + A, + A = 3 (mod 4); P, le plus grand nombre pair tel que P, G 1, (i 2 4). Les relations suivantes sont alors vtrifites:

I,

= A3

+a

avec a

G

3,

et

I, s P, + 1 (i 3 4). D’ou

+ 3)(s + 1) = c 1, Zr+?

(2s

,=I

c ( P , + 1) + A + A: + A + 3 I

1

,a4

ce qui implique:

2 P,+Al+Ar+A32s(2s+3). t

a4

Comme I’intgalitt s (2s + 3) 2 4s + 3 est vtrifiCe pour tout s a 2, on pourra bien dCfinir des entiers pairs Ai (i 3 4 ) teIs que:

en obtenant Zr+3

A,=4s+3, s 3 2 . ,=I

Dans le cas s = 1 il est immtdiat que dans tout partage de 10 cornme somnie de 5 entiers positifs ou nuls, si ces 5 termes ne sont pas tous tgaux i 2, I’un ou moins est supirieur ou Cgal ti 3, et que, I, et l2 ayant 6tC choisis, comme dans le cas s 3 2, parmi les trois plus petits termes, un des trois termes # I , et I2 au moins est

Dicompositions en chaines d'un graphe complet d 'ordre impair

43 1

supCrieur ou Cgal B 3. On prendra alors pour 1, le plus petit entier supCrieur ou Cgal B 3 dans les termes autres que 1, ou 12. II est facile de voir alors qu'on peut trouver des hi vCrifiant les conditions du lernme. 0 Notation. Soit s 3 1 un entier, G,+, dCnote le graphe avec 2s + 3 sornmets x,, x,, . . . ,xzs+3et 4s + 3 ar&tesconstitukes pour toutes les paires contenant x I ou/et xz.

Lemme 2. Soit s 3 1 et A ,,Az, . . . ,A?,+., une suite d'entiers naturels vtrifiant les propritte's suivantes : (i) A l + A, pair; (ii) A I + A z + A 3 = 3 (mod 4); (iii) V i > 3 : Ai est pair ; (iv) A , + A, +. - + = 4s + 3. Alors, le graphe G,,, peut &re dtcompost en 2s + 3 chaines y l , y 2 , .. . , yzr+3 de longueurs respectives A I, A?, .. . ,A2s+3 et telles que xi est 1 'extre'mite' initiale de y, pour chaque i = 1,2,. . . ,2s + 3. Preuve. Aprks une Cventuelle permutation des termes A + A s , . . . ,Az,+l, on place au dCbut les termes Ad, A s , . . . ,A 2 q + 3 qui sont congrus B 2 module 4, on les fait suivre par les termes AzqC4, A 2 q + s , . . . ,A,,, (2q 6 t S 2s) strictement positifs et congrus B 0 modulo 4, et B la fin on place les termes nuls A,,,, A , + s , . . .,A:,+,. Enfin, on posera: A,

+Az+

A1=4r

A, =4r,+2,

+ 3,

i = 4 , 5 ,... , 2 q + 3 ,

i = 2 q + 4 , 2 q + 5 ,..., t + 3 .

A,=4r,,

ConsidCrons la somme zq + 3

z = 2r f

2 21, +

I

=.I

c 1+3

(21, - 1).

i=Zq+l

Cette sornme est Cgale B 2s - t, et ses termes sont tous positifs ou nuls. Par ~ en + nombre ~ tgal 2 2s - t. Ainsi on ailleurs, les sommets x , + ~x,, + ~.,. . ,x ~ sont pourra partitionner cet ensemble de sommets en t + 1 classes P, P,, P5,. .., tellesque IPI=2r, IP,I=2r, ( 4 S i S 2 q + 3 ) , I P , I = 2 r , - l ( 2 q + 4 s i S t + 3 ) . On divisera ultbrieurement P en 3 classes PI,P2 et P,. On dCnotera successivement les ClCments d'une m&me classe P, ( i = 1,2,. . . , t + 3 ) par u',, u:, . . . . On va dCcrire chaque chaine par sa sCquence de sommets. . . , y7r+3 de longueurs nulles. Elles On dtfinit d'abord les chaines Y , + ~ , sont rCduites respectivement aux sommets x ~ +x,,~, ~ , . . . ,x ' ~ + ~ .

Fig. 1. Reprtsentation des chaines y, (-)

et y,+, (-----)

de longueurs respectives A8 = 2 et

A,,, = 6.

Les chaines y, ( i = 2 q comme suit (Fig. 2):

+ 3 , 2 q + 4,. . . , 2 t + 3 ) de

longueurs A, sont dkfinies

y, = ( X , , x I , u ; , x z , u i , XI,

u : , X?, v : ,

mi x.

i1

v ;

3

Fig. 2. Reprtsentation de la chaine y, de longueur A, = 8 .

Dkornposirions en chaines d'un graphe cornplet d'ordre impair

433

Finalement, on d6finit les chaines y l , y2 et 7,. En considerant s6parCment chaque cas dCfini par les diverses valeurs possibles de A I , h2, A 3 (mod4). Compte-tenu des propri6tCs (i) et (ii) et des rbles sym6triques jouCes par h l et A2, ces cas seront: (a) A l = 0, A z = 0, A 3 = 3 (mod 4); (b) A l = 1, A z = 3, A, = 3 (mod 4); (c) A I = 2, A 2 = 2, A, = 3 (mod 4); (d) A , = 0, h2= 2, A 3 = 1 (mod 4); (e) A l = 1, A2 = 1, A, = 1 (mod 4); (f) A l = 3 , A2=3, A 3 = 1 (mod 4). Case (a) (Fig. 3). A I = O , A2=0, h 3 = 3 (mod 4).

y, (------) et ys (. . . . . .) de longueurs respectives A , = 4, A, = 4 e t A , = 3.

Fig. 3. Reprtsentation des chaines y , (-),

Posons A l = 4r1, A 2 = 4rz et A, = 4r, + 3. On a alors r = rl + r2 + r3. Divisons la classe P de cardinalit6 2r en 3 sous-classes, PI de cardinalit6 2r1, P2 de cardinalit6 2rz et P , de cardinalit6 2r,. On d6finit les chaines: y1 = ( X h U I , X 2 , V : ,

7 2

XI,

u:, x2, u:,

x1,

u;+,

xz, UL,, XI);

= (xz, u:, X I , v : , x2,

u:, X I , u:,

x2, & - I ,

XI,

u:,,,

x2);

y3 = (x,, XI, v : , xz, v : , XI,

u ; , x2, u:,

XI, U;,.?-l,

x2,

?I XI, :, x2, ,, xs).

Remarquons que yl est rCduite au sornmet x Isi rI = 0, de mime y2 est riduite au sommet x2 si r2 = 0.

R. Ldpez Bracho

434

Cas (b) (Fig. 4). A , = 1, Az=3, h , = 3 (mod 4).

-y2 (------) et y, (. . . . . .) de longueurs respectives A , = 5 , A2 = 3et A, = 3.

Fig. 4. ReprCsentation des chaines y , (-),

Posons h l = 41, + 1, A 2 = 4r2+ 3 et A 3 = 41, + 3. On a alors r = rl + r2 + r3 + 1. Divisons la classe P de cardinalit6 2r en 3 sous-classes, PI de cardinalit6 2 r l , P , de cardinalitt 2r2 + 1 et P3 de cardinalit6 2r3 + 1. On dtfinit les chaines y l , y 2 et y3 par: 01, Xzr

u:,

XI, 0.4, xz,

u:,

yl = (XI,

Cus (c) (Fig. 5). h i= 2, h2 = 2, h3= 3 (mod 4).

y2 (------) et y3 (. . . . . .) de longueurs respectives A , = 2, A, = 2 et A, = 3.

Fig. 5. ReprCsentation des chaines y, (-),

Dkompositions en chaines d'un graphe complet d'ordre impair

435

Posons A l = 4 r l + 2 , A 2 = 4 r 2 + 2et A 3 = 4 r 3 + 3 .On a alors r = r l + r 2 + r 3 + 1 . Divisons la classe P de cardinalit6 2r en 3 sous-classes, PI de cardinalit6 2rl, P2 de cardinalit6 2r2 1 et P3 de cardinalit6 2r3 + 1. On d6finit les chaines yl, y z et y 3 par:

+

YI=(Xl,u1,X2,u:, XI,

u:, x2, u:,

XI, Uir3-1,X2,

uir3,XI,

uir3+IJ2).

Cus (d) (Fig. 6). A 1 = O , h 2 = 2 , A 3 = 1 (mod 4).

Fig. 6. Reprdsentation des chaines y, (-), y2 (------) et y 3 (. ... ..) de longueurs respectives hI=4,A,=2etA,=1.

Posons A , = 4r1,A z = 4r2+ 2 et A, = 4r3+ 1. On a alors r = r1+ r2 + r3. Divisons la classe P de cardinalit6 2r en 3 sous classes PI de cardinalit6 2rl, Pz de cardinalit6 2r2 et P3 cardinalit6 2r3. On d6finit les chaines y l , y z , et y 3 comme suit: yl = (XI, u t , X2r u:, XI,

u:, Xzr u:,

R . Ldpez Bracho

436 y2 = (x2,

u:, XI, u;,

xz, u:, X I , u:,

CUS

(e) (Fig. 7). A , = 1, A z = 1, A 3 = 1 (mod 4).

Fig. 7. Reprksentation des chaines y1 (-), yz (------) et y3 (. .* .) de longueurs respectives A, = 1, A, = 1 et A3 = 1.

Posons A l = 4rl + 1, A 2 = 4r2+ 1 et A3 = 4r3+ 1. On a alors r = rl + r2 + r3. Divisons la classe P de cardinalitt 2r en 3 sous-classes, Pi de cardinalit6 2rl, P, de cardinalitt 2r2 et P3 de cardinalitt 2r3. On dtfinit ainsi les chaines y i , y-., y3 par: yl = (XI, ut, x27 u:, xi, u : , x2, u:,

y2

= (x2, u:, X I ,

uf,

Cas (f) (Fig. 8). A i = 3, hZ= 3, As = 1 (mod 4).

Decompositions en chaines d’un graphe complet d’ordre impair

437

x

y2 (------) et y3 (... A , = 3, A2 = 3 et A 3 = 1.

Fig. 8. Reprisentation des chaines y , (-),

.. .) de longueurs respectives

Posons A I = 4rl + 3, A 2 = 4r2+ 3 et A, = 4rs + 1. On a alors r = rI + r2 + r3 + 1. Divisons la classe P de cardinalit6 2r en 3 sous-classes, PI de cardinalit6 2rl + 1, P2 de cardinalitt 2 r 2 + 1 et P3 de cardinalitt 2r3. On dtfinit yl, y 2 et y 3 par: y1 = (XI, u1,x*, u : , XI,

u:, xz, u:,

X I , U:,,-l,XZ.

U L , , X I , u1.r,+I,X2rXI);

y2 = (xz, u:, X I , uf, x2,

u:, X I , uf,

x2,

U L - I , XI,

usrL,X?, Ui,,+l,

XI,

x3);

y3 = (XI, xz. u:, X I , u:, Xrt

1 u:3, X I ,

1

Xr,

u;r3-,,xl, U L X 2 ) .

UJ,

0

Preuve du theoreme. On considkre le graphe Kz,+, avec M un partage de M en 2s + 1 parties

= s(2s

+ 1) ar6tes et

M = 11 + 1 2 + . . . + l?S+I. On dtmontera qu’il existe une dtcomposition de K2s+len 2s + 1 chaines C,, C2,.. . , C2q+I de longueurs respectives I , , 1 2 , . . . , 12r+l et d’extrtmit6s initiales deux A deux difftrentes, aprks une orientation de ces chaines. Faisons la dtmonstration par rtcurrence. D’abord, on vtrifie directement la propritt6 pour S = 1 en posant V ( K ? )= {xl, x2, x3}. Trois partages sont possibles pour le nombre d’arites M = 3. On considkre s6parCment chaque cas: (i) soit le partage 3 = 0 + 0 + 3 . On considkre les deux chaines de longueur nulle rtduites aux sommets x I et x2 et la chaine (x3,x1,x2,x3),

R. Ldper Bracho

438

(ii) soit le partage 3 = 0 + 1 + 2. On considerera la chaine de longueur nulle rCduite au sommet x I et les chaines (x2, xj) et (x3, xI,x2); (iii) soit le partage 3 = 1+ 1+ 1. On considkre les chaines (xl, xz), (x2, x,) et (x3,

XI).

On vCrifie aussi directement le cas oh S = 2 et oh le partage de M = 10 est:

10 = 2 + 2 + 2 + 2 4-2. On pose alors V(K5)= {xl,x2,x,, x4, xs}, et on dCfinit les chaines (xl,x2,x4), (x2,x7,xs), (x3,x4, xl), (x4, xs,x2), et (xs,x I ,x3). Maintenant supposons que la propriCtC est vCrifiCe pour un entier S 2 1 et montrons la pour I’entier S + 1 et un partage de M = (s + 1)(2s + 3) en 2s + 3 parties, soit M = lI + l2 + . . . + 12r+3, les termes 11, 1 2 , . . . ,lZr+, n’Ctant pas tous Cgaux B 2. (Le seul cas possible oh les entiers de la suite sont tous Cgaux B 2, est celui qui a CtC considCrC ci-dessus.) On considkre K2,+Icomme un sous-graphe de K 2 r + 3 et on pose

V(Kzst3) = { X I , ~

z ~,

. . ,~ 2 ~ + 7 } ,

3 , .

V(K2,+1)= {x3, x4,. . . ,XZr+3).

Appliquons le Lemme 1. Aprks une Cventuelle permutation des termes ZI, Z2,. . . ,12r+3, on peut trouver une suite de naturels A l , A2,. ..,Az,+, vCrifiant les conditions (i) B (iv) de ce lemme. en 2s + 3 chaines Maintenant appliquons le Lemme 2 et dkcomposons y l , yz,. . . , y 2 * + ,de longueurs respectives A l , h 2 , .. . ,h 2 r + 3 et telle que x, est une des extrCmitts de y, pour chaque i = 1,2,. . . , 2 s + 3. Grice B I’hypothkse de rkcurrence, on peut construire une dCcomposition de K2s+len 2s + 1 chaines C,,C4,. . . ,C,,,,de longueurs respectives 1, - A,, I, A 4 , . . . ,lZr+, - A2.+,. On peut Cgalement supposer que le sommet x, est une des i = 3 , 4 , . . . ,2s + 3. On obtient alors la dtcomposition extrCmitCs de la chaine C,, souhaitCe de Kzr+3 en considerant les chaines y l et y2 et les chaines formCes par la concatknation de yr et C,en leur sommet commun x, ( i = 3 , 4 , . . . , 2 s + 3). 0

B i bliographie [l] C. Berg, Graphes et Hypergraphes (Dunod, Paris, 1970). [2] A. Bouchet et J.L. Fouquet, Graphes universellement dkomposables, i3 paraitre. [3] A. Bouchet and R . L6pez Bracho, Decomposition of a complete graph into trails of given lengths, B paraitre. [4] R. Lopez Bracho, Etude du nombre achromatique des Ctoiles, Thbse de 3tme cycle, Orsay, 1981.


ON

EDGE-CONNECTED DIGRAPHS

~t-

W. MADER Insiitut fur Mathematik, Universitat Hannover, Hannover, West German Federal Republic

Let D be an n-edge-connected digraph and let k,,. . . ,k, be n different edges of D. Subdivide the edge ki by a vertex zi for i = 1,. .. , n and identify z,,. . .,z, to a new vertex z, obtaining a digraph 0,'.Then D' is also n-edge-connected and we say that D' arises from D by the operation 0.. On the other hand, we shall prove that starting from K t we can get all finite n-edge-connected digraphs by alternately adding edges and applying 0,. This successive construction of all finite n-edge-connected digraphs is similar to that given for undirected graphs in [41. The digraphs considered here may have multiple edges and loops. The outdegree and the indegree of a vertex x in D are denoted by d + ( x ;D ) and d-(x; D), respectively. Let V ( D )and E ( D ) mean the vertex set and the edge set of D. The maximum number of edge-disjoint paths from x to y in D is denoted by A (x, y ;D). Let ki E E ( D ) be an edge from xi to y, for i = 1,2, and suppose y , = x2. Then Dklk2may arise from D by deleting kl and k 2 and by adding a new edge from xl to y2. Of course, A (x, y ;D '1~2) S A (x, y ;D ) for all vertices x f y. But, in general, it is not possible to choose k,E E ( D ) ending in a given z E V ( D )and k2E E (D) starting from z in such a way that equality holds for all z # x # y f z, even if d + ( z ;D) = d-(z; D ) and D is n-connected for any given n. (It was proved in [4] that in the undirected case it is always possible to choose k l and k2 such that equality holds for all relevant pairs if z is non-separating and has degree not less than 4.) We have only the following weaker

Theorem 1 [5]. Let D be a finite digraph and let z be a vertex of D with d'(z ;D) = d-(z ;D) 3 1. Then there are edges h ending in z and k starting from z such that min,zxzy#lA(x,y;Dhk)= min,#x+y~zA(x,y;D). This result delivers the successive construction of all n -edge-connected digraphs mentioned above.

Theorem 2 [5]. Every finite n-edge-connected digraph is obtained from K , by alternately adding edges and applying 0.. 439

440

W. Mader

For the proof, it is only necessary to observe that a minimally n-edgeconnected digraph D contains a vertex z with d’(z; D ) = d - ( z ; D ) = n, as proved in [ 3 ] . In [ 1 1 M. Dalmazzo determined the maximum number of edges in a minimally n-edge-connected digraph of given order. Applying Theorem 2, we now get his result easily by induction.

Theorem 3 (Dalmazzo [ 11). For every minimally n-edge-connected digraph D, we haue I E ( D ) I s 2n( 1 V ( D ) l - l), where equality holds if and only if D arises from a tree by replacing every edge [x, y ] by n edges from x to y and n edges from y to x. The following result is also derived from Theorem 1 without difficulty.

Theorem 4 [6]. Given a finite n-edge-connected digraph D and any vertices x, y ofD, there is a path Pfrorn x to y such thatD - E ( P ) is (n - 1)-edge-connected. Being by induction, our proof works, of course, only for finite digraphs, but I should suppose the result valid for infinite n -edge-connected digraphs as well. From Theorem 4 we get the following property of n -edge-connected digraphs immediately by induction: given any n pairs (x,, y,), . . . ,(x”, y.) of vertices in a finite n-edge-connected digraph D, there are n edge-disjoint paths PI,...,P,, in D where P, is a path from xi to yi for i = 1,. . . ,n. This property is also a consequence of Edmonds’s branching theorem (Theorem 5 below) as shown in [7]. It is an astonishing fact that the corresponding problem for undirected graphs has not yet been solved. Let us call a graph G weakly n-linked if, given any n pairs (xl,y,), . . . ,(x,,y,) of vertices of G, there are n edge-disjoint paths P,, . .., P, in G where P, joins xi and y, for i = 1,. .., n. Using Menger’s theorem, it is easily seen that every (2n - 1)-edge-connected graph is weakly n-linked, but, in general, an n-edge-connected graph does not have this property for n even. But Thomassen conjectured in [8] that every (2[n/2] + 1)-edge-connected graph is weakly n-linked. The above mentioned theorem of Edmonds is also an easy consequence of Theorem 1 .

Theorem 5 (Edmonds [2]). Let D be a finite digraph and r E V ( D ) with A ( r , x ; D)a n for all x E V(D - r). Then $here are n edge-disjoint spanning subgraphs T I , .. ., T,, with A ( r , x ; T ) a 1 for all x E V ( D - r ) and i = 1,. . . , n . Proof. The proof follows by induction on the number of vertices. Let I V ( D ) (3 2. The digraph D’ may arise from D by adding n edges from x to r for every x E V(D - r). Obviously, D’ is n-edge-connected. Let D be a minimally n -

n-edge-connected digraphs

44 1

edge-connected spanning subgraph of D’. By Corollary 2 in [3] there is a z E V(D - r ) with d + ( z;D) = d - ( z ;0)= n. By Theorem 1 it is possible to label the edges entering into z, say, h l , .. . ,h. and the edges starting from 2, say, k , , . . . ,k, in such a way that

Do:= ( . . . ( ( p , k * ) % % )

. . .) h A

-z

is n-edged-connected. Let I, be the edge substituted for h,, k,.By induction, Do is decomposable into edge-disjoint spanning subgraphs S1,. . . ,S, with A ( r , x ; S Z ) a l for all x E V ( D , - r ) and i = l , ..., n. For i = l , ..., n, L,:= E ( S , )n { I , , . .. ,In}. If L,# 0, there is an I,, E L , such that h(r,s, ;S, - L , ) a 1 for the starting-point s, of l,,. For i with L, = 0,choose j , so that (i,,. . . ,in}= (1 ,..., n}. For i = l , ..., n, we define so that E , : = { h , , } U { k , : I , E L , }and ?:: = ( V ( D ) ,( E ( S , )- L , ) U E l ) .The edge-disjoint spanning subgraphs TI,.. ., T , of D have the property h ( r , x ; 1 for all x E V ( D -r)and i = 1,. ..,n,and this property is retained if we delete all the edges entering into r, getting in this way T,,.. . ,T,,.

z)3

References [ l ] M. Dalmazzo, Sur la k-connexitt et la k-forte connexitt dans les graphes, Thesis, Universitt P. et M. Curie, 1978. [2] J. Edmonds, Edge-disjoint branchings, in: Combinatorial Algorithms (Academic Press, New York, 1973) pp. 91-96. [3] W. Mader, Ecken vom Innen- und AuRengrad n in minimal n-fach kantenzusammenhangenden Digraphen, Arch. Math. 25 (1974) 107-112. [4] W. Mader, A reduction method for edge-connectivity in graphs, in: B. Bollobis, ed., Advances in Graph Theory, Ann. Discrete Math. 3 (North-Holland, Amsterdam, 1978) pp. 145-164. [ 5 ] W. Mader, Konstruktion aller n-fach kantenzusammenhangenden Digraphen, Europ. J. Combin. 3 (1982) 63-67. [6] W. Mader, On a property of n-edge-connected digraphs, Combinatorica 1 (1981) 385-386. [7] Y. Shiloach, Edge-disjoint branching in directed multigraphs, Inform. Process. Lett. 8 (1979) 24-27. [8] C. Thomassen, 2-linked graphs, Europ. J. Combin. 1 (1980) 371-378.



POLYTOPE DES ABSORBANTS DANS UNE CLASSE DE GRAPHE A SEUIL Ali Ridha MAHJOUB 1.M.A.G., B.P. S3X, 38041 Grenoble, Cedex, France

A graph G is defined to be domishold (Benzaken and Hammer (1978)) if there exist real positive numbers associated to their vertices so that a set of vertices is dominating if and only if the sum of the corresponding weights exceeds a certain threshold b. In this paper, we characterize the polytope of the dominants in this class of graphs, using a polynomial algorithm to find a minimum weight dominating set.

1. Introduction Soit G = ( X , E ) un graphe B n sommets, simple, non orientC, sans boucle, dont X reprksente I’ensemble des sommets, E celui des aretes. Un ensemble A C X est dit absorbant si tout sommet de X - . A est adjacent B au moins un sommet de A. On peut associer B chaque absorbant A de G son vecteur representatif x A dans (0, I}” ( x f = 1 si i E A, 0 sinon). En considerant chacun de ces vecteurs comme un point de R ” , on peut dCfinir I’enveloppe convexe P ( G ) de ces absorbants qu’on appellera polytope des absorbants de G. II est en effet souvent intkressant de caracteriser des structures cornbinatoires par I’enveloppe convexe de ses vecteurs reprksentatifs (cf. les travaux d’Edmonds sur les couplages, de Fulkerson sur les graphes parfaits, de Chvatal, Uhry et Boulala sur les stables dans un graphe sCrie parallkle). Ceci, B notre connaissance, n’a jamais CtC fait sur les absorbants d’une classe de graphes non triviale. De plus, cette caractCrisation est souvent like B des algorithmes de reconnaissance et d’optimisation polynomiaux. Dans cet article, nous caractkrisons le polytope P ( G ) dans la classe des graphes absorbants B seuils dCfinie par Benzaken et Hammer [l] et, pour ce faire, nous donnons un algorithme polynomial en O(n) de construction d’un absorbant de poids minimum dans un tel graphe, alors que ce problkme est NP-complet dam le cas gCnCral. Definition 1 (Benzaken et Hammer [l]). Soit G = ( X , E ) un graphe fini, sans boucle, simple, non orientC, X = {1,2,. . . ,n}, G est dit absorbant ci seuil s’il 443

A . R . Mahjoub

444

existe des poids ai SO associts aux sommets de G et un seuil b SO tel que A C X est un absorbant (ou dominant) de G si et seulement si a, 1 b.

c,,,

Parmi les caracttrisations des graphes absorbants 9 seuil ([ l]),nous utiliserons essentiellement la suivante.

Theoreme 1. Les deux propriktks suivantes sont kquivalentes : ( 1 ) G est absorbant a seuil. (2) G peut se ramener 2 un graphe rkduit a un sommet en enchainant un nombre fini de fois les opkrations suivantes (cf. Fig. 1): 4 , : Supprimer un sommet isolk. 42 : Supprimer un sommet universe1 (sommet adjacent a tous les autres sommets ). 4, : Supprimer une paire de pdes (deux sommets non lit% entre eux, mais adjacents ci tous les autres sommets).

(a)

Sommet isolk Fig. 1.

(b)

Sommet universel

(C)

Paire de p81e

6 est le graphe restant a p r t s avoir fait I’une des opkrations 4i,

ou &

Remarque 1. L’operation 41va dtfinir des ttapes dans le ‘dtmontage’ de G. Chaque &ape consiste 9 supprimer des sommets universels et des paires de p81e adjacents 9 un sommet qui, par la suite, devient un sommet isolC. Autrement dit, il s’agit dans chaque ttape d’enchainer des optrations de type & ou 43suivies d’une opkration de type 4 , . Soit n l le nombre d’optrations de type 41,9 faire, pour ramener G i un sommet. On numtrotera n l + 1 ce sommet, et i le sommet isolt definissant la ibme operation 41.Pour tout sommet j # n l + 1, on notera V, I’ensemble des sommets de G qui sont adjacents h j et V.i+l= X -{l,. . . , nl}. D’aprbs ce qui prichde, on peut reprksenter G par le schema symbolique de la Fig. 2.

Polytope des absorbants

On posera dans la suite

w k

= V, -

445

vk-I, 1 S k S n l .

!kIjl In '1"71 v",+,-v", 0---0 vk-vk.l 0---0

V*-V1

0

v,-vo

Fig. 2 . V , - V , - , = { i ( i E V , et i P V , _ , } e t VO=O.

Remarque 2. Dans la suite on considbre le sommet n l + 1 comme un sommet universe1 dans Wml+l.

Soit C = ( C , ,..., C,) un vecteur de R". On designera par P ( G , C ) le problhme de la recherche d'un absorbant A,, de G tel que Ci soit minimum (on dit que A,, est de poids minimum dans G). Dans ce qui suit, nous allons donner un algorithme de rCsolution de P(G, C) qui exige seulement un nombre d'operations ClCmentaires proportionnel i n. Pour ce faire on peut supposer que C, 3 0 pour i E X, car les sommets ayant des poids nCgatifs appartiennent forcCment i A,,.

xi,,,,

2. Algorithme de resolution de P(G,C )

2.1. Algorithme

On notera (Vk). et (wk)y (resp. (Vk), et ( w k ) p ) les ensembles de sommets universels (resp. paires de p6les) de v k et wk. Pour k = 1,. . . , n l + 1, soit pk (resp. uk) un sommet de ( w k ) p (resp. ( w k ) u ) tel que c,, = min{C, i E ( w ~ ) , , ) (resp. c., = min{C, i E (wk),,}.Soient i t et j * respectivement les sommets teis que

I

I

C,;= min{C, I i

= k,.. . , n l )

et

c,.= min{C, I i E { p l , .. . , P . ~ + u~ I, , .. . , un,+l}} et soit k,,tel que j * E W,,,,. D'aprks les notations prCcCdentes, I'absorbant A,,de poids minimum parmi les absorbants suivants est solution de P(G, C): (a) {l,2,.. ., r - 1 , P,, j * } , r = 1 , . . ., ko, (b) {1,2,.. ., r - 1, u,}, r = 1,. .. , ko+ 1, (c) {1,2,..., r-l,P,,i:}, r = l , ..., ko, (d) {1,2,...,ko,P~+1,~}9

A.R. Mahjoub

446

oij s est un sommet tel que

2.2. Justification Soit A. une solution de P(G,C), et soit r le plus petit entier compris entre 1 et ko (s’il existe) tel que A. r l W,# 0. (1) r existe. Donc {1,2,..., r - l } C A o . Si A o n ( W , ) , # O alors A o = { 1 , 2,..., r - l , u , } , sinon, il existe i# p, tel que A , = { 1 , 2 , . . . , r - 1,p., i}, donc i E {it,j } . (2) r n’existe pas. Donc A. f l W,, = 0 et par suite { 1,2, . . . , ko} C Ao.D’apr2s la ddfinition de j *, il existe dans ce cas, un absorbant A. de poids minimum tel que A. fl Whl+l# 0, donc A. = {1,2,. . . ,ko, uq,+,}ou bien A. = {1,2,.. . , k o , p b f I ,s } . 2.3. Exemple Soit G le graphe ci-dessous (Fig. 3), dCfini par I’enchainement des opirations $2,

41, $3,

$1,

42. w3

92

91

w2

Wl

93

92

Reprtsentation schkmatique de G .

5

7

6

Fig. 3.


447

On affecte aux sommets de G les poids C,= 2 , C2=3, C 3 =12, C 4 = 2 0 , Cs= 8, C,= 16, C, = 4. D’aprhs l’algorithme, un absorbant A. de poids minimum dans G est parmi les absorbants {4}, {1,5,7}, {1,5,2}, {1,2,7} donc A,, = {1,2,7} de poids 9.

3. Polytope des absorbants d’un graphe absorbant a seuil 3.1. Une famille de facette de P ( G )

Lemme 1. Si G n’a pas de sommets isolks (i.e., V , # 0), alors P ( G ) est de pleine dimension. Demonstration. Si VI# 0, les ensembles X - { i } , i = 1,.. . ,n et I’ensemble X hi-mCme forment une famille de n + 1 absorbants affinement indkpendants. Remarque 3. Pour caracttriser P ( G ) , on supposera, sans perte de gCnCralitC, que G n’a pas de sommets isolCs.

cYGI

Definition 2. L’inCquation a x i *a(,,ai E R , Vi, est dite facette (face de dimension n - 1) de P ( G ) si et seulement si (1) elle est valide pour tout absorbant de G ; ( 2 ) il existe n absorbants affinement independants vCrifiant cette inkquation avec CgalitC. Lemme 2. Pour tout sornmet i E X , la contrainte xi s 1 dkfinit une facette de WG). Demonstration. D’une part, la contrainte xi S 1 est satisfaite par tous les absorbants, d’autre part les n absorbants suivants X , X - { l}, ...,X - { i - l}, X - { i + l}, . . . , X - { n } sont affinement indipendants et virifient la contrainte avec CgalitC. 3.2. Le polytope P ( G ) 3.2.1

Theoreme 2. Le polytope P ( G ) est dkfini par les inkgalitks suivantes:

A.R. Mahjoub

448

(1 - 8,,l+l.,)x,+

C x, 2 1,

IEV,

9x,+(j-k+2) t=k+I

i = I , . . . , n r + 1,

(1)

i E ( W k ) p , k = 1,. . . , n l + 1,

x,+k(j+l-t) r E \'k u ( W ,

+

I)u

1=

k

2

x,

I E ( W , , I),ucW,

Q

OU

1 si n l + 1 = i,

&,+,.,

=

0

sinon.

I1 est clair que les contraintes (l), ( 3 ) et (4) sont satisfaites pour tout absorbant du graphe G. Pour une contrainte ( C ) de type ( 2 ) , dkfinie par j , k tels que 0 S k S j S n l , on considkre un absorbant A de G. Si A n (Vk U ( W k + l ) u )0, # alors (C) est vCrifiCe, sinon soit tl,, k s tl,sn l , le plus petit entier tel que A n V,, = 0 et A i l V,,,, # 0, donc {1,2,. . . ,ti,} C A. Suivant que t,,s j ou ti, > j on peut voir fecilement que la contrainte (C) est bien v6rifiCe. Remarque 4. Les contraintes de type (1) et ( 2 ) correspondantes respectivement Q i = n l + 1, et ( k ,j ) tel que k = j = n , , sont les mCmes.

3.2.2. Dtmonstration du Thiort?me 2 Pour demontrer le t h C o r h e , il suffit de montrer que les points extremes du polytope Q reprksentent tous des absorbants de G. Pour ce faire, considkrons le programme IinCaire suivant:

PG

=

[IiIZi,

CERn.

On sait que pour tout point extr&me de Q, il existe un vecteur C tel que ce point soit solution optimale unique de PG. Par consequent, il suffit de montrer que pour tout systkme de poids C = (C,, . . . ,Cn)E R ",sur les sommets de G, il existe une solution optimale de PG qui est le vecteur representatif d'un absorbant de G. Pour cela, nous montrons que pour tout absorbant A. de poids


349

minimum dans G et de vecteur reprdsentatif xA,, donne par I’algorithme prickdent, il existe une solution yAo du dual DG de PG qui vCrifie avec x A o les conditions du thCorbme des Ccarts complCmentaires. Puisque on a suppost Ci 3 0, V i E X , on peut omettre les contraintes (3). A une contrainte de type (l), correspondant 5 un sommet i (is016 ou appartenant 2 une paire de p61e), on associe la variable duale yi, et 5 une contrainte de type (2) dCfinie par les entiers k, j on associe la variable duale &k,j. Le dual DG s’Ccrit:

Soit P(Ao)le poids d’un absorbant Ao,solution de P(G,C), alors d’aprks I’algorithme prCcCdent, il existe r, 1 d r 6 k o + 1, tel que:

P(Ao)3 Cj*+ CI + Cz + *

*

+ C,-i,

et

P(Ao)d C,. + Cl + Cz + * Done, il existe t, r - 1S t

. + Cb + C, .

6 ko, tel

P(Ao)= Cj. + C1+ Cz+. .

*

que

+ C, + 6

avec 0 s 6

G

C,,,.

D’oh le lemme suivant. Lemme 3. Si A,, est solution de P(G, C ) , alors il existe t, r - 1 d t 6, 0 s 6 c Cl+I,tel que

d

ko, et un riel

A.R. Mahjoub

450

P(Ao)= C,*+ Ci + C2+

9

* *

+ Ct + 6.

(9)

Soit

C: = min{Ci,.. . ,C,, Co}, i = 1 , . . ., ko, auec C: = 0, Cz+l=Co, ou

c0= min{C,., c,+,, . . . ,c,,,).

Nous avons C: s Ci*clpour i = 1 , . . .,ko. Alors, si A. est solution de P(G, C ) et de poids donne par la relation (9), il en rksulte: c h c h

+ ’ ’ + c,+ C,*+ 6 s c,,+ C t , 4- ch+l+ + cf -I-6 c,,, pour h tq, 1 G h d t,

+

ch+l

*

(1 1)

* * *

et si t = ko,

p et

ah

6 C C,.

(12)

sont dCjh dkfinis dam I’algorithme au Section 2.1. Soit io te1 que O c i o S t , s i O s 6 s C , : l et C-,*GSGC-,*,~,

zo

=t

+ 1,

sinon.

Lemme 4. Si P ( A o )= C,. + C1+ C, + Cl+I,alors

- + C,+ 6 auec r - 1s t d ko et 0 s 6 s +

*

y*o =

OU

1 si 6 > C,:,,

1 si t = ku,

a,,,

et A =

=

0 sinon,

est une solution rialisable du dual DG.

0 sinon,


45 1

Demonstration. Simple 2 verifier B partir de (lo), (11) et (12). D’aprks les Lemmes 3 et 4, on dCduit que pour tout absorbant A. solution de P(G, C ) , on peut associer une solution rCalisable y % du dual DG.

Lemme 5. Pour tout absorbant Ao, solution de P ( G , C ) , dktermine‘ par i’algorithme, xAo est une solution optimale de PG. Demonstration. I1 est facile de verifier que pour tout absorbant Aorde poids minimum propose par l’algorithme, x A o vtrifie avec y A o les conditions du theorkme des Ccarts complkmentaires. Le Lemme 5 achkve la dkmonstration du ThCorbme 2.

Exemple. On reprend le graphe G donne par la Fig. 3, le polytope P ( G )sera dans ce cas:

+ x4

XI XZ

x2+

+

x4+

x5+

Xh

x3+ x4+

X5-k

x6-k

x4+

x5+

x7

x3+

3

1,

3

1,

x731, 3

1,

Les quatres contraintes marquees d’une astCrisque sont redondantes, ce qui montre que les contraintes de type (2) ne sont pas toutes des facettes. La solution optimale du dual DG associee i I’absorbant de poids minimum A,, est: y Ao =

y3= 1, Eo.1 = 2, El.1 =

1,

y, = &j,k = 0, ailleurs.

452

A.R. Mahjowb

References 111 C. Benzaken and P.L. Hammer, Linear separation of dominating sets in graphes, Discrete Math. 3 (1978) 1-10. [2] C. Berge, Graphes et Hypergraphes (Dunod, Paris, 1973). [3] M. Boulala and J.P. Uhry, Polytope des indbpendants d'un graphe sirie-paralltle. Discrete Math. 27 (1979) 225-243. [4] V.Chvatal, On certain polytopes associated with graphs, J. Comb. Theory 18 (1975) 138-154. [5] M.W. Padberg, On the facial structure of set packing polyhedra, Math. Programming 5 (1973) 199-2 15. [6] U.N. Peled, Properties of facets of binary polytopes, Discrete Math. 1 (1977) 435-456. [7) G.L. Nemhauser and L.E. Trotter, Properties of vertex packing and independence system polyhedra, Math. Programming 5 (1974) 48-61. [8] G.L. Nemhauser and L.E. Trotter, Vertex packing structural properties and algorithms, Math. Programming 8 (1975) 232-248. [9] L.E. TrotterrA class of facet producing graphs for vertex packing polyhedra, Discrete Math. 12 (1975) 373-388. [lo] L.A. ' Wolsey, Further facet generating procedures for vertex packing polytopes, Math. Programming 1 1 (1976) 158-163.


AN UPPER BOUND ON COVERING RADIUS H.F. MATTSON, Jr. School of Computer and Information Science, Syracuse University, Syracuse, New York 13210, USA We present a new upper bound on the covering radius of any binary linear code. For some codes the new bound improves on the Delsarte bound.

1. Introduction The covering radius of a code A is defined as the least integer p such that the spheres of radius p centered at the codewords of A cover the space 2;. (By code we mean a non-empty subset of Z ; . ) The packing radius of A is the largest integer rr such that the spheres of radius rr centered at the codewords are mutually disjoint. Thus rr S p , and when rr = p the code is called perfect. Coding theorists have made intensive studies of the packing radius since the beginning (1949), but not of the covering radius. The little that is known about the covering radius consists mostly of lower bounds, some of which apply only to special codes. For a survery on the covering radius the reader may see [4]. Only a few upper bounds on the covering radius are known. That of Delsarte [2] has the most general applicability, holding even for nonlinear codes. Another upper bound [3] holds for a restricted class of nonlinear codes and appears to be useful only for codes of low rate. (The rate of A is defined as (l/n)log,(A I .) An elementary upper bound for linear (n,k ) codes A is cr(A ) s n - k,

(1)

where we denote the covering radius of A by cr(A). The purpose of this paper is to present a new upper bound on the covering radius for linear codes, to compare it to an extent with the bound of Delsarte, and to apply it to construct a code with ‘good’ covering radius. 2. Main definitions The code A is called linear if it is a linear subspace of the vector space 2; over the field GF(2) = (0,l) of two elements. From now on we assume that all our codes are linear. 453

H.F. Mattson, Jr.

454

If k is the dimension of A, then we call A an (n,k ) code.' A can be presented by means of a generator matrix, a k X n matrix with A as row space, or by means of a parity-check matrix, pcm(A), a matrix H of n columns and rank r = n - k each row of which is a vector in the orthogonal, or dual, code A'. Thus the number of rows may be greater than r, and A is the set of all linear relations on the columns of H. For x E Z;, a vector in the coset x + A is called a coset leader if it has the least (Hamming) weight among all elements of its coset. The covering radius of A is the weight of the coset leader of maximum weight among all the cosets of A. For any x EZ1, we define the syndrome of x to be Hx', where x ' is the column vector obtained by transposing x . ( H is pcm(A), and the syndrome of x depends not only on A but also on the choice of H . ) Two vectors are in the same coset if and only if they have the same syndrome, so there is a 1-1 correspondence between the 2"-' syndromes and the 2"-k cosets. The code is the coset with syndrome 0. The covering radius is the least integer p such that every syndrome is a sum of p or fewer columns of H. An information set for the ( n , k ) code A is a linearly independent set of k coordinate-places of A. Thus a k-subset I of coordinate places is an information set if and only if every vector of 2: appears on I. Two (n, k ) codes are called equivalent whenever there is a coordinatepermutation of 2; which carries one code onto the other. Since equivalence preserves the covering radius, among other things, we will freely pass from a code to an equivalent code, usually without explicit mention. The redundancy of the code is r = n - k. Consider a code A given by its parity check matrix in the form H = I, ; D, where I, is the r X r identity matrix and D is any r X k matrix. Thus A is an ( r + k, k ) code. Let j denote the rank of D.Then the column space A 2 of D is an (r, j ) code, of covering radius pz. We also consider the ( k , k - j ) code A I of which D is a parity-check matrix, and call its covering radius p l . (A1is a 'shortened' version of A, in that A I is obtained from the subcode A. of A having 0's on the first r coordinates by projecting A. onto its last k coordinate places.) Then we have the following result.

Theorem 1. Under the above definitions, the covering radius of A is at most pt

+ p2: cr(A ) C P I + p2.

* The reader may consult any book on coding theory for basic terms and results, e.g. [l], [5], [6] and [8].

A n upper bound on covering radius

455

Proof. Let z E 2;be any syndrome (we consider z a column vector). We shall show that we can express z as a sum of at most p l + pz columns of H . Now there is a vector a of A, at distance p2 or less from z. Thus z + a = e, where the Hamming weight of e is at most p2. Ignoring, as we may, the redundancy if j < r, we regard a as a syndrome of the code A 1 .Then a is a sum of at most p I columns of D. Since e is a sum of at most p2 columns of I,, z = a + e is a sum of at most pl + p2 columns of H. 0 Remark. The rank of D is not necessarily constant over the set of codes equivalent to A. Since, when j = k, pl is necessarily k, it is sometimes helpful to search for an equivalent code having D of lower rank. Example. Consider the two matrices H :

1

1 0 0 1 1 1 0 1 0 0 1 1 . 0 0 1 1 0 1 On the left D has rank 2, on the right rank 3, yet A is equivalent to A ’, as one sees by interchanging columns 3 and 4 and performing row operations. A and A ’ are both (6,3,3) codes which are known to have covering radius 2 (and this fact is obvious). Let us calculate pl + p2 for each presentation. On the left D is

and it is the pcm of the (3,1,3) code (000,l ll}, which tells us that pl = 1. The ‘column’ code A z is the (3,2,2) code of all even-weight vectors; it has covering radius 1. Thus pl + p2 = 2, and the bound is the true value. On the right pl = 3 and p2 = 0. In this example p l + p r = n - k, the upper bound mentioned in (1).

3. The Delsarte bound

We now compare the p , + pz bound to the Delsarte bound. Since it is not easy to calculate ‘either bound in any generality we will concentrate on some small codes, but first we express the Delsarte bound in the linear case. Delsarte’s Theorem [2]. If H is the parity -check matrix of the ( n , k ) code A , then cr(A) S s’,

H.F. Mattson, Jr.

456

where s ' is the number of different nonzero weights occurring among the vectors of the row space of H, an (n, n - k ) code culled the orthogonal, or dual, code of A and denoted A ' . Delsarte's bound is exact for perfect codes [6]; the new bound is not, for it gives p , = 2, p2 = 0 for the (7,4,3)perfect Hamming code of covering radius 1, as the reader may easily verify. We now give an extensive sampling of (10,4,4) codes, having calculated both bounds for each sample, and the covering radius. We chose the lowest rank for D among the equivalence class of each sample code. We first calculate one example in some detail. Let

H=I6,

[1.:J

1 1 0 0 1 0 1 0 D=Z6 1 0 0 1

=pcm(C,).

The columns of D have sum 0, so j = 3. A I is the (4,1,4) code {04,l'} with cr(A,)=2 = p l . A2 is a (6,3,3) code as we can see by direct verification. Thus p2 = 2, and cr(C,) 6 4. We get the Delsarte bound as follows. The sum of all rows of H is 1" = (1 1 . . * l), the all-1 vector of weight 10. The minimum nonzero weight in C: is 3; row 1 +row 2 is a vector of weight 4; to that add row 5 for a vector of weight 5. Therefore the nonzero weights of vectors in C: are 3 , 4 , 5 , 6 , 7, 10. Thus the Delsarte bound is 6. The covering radius is 4, because the syndrome 1' (now a column vector) is, by.an obvious parity argument, not the sum of an odd number of columns of H. Since it is evidently not the sum of any 2 columns of H, its coset has weight 4. We now list (Table 1) 13 (10,4,4) codes G,.. .,C13by displaying the 6 rows of D as integers written in base 10; their base-2 representatives are the rows. Thus for C,the first row of D 'is' 12 and the last 'is' 3. Although we do not claim that this table contains all (10,4,4) codes up to equivalence, it may be representative of the values of the covering radius, pl + p2, and 'Del', the Delsarte bound, for such codes. Because we chose the minimum value of p i + p2 for the equivalence class in each case worked out, we know that the 13 codes Ciare mutually inequivalent, since covering radius and Delsarte bound are invariants. The codes Ci for i = 10,11,12 are mutually inequivalent because the Ct have, respectively, exactly 1, 0, and 2 vectors of weight 1. Although it may not be fair to quote the Delsarte bound when it exceeds n - k, here 6 , it is interesting to note that the pl + p2 bound improves on the Delsarte bound in 12 of the 13 cases.

A n upper bound on covering radius

457

Table 1 Some (10,4,4) codes C,, . . . , C,,

1 2 3 4 5 6 7 8 9 10 11 12 13 ~~~~

1 2 15 1 5 1 4 12 1 5 15 14 15 15 13 14 14

1 0 12 1 2 1 3 11 1 5 14 13 12 14 13 13 13

9 10 Y 8 11 8 13 11 10 13 11 11 11

6 6 6 6 6 4 11 8

9 11 4 7 8

5

5 6 3 5 2 7 7 7 7 3 0 7

3 3 3 1 4

4 3 3 4 3

4 4 4 4 5

I

5 3 3 3 4 4 4 3

5 5 6’ 6‘ 6

3 7 7 0 3 0 0

6‘ 6 6

6 7 8 8

8 6 6 4

8 8 8 8 Y

~~

Added in proof: Here 6 can be replaced by 5 , because code is equivalent to code with p2 = 1.

In working out this table, which we did by hand, we were aided by the following.

Lemma 1. No (6,3) code with covering radius 3 is the ‘column-code’ A2 of a (10,4,4) code (in the present notation). Lemma 2. The 6 X 4 matrix D yields a (10,4,4) code if each column of D has weight at least 3, and each two columns of D are at distance at least 2, and no three columns of D have sum 0. Lemma 3. A (10,4,4) code has a parity-check matrix of the form I,; D with D of rank 3 iff the code has a vector of weight 4 such that the complementary 6 places do not contain the support of any nonzero codeword. Corollary. If the code has a vector of weight 10, then it is a ‘rank -4’ code. If it has a vector of weight 9 and a vector of weight 4 not ‘contained’ in the former, then it is a ‘rank-3’ code. Consider the (36,13) code A with parity-check matrix

D, where

H.F. Mattson, Jr.

458

the columns of D being 13 vectors of the (23,12,7) Golay code filled in as indicated on an information set consisting of the top 12 rows. Here pl = 6 and pz = 3, so cr(A) S 9. But the sphere-covering lower bound on covering radius [4,7] is 7 for (36,13) codes.

4. (Added after the conference (23 July 1981))

The following result in some cases improves the bound of Theorem 1.

Theorem 2. With the hypothesis of Theorem 1, the covering radius of A is at most u + p2, where u = min{pI, L4W11,

and

W = max{la

1;

a E A, 1 a 1 S r +pi}.

When p l > r, the bound of Theorem 2 is better than that of Theorem 1. We will present the proof of Theorem 2 and some applications in a future paper.

References [ 11 E.R. Berlekamp, Algebraic Coding Theory (McGraw-Hill, New York, 1968). [2] P. Delsarte, Four fundamental parameters of a code and their combinatorialsignificance,Inform. and Control 23 (1973) 407438. [3] T. Helleseth, T. Kldve and J. Mykkeltveit, On the covering radius of binary codes, IEEE Trans. Inform. Theory IT-24 (1978) 627-628. [4] M. Karpovsky, H.F. Mattson, Jr. and J.R. Schatz, A survey on covering radius, to appear. [5] J.H. van Lint, Coding Theory, Lecture Notes in Math. 201 (Springer, Berlin, 1971). [6] F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes (North-Holland, Amsterdam, 1977). [7] H.C.A. van Tilborg, Uniformly Packed Codes (Technical University, Eindhoven, 1976). [8] W.W. Peterson and E.J. Weldon, Jr., Error-Correcting Codes (MIT, Cambridge, 1972).


SUR UNE PROPRIETE EXTREME DES PLANS PROJECTIFS FINIS Jean FranGois MAURRAS Universite‘ de Paris X U , Paris

We show that p lines of the finite projective plane of order n cover at least ( n + 1)2p(n + p ) - ’ points and we construct some uniform regular hypergraphs having similar properties.

1. Introduction Soit P un plan projectif fini d’ordre n, deux droites ayant un seul point d’intersection, il est aisC de constater que n + 1 droites recouvrent au moins :(n + 2 ) ( n+ 2 ) points. Cette constation permet de montrer que, B priori, une quelconque methode directe d’inversion de matrice (du type de Gauss par ex.) ne peut pas conserver le creux de la matrice initiale (prendre une matrice dont tous les ClCments non nuls sont linkairement indkpendants sur Q, le corps des rationnels, ClCments non nuls dont le support est la matrice d’incidence de P ) . Un argument de comptage de De La VCga [l] permet de montrer que mZme pour des matrices (n,n ) B k ClCments non nuls par ligne et par colomne ( k 3) cette propriCtC de conservation du creux n’est pas vCrifiCe, le nombre d’kltments non nuls de I’inverse Ctant en O(n2). Dans cet article on dtmontre tout d’abord que p droites du plan projectif d’ordre n couvrent au moins (n + l)’p(n + p ) - ’ points, on construit alors un hypergraphe 3-rtgulier et 3-uniforme sur n points tel que n / 2 arktes couvrent au moins n / 2 + 6 points, combinant les deix rksultats prCcCdents on peut gCnCraliser ce rtsultat aux hypergraphes k -rCguliers.

2. Les plans projectifs finis

Soit P un plan projectif fini d’ordre n [3], E son ensemble de points. Lemme. p droites coururent au rnoins ( n + l)’p(n

En effet ces p droites s’intersectent deux 459

+ p ) - ’ points.

deux en 4p@ - 1) couples de

J. F. Maurras

460

droites. Soit F C €, I F / = q, I’ensemble des points couverts par ces droites. V e E F soit qt le nombre de droites contenant e. On a

Remarquons que EeEFqc (qc - 1)/2 est une fonction convexe minimum, en continu, pour V e E F, q. = ( n + l ) p / q . D’ou

q ( n + l ) ( p / q ) ( ( n+ l ) ( P / q ) - 1 ) s P ( P - 1) I’inCgalitC cherchCe.

Remarque. L’on aurait pu obtenir ce rCsultat au moyen de I’inCgalitC classique de Reiman [2]. Corollaire. Soit pour n, ordre d ’ u n plan projectif, n + w ( n + 1 ) f ( n ) droites couurent presque tous les points.

alors f ( n ) +

m,

alors

Nous allons i! prCsent ‘tronquer’ le plan projectif de la f a w n (classique) suivante: Tout d’abord on numbrote les points d’une droite, que I’on privilCgie, de 1 Q n + 1; puis, successivernent ceux des droites rencontrant le point 1 (droite par droite). On numirote ainsi tous les points. D’autre part on numerote la premikre droite 1, les n suivantes de 2 Q n + 1. On numCrote alors les n 2 droites restantes en commengant par celles recontrant le point 2, puis celles rencontrant le point 3, jusqu’Q celles rencontrant le point n + 1. On considbre Q prCsent I’ensemble Ek des points indexts par K (k E N )

K

= { j E N,

n

+ 1 + kn < j s n ’ +

n

+ l},

et I’intersection des droites indexCes par K avec Ek. Soit p k une telle configuration. Par le lemme, le nombre de points de E non recouverts par p droites est au plus

+ + 1- ( n + 1)2p/(n+ p ) .

g@) = n 2 n

ConsidCrons pk lorsque n +. 03; si le nombre de points de pk est grand devant g @ ) , alors presque tous les points de Pk seront recouverts par p droites quekonques de pk,d’oh en remarquant que g @ ) est minimum pour p = rn/2+n.

Soit donc s+

C...

,=lA

rj d. Le Lemme 2 fournit une condition suffisante pour construire un (n, d)-code sphCrique par la reunion de plusieurs (n, d)-codes sphkriques extraits des classes CI, cz,.. . ,

c..

Une construction de codes sphiriques

471

5. Codes sphkriques extraits de C, L’idCe de construction utilisCe est la suivante: Un (n,d)-code sphkrique optimal est tout d’abord extrait de C:: IxIJ,Ix*J,. - 3 lxm I.

De chaque classe f l ,i2, . . . ,f sont ensuite extraits les (n, d)-codes sphCriques U s,. optimaux s l ,s2,. . . ,s,. Enfin, on forme le code sphCrique Sk = sl U sz U D’apres le lemme 1 on a: i# j , d(si, s j ) = d(lxi 1, Ixj I) et cette distance est au moins Cgale i d de par la construction de I’ensemble des IxI I,. . . ,I x, 1. Ainsi les ensembles sI,.. . , s, sont deux i deux disjoints et leur rtunion S, est un (n, d)-code sphtrique.

--

Conjecture. Avec cette construction le code

s k

est optimal dans C,.

5.1. Code sphe‘rique optimal sur C;

Les vecteurs de C: ont k composantes non nulles Cgales h l / d . Ces vecteurs sont binaires au sens de la dtfinition donnte au Section 1. La distance entre deux vecteurs de C;est donc directement fonction du nombre de leurs composantes distinctes, c’est A dire de leur distance de Hamming:

Ainsi, rechercher un (n, d)-code sphCrique optimal dans C; revient h rechercher le code binaire optimal en longueur n, dont les mots ont pour poids de Hamming k et dont la distance minimale de Hamming est:

dH= rkd’]. ([x 1 : partie entibre par ex&.)

Par definition, ce code optimal contient A (n, [kd‘], k ) vecteurs. On peut donc extraire de C; un (n,d)-code sphCrique optimal

I

1 x;! 1

~ 1 1 ,

7

. . ., I x m I

avec m = A (n, [kd21,k).

5.2. Code sphe‘rique optimal dans une classe d ‘kquivalence i Soit x E ck. Les vecteurs de la classe k ont leurs composantes non nulles qui occupent toujours les mtmes positions: celles de 1 x I. Ces vecteurs ne different entre eux que par les signes de ces k composantes non nulles. La distance entre deux vecteurs de k, xI et xz, est directement fonction du

B. Montaron

478

nombre de signes diffCrents, composante 2 composante, entre ces deux vecteurs, c’est ii dire fonction de leur distance de Hamming au sens large: d ( X i , Xz) = d ( 4 / k ) d ~ ( X Xz). i,

Finalement, rechercher un ( n ,d)-code sphCrique optimal dans i,revient rechercher un code binaire optimal en longueur k , dont la distance minimale de Hamming est:

dH = r k d 2 / 4 ] . Par dCfinition, un tel code contient A ( k , r k d 2 / 4 ] )vecteurs.

5.3. (n,d)-code sphe‘rique extrait de Theoreme 1. I1 existe duns

ck

ck u n ( n , d)-code sphe‘rique sk de cardinal

A ( n , rkd’], k ) . A ( k , [ k d 2 / 4 ] ) . A la Section 5.1 nous avons extrait de C; un (n,d)-code sphCrique

IXI I, I I x2

f

...

9

IXm I

avec rn = A ( n , [kd’], k ) . De chaque classe 5,, .. .,5, ont CtC extraits ii la Section 5.2 les (n,d)-codes sphbriques sl,. . . ,sm ayant chacun pour cardinal

A ( k , 1kd */41). Leur r6union est (Lemme 1) un (n,d)-code sphCrique cardinal A(n, rM2], k ) A ( k , [k(d2/4)1).

sk

= s IU

Us,

de

Remarque. D’aprbs la conjecture Cmise ce code est le plus grand ( n ,d)-code que I’on puisse extraire de C,. Exemple. Les cardinaux des codes sphCriques en distance d = 1 sont des nombres appelQ ‘kissing numbers’. En appliquant le ThCorbme 1, on a

V k , k = l,.. . ,n,

T,,2

A ( n , k, k ) A ( k , r k / 4 ] ) .

Par exemple, pour n = 4, si I’on choisit k = 2, on a T~ 2 A (4,2,2)A ( 2 , l ) avec A (4,2,2) = 6 et A (2,l) = 4 d’ou T~ 2 24. C’est d’ailleurs la rfieilleure valeur connue. On sait par ailleurs que T~ S 25 (cf. [2]).

Nous avons tent6 de construire 25 vecteurs norm& de 0,avec une distance minimale 1. Jusqu’i prCsent nos efforts n’ont abouti qu’8 une distance minimale de 0.92. Cette distance peut certainement t t r e sensiblement amtliorCe, mais il

Une construction de codes sphiriques

479

semble peu probable qu’elle puisse atteindre 1, si bien que T~ est tr6s probablement Cgal h 24. La construction du code sphkrique de 24 ClCments correspondant ri la borne donnee est trks simple: Le code binaire associC au code sphCrique extrait de Ct C 0,correspondant ri A (4,2,2) = 6 est

Le code binaire associC au code sphkrique extrait d’une classe d’un vecteur de C;, correspondant h A (2,l) = 4, est

Le code (2) dkfinit toutes les possibilitks permises pour les signes des composantes non nulles des vecteurs du code (1). La composition de ces deux codes produit un (4,l)-code sphkrique: (-x,x,O,O)

(-x,

(-x,O,x,O)

(-x,o, -x,O)

(x,O,O, - x )

(-x,O,O,x)

(-x,o,o, - x )

(0,x, x, 0)

(0, x, - x, 0)

(0, - x, x, 0)

(0, - x, - x, 0)

(O,x,O,x)

(O,X,O,

(0, -x,O,x)

(0, -x,o, - x )

(O,O,X,X)

(O,O,x,-x)

(O,O,--x,x)

(O,O,-x,-x)

(X,X,O,O)

(x,

(x,O,x,O)

k0,

(x,O,O,x)

-X,O,O) -X,O)

-x)

-X,O,O)

oh x = l/G car les vecteurs doivent Ctre normks. Cette construction est

ginkrale. Elle s’applique aux codes sphCriques S, dCcrits au Section 5 . 6. Codes sphCriques extraits de U.

Pour les distances d infCrieures ou kgales ri d?, le Lemme 2 fournit un critttre pour rCunir certains codes parmi les (n,d)-codes sphkriques optimaux S , , S 2 , .. . ,S. de manikre h former un (n,d)-code sphkrique plus grand, extrait de U,. C’est 1’Ctape finale de la construction, sanctionnke par le thCor6me suivant.

B. Moniaron

480

ThCoreme 2. I1 existe duns U,, C R, un (n,d)-code sphe‘rique de distance d s fi,de cardinal:

oir E est une famille d’entiers { e l , .. . , e l } telle que E C{1,2,.. . , n } et Vi, i = 1 , ..., I - 1 ,

Ce thkorbme se dCduit directement du ThCor&me 1 et du Lemme 2. Exemples. Le ThCorCme 2 permet d’obtenir les bornes suivantes pour les ‘kissing numbers’: ( 1 ) T,, b A(4,1)A(n,4,4)+A ( 1 6 , 4 ) A ( n ,16,16)+ * *, ( 2 ) T. b A ( 2 , l ) A(n,2,2) + A (8,2)A(n,8,8) + * * , (3) 7” 3 A ( n , [n/41)+A (n,[ n / 4 ][, n / 4 ] ) A( [ n / 4 ][a[n/4]1>+ , -. . Ces bornes sont difficiles A Cvaluer asymptotiquement. En particulier, comment doit-on choisir la famille E pour que la borne soit la meilleure possible asymptotiquement? Pour les petites dimensions, les risultats sont relativement bons. Pour n s 10, A I’exception du cas n = 6, les valeurs obtenues par les bornes correspondent aux meilleures valeurs connues. C’est aussi le cas (cf. Tableau 1)pour la plupart des dimensions infbrieures A 25.

-

-

Tableau 1 Valeurs de n

1

2 3 4 5

6 7 8 9

10 11 12 13 14 15 16

T”,

n

16

meilleure valeur connue [2] 2 6 12 24 40 72 126 240 306 500 582 840 1130 1582 2564 4320

valeur donnCe au thCorhe 2 2 6 12 24 40 60 126 240 306 500 582 840 1066 1484 2340 4320

borne sup. (cf. P I ) 2 6 12 25 46 82 140 240 380 595 915 1416 2233 3492 5431 8313

Une construction de codes sphe'riques

En particulier pour n T~ 2

=8

38 1

le ThCorbme 2 donne:

A (2,l)A (8,2,2)+ A (8,2) soit

T~ 3 240.

On sait qu'en fait T~ = 240 et que le code sphirique associC est unique, aux symitries et rotations prks (cf. [3]). La construction de ce code est fort simple par la mCthode dCcrite aux Section 5 et Section 6, en extrayant un code sphtrique de C2et de C8et en les rkunissant.

Refbrences

111

F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes I, I1 (NorthHolland, Amsterdam, 1977). [2] A.M. Odlyzko and N.J.A. Sloane, New bounds on the number of unit spheres that can touch a unit sphere in n dimensions, J. Comb. Theory, Ser. A 26 (1979) 210-214. [3] E. Bannai and N.J.A. Sloane, Uniqueness of certain spherical codes, Canad. J. Math. [4] J. Berman and K. Hanes, Optimizing the arrangement of points on the unit sphere, Math. Comput. 31 (1977) 1006-1008. 151 P. Delsarte, J.M. Goethals and J.J. Seidel, Spherical codes and designs, Geometriae Dedicata 6 (1977) 363-388.



SYMMETRIC INSEPARABLE DOUBLE SQUARES Annie-Francoise MOUYART Universiri de Lille I. France

1. Definitions

A double square of order n is an n X n array based on an n-set S, satisfying the following rules: (1) each cell contains two symbols; ( 2 ) every symbol occurs exactly twice in each row and in each column; ( 3 ) every unordered pair of different symbols occurs exactly twice in the square; (4) for every symbol s, the pair (s,s) occurs once in the main diagonal, and nowhere else. If a double square can be obtained on superimposing two latin squares, we call it a separable square. Otherwise, it is an inseparable square. A double square is symmetric if the cells ( i , j ) and 0,i)contain the same unordered pair, for every i and j . We shall study symmetric inseparable double squares of order n, calling them SIDS(n).

2. Similar squares

Graeco -latin squares, obtained on superimposing two orthogonal latin squares, satisfy conditions (l), (2) and (3). If the two latin squares have a common transversal, the graeco-latin square can be changed into a separable double square. A symmetric separable double square can be constructed from a selforthogonal latin square (latin square orthogonal to its transpose). Brayton, Coppersmith and Hoffman [l] proved the existence of self-orthogonal latin squares of all orders except 2, 3 and 6, so we can say the following. Theorem 2.1. There exist symmetric separable double squares of all orders except 2, 3 and 6 . 483

A.-F. Mouyart

484

Heinrich and Wallis [5] call N * ( n , 2 ) a double square, and N ( n , 2 ) a double square which cannot be separated into a pair of orthogonal latin squares. So an N ( n , 2 ) is either an inseparable square or a square which can be separated into a pair of non-orthogonal latin squares. They give examples and direct constructions of N ( n , 2 ) . Except the N ( 7 , 2 ) , separable into a pair of nonorthogonal latin squares, all are inseparable double squares, but only an N ( 6 , 2 ) is symmetric. They prove the following. Theorem 2.2. For every n

3

5 , there exists a n N ( n , 2 ) .

3. Examples

We can study the existence of double squares of small orders. (i) n = 1. There exists a symmetric separable double square (Fig. 1):

Fig. 1

(ii) n = 2 and n = 3 . There exists no double square. (iii) n = 4. The only double square of order 4 is symmetric and separable into a pair of orthogonal latin squares (Fig. 2).

34

22

14

13

24

14

33

12

23

13

12

44

Fig. 2.

(iv) n = 5 . Every double square of order 5 is equiualent to one of the squares of Fig. 3 and Fig. 4. It can be obtained from one of these squares by permuting rows, columns and symbols, and sometimes transposing. 11

34

45

25

23

11

34

34

22

15

35

14

35

45

15

3-

:2

24

45

25

35

12

14

13

23

14

24

13

55

Fig. 3. Separable, symmetric.

25

35

24

22

14

15

34

45

33

12

12

23

15

25

44

13

24

13

14

23

55

Fig. 4. Inseparable, not symmetric,

Symmetric inseparable double squares

485

(v) n = 6. Euler’s problem of the thirty-six officers cannot be solved because there exists no graeco-latin square of order 6. But there exists inseparable double squares. The following ones are symmetric. The square of Fig. 6 was constructed by permuting the symbols 1 and 2 along two symmetric cycles in the square of Fig. 5. 11

34

25

26

36

45

11

34

25

26

36

45,

34

22

16

15

46

35

34

22

16

15

46

35

33

56

2 -1

56

44

13-23

25

16

33

56

1 -2

26

15

56

44

23-13

P P

25 26

16 15

Fig. 5.

P P

Fig. 6.

(vi) n = 7. Fig. 7 is an example of SIDS(7). 11

34

26

57

24

37

56

34

22

15

17

67

35

46

26

15

33

16

27

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45

57

17

16

44

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25

23 13

24

61

27

36

55

14

37

35

47

25

14

66

12

56

46

45

23

13

12

77

Fig. 7

From this example and Theorem 2.2, we prove the following. Theorem 3.1. For every n 2 5, there exists an inseparable double square of order n. We shall now prove from small squares the existence of SIDS with recursive constructions. First we give examples of SIDS of orders 8 to 16 (Figs. 8-16). Numbers of figures are orders of squares. 11

38

56

68

24

21

34

57

38

22

17

15

36

58

46

47 45

56

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33

12

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78

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25

13

34

46

28

35

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25

77

16

57

47

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88

Fig. 8.

A.-F. Mouyart

486 11

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Fig. 9.

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00

10

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Fig. 10.

11 34

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68 9B

8A

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3A

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5A

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67 68

78 6A

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8B

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Fig. 11.


487

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9C

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AB

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59

6B

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9C

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ED

55

1D

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9C

ID

59

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89

CD

BD

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18

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28

38

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37

6C

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99

18

45

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27

89

7C

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18

AA

35

26

25

CD

9D

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78

3A

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29

46

45

35

BB

17

16

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AD

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39

28

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36

26

17

CC

15

37

27

25

16

15

DD

BC

AC

9B

69

4A

48

.38

Fig. 13. 11

34

24

23

67

57

56

9A

8D

89

AC

DE

BE

BC

34

22

14

13

68

58

59

6E

7B

7D

9C

AB

RE

CD

24

14

33

12

7A

8E

6A

5C

5D

6B

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98

44

9E

5E

7C

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78

69

AD

23

13

12

BD

8B

67

68

7A

9E

55

1B

1D

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39

4E

4C

8A

57

58

8E

BD

1B

66

1C

2C

2A

3E

3D

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79

49

56

59

bA

8B

1D

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77

48

CE

4D

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29

3A

38

9A

6E

5c

5E

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4B

88

1A

19

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37

38

47

ED

7% 5D

7C

28

2A

CE

1A

99

1E 48

36

46

35

89

7D

68

6C

3C

3E

4D

19

1E

AA

45

58

28

27

AC

9C

7E

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39

3D

2E

60

48

45

BB

18

17

26

DE

A%

9D

78

4E

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29

37

36

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18

CC

25

16

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RE

8C

69

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79

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46

28

17

25

DD

15

BC

CD

9B

AD

8A

49

38

47

35

27

26

16

15

EE

Fig. 14.

488

A.-F. Mouyarr 11

34

24

23

67

57

56

9A

8B

89

AD

BD

EF

CF

CE

34

22

14

13

68

58

59

6C

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79

AE

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DE

24

14

33

12

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AC

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13

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44

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8C

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68

78

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55

1D

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57

58

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66

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59

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29

38

38

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88

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47

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28

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29

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17

35

28

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69

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25

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16

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47

31

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35

18

26

EE

15

CE

DE

9D

7A

98

7B

4C

3A

36

48

28

25

16

15

FF

Fig. 15.

BC

DF

EG

7B

7C

AE

BF

AG

CF

DG

DE

5G

6B

79

8F

BE

9G

AD

AC

6A

9F

78

7D

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5A

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2A

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3G

49

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39

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AB

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88

IF

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37

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38

59

69

8C

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FG

CE

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22

14

13

68

14

33

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23

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44

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67

68

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55

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66

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25

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36

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45

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DE

AC

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8B

38

38

47

46

27

25

16

15

GG

Fig. 16.


489

4. Pairwise balanced designs

Definition. Let n be a positive integer and K a set of positive integers. A PBD(K, n ) is a hypergraph with n vertices (points) such that: (1) the cardinality of every edge (block) is in K ; ( 2 ) every pair of points is contained in exactly one block. Proposition 4.1. If there exists a PBD(K, n), and if every k in K is the order of a double square, then there exists a double square of order n.

Proof. With each block B of the P B D ( K , n ) , we associate a double square of order I B 1, whose indices and symbols are the points of B. The double square S of order n is constructed in the following way: Main diagonal: S : = ( i , i ) . Cells (i,j ) with i # j : Let B be the only block containing both i and j . In the double square associated with B, let ( a , b ) be the pair of symbols in the ( i , j ) cell. Let SI = ( a ,b ) . S is clearly a double square: For every i and j , Si is defined and contains two symbols; the main diagonal contains the pairs (i, i ) ; every row S, contains every symbol a exactly twice: if a = i : S : = (i, i ) ; if a # i : a and i occur together in exactly one block. Symbol a occurs twice in the row i of the double square associated with that block, viz., in cells ( i , j ) and (i, k). So a occurs in S1 and in S:. In the same way, every column contains every symbol exactly twice. Every pair of symbols ( a , b ) occurs twice in the double square associated with the only block containing both a and b, viz., in (i,j ) and in ( k , I ) . So (a, b ) occurs in S;l and in S : . 0 S is symmetric if and only if every double square associated with a block is symmetric; S is inseparable iff at least one of those double squares is inseparable. As there exists symmetric double squares of all orders except 2 and 3, we can construct bigger SIDS. Theorem 4.2. If there exists a PBD(K, n ) without any pair or triple, and if the cardinality of one block is the order of a symmetric inseparable double square, then there exiscs a SIDS(n).

A.-F. MOUYUTI

490

Wilson [8] proved the existence of PBD(K, n ) when n is sufficiently large and satisfies some properties. Theorem 4.3. Let K be a set of positive integers. W e define a ( K ) as the greatest common divisor of the integers k - 1, and p ( K ) as the greatest common divisor of the integers k ( k - 1). There exists a PBD(K, n ) for all sufficiently large integers n satisfying :

n = 1 mod a ( K ) and

n ( n - 1)=0 mod P ( K ) .

If K is the set {6,7,8}, (Y ( K ) is 1, as the GCD of 5, 6, 7 and p ( K ) is 2, as the GCD of 30,42, 56. n = 1 mod 1 and n ( n - 1) = 0 mod 2 are true for every n, so there exists a PBD({6,7,8}, n ) for all sufficiently large integers n. As we constructed SIDS of orders 6, 7 and 8, we can say the following. Theorem 4.4. There exists symmetric inseparable double squares of every s u f i ciently large order.

5. Transversal designs Definition. A transversal design TD(r,s) is a set of rs points classified in s disjoint groups of r points, and in blocks of s points, one from each group, such that every pair of points from different groups belongs to exactly one block.

The blocks and groups of a transversal design can be regarded as the blocks of a pairwise balanced design. Proposition 5.1. If there exists a TD(r, s), then there exists a PBD({r, s}, rs).

Removing some points from a transversal design, we can construct a pairwise balanced design whose blocks are the truncated blocks and groups. Proposition 5.2. lf there exists a TD(r, s), then, for every r , , r z , . . . , r, at most equal to r, there exists a PBD({r,, r?,. . . ,r,, 2 , 3 , . . . ,s}, r , + rr + . . + rs).

-

Transversal designs and orthogonal latin squares are linked. Proposition 5.3. There exists a TD(r, s ) if and only if there exists s - 2 mutually orthogonal latin squares of order r.


19 I

Let us recall some known results about the existence of orthogonal latin squares. Proposition 5.4. For every n 2 15, there exists three mutually orthogonal latin squares of order n. Proposition 5.5. If n is a prime power, there exists n - 1 mutually orthogonal latin squares of order n.

Now we can use these results to prove the existence of SIDS. Proposition 5.6. There exists a SIDS(n) for every n

5 66.

Every integer n 2 66 can be written in the form n = 4x + y where y = 6,7,8 or 9 and x 2 15. From Propositions 5.3 and 5.4 we know that there exists a TD(x,5). If we truncate one group and leave only y points in it, we construct a PBD({4,5,x,y},4x + y ) with a block of size y . As there exists a SIDS(y), we know from Theorem 4.2 that there exists a SIDS(4x + y ) . Proposition 5.7. There exists SIDS of euery order between 26 and 65, except maybe 31.

Proof. From Propositions 5.3 and 5.5 transversal designs T D ( n ,n + 1) exist when n is a prime power. So there exists a TD(9, lo), a TD(8,9),a T D ( 7 , S )and a TD(5,6). Keeping seven of the ten groups of the TD(9, lo), we obtain a T D ( 9 , 7 )which is a PBD({7,9},63),and we construct a SIDS(63). Leaving 0, 1, 4, 5, 6, 7 or 8 points to one group of the T D ( 9 , 7 ) ,we obtain PBD with 54,55, 58, 59, 60,61 or 62 points, six blocks of 9 points and blocks of 4 to 8 points. So there exists SIDS of orders 54, 55, 58, 59, 60, 61 and 62. In the same way, with five groups of the TD(9,IO)and a truncated group, we construct SIDS of orders 45, 46, 49, 50, 51, 52, 53. With the TD(8,9), we obtain the following orders: 64, 65 : 8 groups+O or 1 point, 56, 57 : 7 groups+O or 1 point, 48 : 6 groups, 47 : 5 groups + 7 points, 32, 33, 38, 39 : 4 groups+O, 1, 6, 7 points. With the TD(7,S): 35 : 5 groups, 28, 29, 34 : 4 groups+O, 1, 6 points.

A.-F. Mouyart

492

With the TD(5,6):. 30 : 6 groups, 26 : 5 groups+ 1 point, 27 : 3 whole groups+3 groups of 4 points, but the three removed points must not belong to the same block, or the truncated block would have only three points. 0

6. Marshall Hall's construction

We can imitate the construction of orthogonal latin squares described by Hall in [4]. Theorem 6.1. If there exists a SIDS(n), then there exists a SIDS(3n + 1).

Proof. We construct a 4 X (3n

indices

symbols

{'

f -'

+ 1)2 orthogonal array:

D

A

B

C

X

L

D

B

A

X

C

K

D

C

X

A

B

D

X

C

B

A

D is ( O , l , . . . , 2 n ) . For i = 0 , 1 , .. . , 2 n , define vectors of length n of residues modulo 2 n + 1 : a, = (i, i,. . . ,i), b , = ( i + l , i + 2 ,..., i + n ) , ,..., i - n ) . Now construct the vectors of length n ( 2 n c, = ( i - 1 , i - 2

. ., a 2 n ) r B = (bo, b i , . . . , b n ) , c = (c,,,CI,. . . , C2").

A

+ 1):

= (ail, ai,.

Vectors X , K , L are formed with symbols x I , .. . ,xn: X = ( x l , .. . ,x., . . . ,x,, . . . ,x,) 2 n + 1 times, K = ( x l , .. . , x n , .. . , x l , .. . , x n ) n times, L = ( x i , .. . ,x,, . . . ,x , , . . . ,x , ) each x, n times.


403

We construct a symmetric double square of size 3n + 1 from the orthogonal array, and put a S I D S ( n )in the empty subsquare. So we obtain a SIDS(3n + 1).

0 From SIDS of orders 6 , 7, 8 and 10, we can construct new SIDS. Proposition 6.2. There exists SIDS of orders 19, 22, 25 and 31.

7. The singular direct product

The singular direct product, introduced by Sade [7] to give counter-examples to Euler’s conjecture, was used to construct self-orthogonal latin squares, particularly by Lindner [6] and Crampin and Hilton [2].It will be useful now to find new symmetric inseparable double squares. Theorem 7.1. If there exists a SIDS(p + q ) with p f 2,3,6 and with, in the top left corner, a subsquare which is a symmetric double square of order q (possibly zero), then, for every n # 2,3,6, there exists a SIDS(np + 4).

Proof. Let A and B be self-orthogonal latin squares of orders n and p, whose symbols are respectively xI,.. ,x,, and y l , . . . ,y,. Let C be the SIDS(p + q ) and y l , . . . , y p ,zI,. . . , z, its symbols. Let D be the symmetric double square of order q which is a subsquare of C, and z,, . . . , z, its symbols. Construct S , SIDS(np + q ) whose symbols are zl, ..., z, and ( x # , y , ) ,i = 1 , . . . , n, j = 1 , . , . , p , in the following way: Dilate the latin square A and, in each cell A ! = x ( i # j ) , put the latin square B with every symbol y replaced by ( x , y ) . Do the same construction with the transposes of the two squares A and B. Superimpose the two squares (the diagonal is empty). Add a ‘big top left corner’ A! containing D, a ‘big first row’ A. of n q X p rectangles, and a ‘big first column’ A” of n p X q rectangles. In the ‘big cells’ A:, Ah, A: and A f ,put the square C (in four pieces) with every symbol y replaced by (x, y ) , where x is the symbol in the (i, i ) cell o f the latin square A. Using the squares of Fig. 17, Heinrich constructed the SIDS(21) of Fig. 18. Theorem 7.2. There exists a SIDS(21).

A.-F. Mouyart

494

x

1

= 1

x2 = 2 x

3

= 3

x4 = 4

5

7

9

6

8

Y1 = 5

9

6

8

5

7

Y2 = 6

8

5

7

9

6

Yj = 7

7

9

6

8

5

Yq = 8

6

8

5

7

9

Y5 = 9

67

58

59

69

78

67

55

Z9

Z8

79

68

58

29

66

89

27

57

59

28

89

77

56

26

69

79

27

56

88

25

78

68

57

26

25

99

2

1

= z

Fig. 17.

8. Conclusion

As a result of all the above we can now say the following.

Theorem 8.1. There exist symmetric inseparable double squares of every order L 6, except maybe 17, 18, 20, 23 and 24. And perhaps the following.

Conjecture. For every n 2 6, there exists a SIDS(n).


-

49.5

-

46 47

zz

1

45 I 4 5 I46 I 4 7 48 49 49 48

16

17 15 18 15

-19 16 19

-.

1

18 z 1 8 ! 15

4 26

1

? 25

29

~

28

27

17 18

26 27

~

-

-m36

35

39

38

37

26 38

28 37

25 36

27 35

29 39

, 25

29

-.

25 28

39

38

29

26

29

27

27

-.

25 29

T 9 M z - b % 5 k

-_

26 29 27 28

36

35

37 35 38

39

38

39

38

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27

26

16

18 28

15 27

15 26

17 25

19 29

47 49

46 48

T-p-ptqY

-.

35 36 39

38 39 -t* Z 37 39 37

37 38

36 38

35 37

Z

Z

36

35

39 39

16 28

18 27

46 47

25 15

27 19

29 18

26 17

28 16

45 45

Z

Z

49

48

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1

._

-.

-_

37 1 3 6 4---i-

1 35 36

38 38

-

36

Z

17

26 -35- ._-

19 25

-.

29 -

45 48 _45 49 __ 46 49 47 48

Fig. 18.

Acknowledgement

The author wishes to thank FranGois Sterboul and Katherine Heinrich for their help.

A.-F. Mouyart

496

References [ I ] B.K. Brayton, D. Coppersmith and A.J. Hoffman. Self-orthogonal latin squares of all order3 n # 2 . 3 , 6 . Bull. Amer. Math. SOC.80 (1974) 116-118. [2] D.J. Crampin and A.J.W. Hilton. Remarks on Sade’s disproof of the Euler conjecture with an application to latin squares orthogonal to their transpose, J. Comb. Theory, Ser. A 18 (1975) 47-59.

[3] J. Dines and A.D. Keedwell. Latin Squares and their Applications (Academic Press, New York. 1974). 141 M. Hall, Combinatorial Theory (Blaisdeli, Waltham, Massachusetts, 1967). [5] K. Heinrich and W.D. Wallis, Almost graeco-latin squares, Ars Combin. 10 (1980) 55-63. 161 C.C. Lindner. The generalized singular direct product for quasigroups. Canad. Math. Bull. 14 (1971) 61-63. [7] A . Sade, Produit direct-singulier de quasigroupec orthogonaux et anti-abeliens, Ann. Soc. Sci. Bruxelles Sir. I 74 (1960) 91-99. [81 R.M. Wilson, An existence theory for pairwise balanced designs, 111: Proof of the existence conjectures, J. Comb. Theory, Ser. A 18 (1975) 71-79.


THE NUMBER OF EDGES IN A k-HELLY HYPERGRAPH Henry Martyn MULDER Subfaculteit Wiskunde, Vrije Uniuersiteit, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands An upper bound is derived for the number of edges in a k-Helly hypergraph. The k-Helly hypergraphs attaining this upper bound are characterized.

1. Introduction

We first introduce some terminology. Terms and notations that are not introduced here can be found in Berge [l].In this note 8 is a simple hypergraph on the finite set X of size n (i.e., 8 is a family of non-empty subsets of the n-set X). A hypergraph %‘ is k-linked if any k edges, not necessarily distinct, have a non-empty intersection. Clearly every hypergraph is both 0-linked and 1-linked. A hypergraph 8 is a k-Helly hypergraph if any k-linked subhypergraph 8’C 8 has a non-empty intersection. Such hypergraphs have been characterized independently by Berge and Duchet [2] and Sierksma [6]. A 2-Helly hypergraph is just a Helly hypergraph in the usual sense. A hypergraph 8 is a I-Helly hypergraph (and simultaneously a 0-Helly hypergraph) if and only if 8 # 0. It is easily verified that, for k 2 2, any subset of X of size less than k can be added to a k-Helly hypergraph without destroying the ‘k-Helly property’. A k-uniform hypergraph has all its edges of size k. By Pk(X) we denote the family of all k-subsets of X . For x in X , the family of subsets of X containing x is denoted by {x} r . The problem of determining an upper bound for the number of edges in a k-Helly hypergraph discussed here is related to problems treated by Frankl in [4].Frankl studies hypergraphs that do not contain an r-simplex, where an r-simplex is a family of r + 1 edges E l , .. . , E,,,, such that

n

,+I

nE,=O ,=I

and

nE,#0

f o r j = I , ..., r + l .

I#/

His main result is the following. Let 8 be a p-uniform hypergraph on the n-set X containing no r-simplex, with p > 31, n 2 n,,(p, r ) . Then either every edge of E contains a fixed vertex x in X , or we have 497

H . M . Mulder

498

It is easily verified that a hypergraph 8 is a k-Helly hypergraph if and only if 8 does not contain an r-simplex for any r 5 k. Hence, if p > 3k and n 3 no(p, k), then Frankl's theorem gives an upper bound for the number of edges in a p-uniform k-Helly hypergraph on an n-set. The object of this note is to prove that, if 8 is a p-uniform k-Helly hypergraph with p > k, then either every edge of 8 contains a fixed vertex x in X , or we have 1 81< (;:l). Some consequences are also given. The results given below are contained in [5].

2. k- Helly hypergraphs Let 8 be a hypergraph. A pointer of an edge B of 8 is a subset of B of size I B I - 1 that is not contained in any other edge of 8. The crux of the proofs of the theorems below is given in the next lemma.

Lemma. Let p be a n integer with p 2, and let 8 be a p-uniform hypergraph on the n-set X . If each edge of 8 has a pointer, then

Furthermore, equality holds if and only if there is a vertex x in X such that

(x).

8 = { x } ~n pP

Proof. For each edge B of 8 we fix a preferred pointer B' in 9p-l(X).Let $23 be the family of preferred pointers. We define a bipartite graph G as follows: 8 U (!iPp-l(X)\ 9)is the vertex-set of G and there is an edge between B in 8 and A in !iPp-,(X)\9 whenever A is contained in B. Then each vertex B in 8 has degree p - 1 in G, and each vertex A in Pp-,(X)\93 has degree at most n - p + 1 in G. By counting the edges in G between 8 and !iPp-l(X)\% twice, we get

~ p - ~ ) ~ 8 ~ ~ ~ ~ - p + ~ ~ l ~ p ~ l ~ ~ ~

from which the upper bound follows.

The number of edges in a k-Helly hypergraph

499

If equality holds, then each vertex A in PP-,(X)\9 must have degree exactly n - p + 1 in G, and so A U { y } is an edge in 8 for each y in X \ A. Furthermore, any edge B in 8 has exactly one pointer: its preferred pointer. To complete the proof we use a 'shifting-technique'. Let B be an edge in 8, and let B' = B \ { x } be its pointer. Then we have to prove that any k-subset of X containing x is in 8. Let A = B \ { z } be a ( p -1)-subset of B containing x. Then A is not a pointer and has degree n - p + 1 in G. Let y be an arbitrary vertex in X \ B. Then C = A U { y ] is in 8. If C \ { x } is not the pointer of C, then ( C \ { x } ) U { z } would be an edge in 8. Since

B' = B n ((C\{x}) U { z } ) , this contradicts the fact that B' is the pointer of B. Hence C \ { x } is the pointer of C. So we have proved that any p-subset C of X containing x with ( B A CI = 2 is an edge of 8. Applying the preceding argument on each edge containing x, we get the required result. Theorem 1. Let 8 be a p-uniform k-Helly hypergraph on the n-set X . If p > k, then

Furthermore, equality holds if and only if there i s a vertex x in X such that 8

={XI? nPp(x).

n

Proof. If k s 1, then 8# 0, and the assertion follows immediately. If k 3 2, then it follows from the fact that p > k that each edge in 8 must have a pointer. Hence the result follows from the lemma. 0

The problem of establishing an upper bound for the number of edges of a k-uniform k-Helly hypergraph is a special case of TurBn's problem (see [7]), which is still open. The case k = 2 in the following theorem is due to Milner (see [3]). Theorem 2. Let 8 be a k-Helly hypergraph on the n-set X . Then

1%

H.M. Mulder

500

Furthermore, equality holds if and only if there is a vertex x in X such that

8 ={x}'

u

k-l

u

P,(X).

i=l

Proof. Write gi = 8 n S i ( X ) . Then

Furthermore, by Theorem 1, we have

on X . Each edge in must have a Consider the hypergraph 8,U pointer that is not an edge in 8,. Let d be the family of non-edge pointers of edges in Then we have l g k

u8,+11=)8,1-c)8,+i)~1$k)+)~l

n-1

= l ~ k u 4 . ( k " ) = ( k - 1 ) + (

n-1

).

Combining the three inequalities we get the upper bound for 18 1. To get a maximum k-Helly hypergraph we must have equality in all of the above inequalities (if possible). So gi = 9, ( X ) for i

=

1, . . . ,k - 1.

Moreover, there must exist a vertex xi in X such that

8,= {xi}' n Pi ( X ) for i

=k

+ 2,. . . , n.

Finally, we have

If k 2 n - 1, then the assertion is easily verified, and is left to the reader. If k =sn - 3, then it follows that x = xk+?= * = x,, for some vertex x in X . Furthermore, we can easily deduce that every edge in 8, U 8 k + I must contain x, and so we have " , = { X } ~ ~ P , ( Xf o) r i = k ,..., n.

Let k = n -2, and consider the hypergraph 8, U i Y k + l . Then it follows that any k-subset of X is either an edge in g k or a pointer of an edge in Furthermore, any edge in has exactly one pointer. Clearly not all k-subsets and let €3 \ ( x ) be #0. Let B be an edge in of X are edges in %,and so

The number of edges in a k-Helly hypergraph

50 1

its pointer. Then using the k-Helly property of 8kU and the ‘shiftingtechnique’ from the proof of the lemma, we can deduce that all edges in 8, U &+, contain x. This implies that 8 ={x}+

u

*-I

u

Si(X),

i=I

and the proof is complete.

0

References [ 11 C. Berge, Graphs and Hypergraphs (North-Holland, Amsterdam, 1973). (21 C. Berge and P. Duchet, A generalization of Gilmore’s theorem, in: M. Fiedler, ed., Recent Advances in Graph Theory, Proc. Symp. Prague, June 1974 (Academia, Praha, 1975). [3] P. Erdiis, Topics in combinatorial analysis, in: R.C. Mullin et al., eds., Proc. 2nd Louisiana Conf. Combinatorics, Graph Theory and Computing (Baton Rouge, 1971). [4] P. Frankl, On a problem of Chvital and Erdos on hypergraphs containing no generalized simplex, J. Comb. Theory, Ser. A 30 (1981) 169-182. [S] H.M. Mulder, The interval function of a graph, Ph.D. thesis, Vrije Universiteit (Mathematisch Centrum, Amsterdam, 1980). [6] G. Sierksma, Axiomatic convexity theory and the convex product space, Ph.D. Thesis, Rijksuniversiteit Groningen, Groningen, 1976. [7] P. Turin, An extremal problem in graph theory, Mat. Fiz. Lapok 48 (1941) 436-452 (in Hungarian).



PROJECTIVE GEOMETRIES AND THEIR TRUNCATIONS U.S.R. MURTY University of Waterloo, Canada and Universidade de Sao Paulo, B r a d Dans cet article on developpe une definition synthitique des geometries projectives en termes d’espaces IinCaires ( = matroides simples, gkomktries combinatoires). On parle alors du t h e o r h e bien connu de Veblen-Young dans ce contexte. Nous relions ensuite notre definition avec la definition habituelle des geometries projectives comme structure d’incidence entre points et droites.

1. Introduction In this paper we develop a synthetic definition of a projective geometry in terms of linear spaces, viz., simple matroids, combinatorial geometries. We then discuss the well-known Veblen-Young theorem in this context, thereby relating our definition with the customary definition of a projective geometry as a certain type of point-line incidence structure. We restrict ourselves to the finite case; thus all sets considered are finite. To facilitate the reading of the paper we shall recall in Section 2 a few pertinent definitions and results from matroid theory. Those not given here can be found in Welsh [ 2 ] . 2. Matroids and linear spaces

Let E be a set and 9 be a family of sub-sets of E called independent sets. Then M = (E,9) is called a matroid on E if 11. 0E4;. 12. A E 9 , B C A j B €9.

13. For any X E any two maximal independent sub-sets of X have the same cardinality.

A prototypical example of a matroid is obtained by considering a finite set E of vectors in a vector space together with the family of linearly independent sub-sets of E. Let M = (E, 9) be a matroid and let X C E. A maximal independent sub-set of X is called a basis of X. Axiom I3 says that all bases of X have the same 503

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U.S.R. Murty

cardinality; this common cardinality of the bases of X is called the rank of X and is denoted by r(X). (The bases of E are called the bases of M and r ( E ) is called the rank of M . ) It can be shown that there is a unique maximal superset cl(X) of X such that r(cl(X)),= r(X); this set cl(X) is called the closure of X, and the function cl : 2" 3 2E is called the closure operator associated with M. The closed sets, that is, sets X for which cl(X) = X7are called the flats of M. A k-flat is a flat of rank k. We often use the terms point, line, plane, hyperplane instead of 1-flat, 2-flat7 3-flat, and ( r ( E ) - 1)-flat, respectively. If cl(0) = 0 and cl((e}) = { e } for all e E E, we shall refer to M as a linear space. The following theorem gives a characterization of functions from the power set of E into itself which are closure operators associated with matroids on E.

Theorem 1. Let E be a set and el be a function from into itself, Then cl is the closure operator of a matroid on E if and only if C1. X C cl(X) for all X C E. c 2 . Y c x j cl( Y) c cl(X). C3. cl(cl(X)) = cl(X) for all X C E, C4. For every closed set F (i.e. F = cl(F)) and any two elements p, q not in F, q E cl(F U { p } ) 3 p E c V U ( 4 ) ) . 0 Usually, in place of C4, a seemingly stronger condition is used, but it can be seen that C4 suffices. Given the closure operator cl of a matroid M on E, it is easy to recover the independent sets of M : subset I of E is independent in M if and only if efZ cl(I\{e}) for all e E I. The following theorem characterizes families of subsets of E which are families of flats of matroids on E.

Theorem 2. Let 9 be a family of subsets of a set E. Then 9is the family of flats of a matroid M on E if and only if F1. E E 9. F2. 9 is closed under intersection. (Thus is a lattice under inclusion with F, A F~= F, n F~.) F3. The lattice 9satisfies the Jordan-Dedekind chain condition. (Call F E 9a k-flat if F has rank k in this lattice.) F4. If S C E, and I S I = k , then either S is contained in a ( k - l)-flat or there is a unique k-flat which contains S. Given the family 9 of flats of a matroid M on E, it is easy to recover the closure operator cl of M : For X c E, cl(X) is the intersection of all flats that contain X. In a linear space defined on a set E, the elements of E may be identified with

Projective geometries and their truncaiions

505

points of M . Linear spaces can be viewed, and it is convenient to do so for the purposes of this paper, as abstractions of the incidence relations holding among ordinary geometrical objects: any two points lie on a unique line, any three non-collinear points lie on a unique plane, etc. Let M = (E,9) be a matroid and let S E. Then (S,9s),where 9:is the set of independent subsets of M contained in S, is another matroid. It is called the restriction of M to S, or the submatruid of M induced by S, and is denoted by MIS. If S is a flat of M, then the flats of M / S are simply those flats of M which are contained in S. Now let r be an integer such that 0 6 r 6 rM ( E ) .(Here r M ( E )denotes the rank of E in matroid M.) Define 9'')to be {IE 9 I1 S r } . Then N = (E, 9(r))is a matroid on E and is called the truncation of M to rank r. It is convenient to visualize this operation of truncation in terms of the lattice of flats. If 9 is the lattice of flats of M and %(') the lattice of flats of N, 9'' is' obtained from 9 by simply deleting all those flats which have ranks r, r + 1,. . . ,n - 1. It is easy to see that rN ( X ) = min{rM(X), r } for any X C E. Note that two different matroids of rank n on E may have the same truncations to rank r for every r < n. Given matroids N and M on E of ranks r and n, r S n, we say that M is an erection of N to rank n if N is a truncation of M . Truncations are easy, but erections are difficult; in fact, a matroid need not have erections to higher ranks. The operation of erection, which is due to Crapo [2], plays a central r81e in this paper.

11

3. Projective geometries

We define a linear space M = ( E , 9 ) of rank n ( n geometry if

3 ) to be a projective

P1. If H is any hyperplane of M and L is any line of M,then H P2. For any line L of M , IL 1 3 3 .

fl

L # 0.

Examples of projective geometries may be obtained as follows. Let K be a finite field and let V be a vector space of n-dimensions over K (n 3 3). Each subspace of V can be regarded as a subset of the set E of 1-dimensional subspaces of V. The lattice of subspaces of V viewed this way is the lattice of flats of a linear space on E of rank n. Indeed, it is a projective geometry. It is called the projective geometry of rank n ouer the field K and is denoted by PG(n, K ) . A classical theorem states that for n 3 4, every projective geometry of rank n is isomorphic to some PG(n, K ) . Projective planes (geometries of rank 3 ) exist which are not isomorphic to any PG(3, K ) . See Hall [4]for such examples. If M is a projective geometry of rank n ( n 3 3 ) and F is any flat of rank r

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U.S.R. Murty

( r 3 3), then F is itself a projective geometry, that is, the submatroid of M induced by F is a projective geometry. This is the content of the following theorem.

Theorem 3. Let M = ( E , 4 ; ) be a projective geometry of rank n ( n 3 3 ) and let F be an r-pat of M ( r 3 3). If G is a hyperplane of F and L is a line of F, then G nL Z 0 . Proof. We may assume that L is not contained in G. Let { x I , x 2 , .. . ,x,-,} be a basis of G, and let x, E L \ G. Then {xl,x,, . . . ,x,} is a basis of F. This basis of F can be extended to a basis, say, B = { y , , y 2 , . . . ,y "-,,xI,x 2 , . . . ,x,} of E. Consider the hyperplane H of M which is the closure of B\{x,}. It is easy to see that H n F = G . By P1, H n L Z O . Hence G r I L Z O . 0 The above theorem is a special case of the following more general result.

Theorem 4. Let FI and F2 by any two pats in a projective geometry of rank n. Then r(FIUF2)+r(FrnF2)= r(FI)+r(Fz).

Proof. We may assume that neither flat is a subflat of the other and, also, that r ( F I )and r(F2)are both at least 2. Otherwise the result is trivial. First let us consider the case in which FI is a hyperplane. Since F2 is not contained in FI,r(F, f l FJS r(F2)- 1. If r(FIf l F 2 ) < r ( F 2 ) - 1 then there will be a line of F2that misses F , f l F2.(A general fact: If F is a flat of rank 6 r - 2 in a linear space of rank r ( r 2 2), then there is a line of the linear space that misses F.) Such a line would miss Fl.A contradiction. Hence r ( F 1fl F2)= r ( F J - 1 and the result follows. Proof in general is by induction on r(Fl U F2).If r(Fl U F2)d 3, the statement is easily verified. Assume that the statement is valid if r(Fl U F2)= m - 1 (rn 3 4) and consider the case in which r(F, U F2)= m. Let H be a hyperplane of cl(FI U F,) which contains F , but not Fz. Let FI = FIand F ; = H f l F?. We have

r(FI U F:) = r ( H ) = r(F1U F 2 ) - 1. Therefore, by induction hypothesis, r(FiUF;)+r(FinFI)=r ( F i ) + r ( F ; ) ,

or

r(H)+r(FlflF2)=r(FI)+r(HnF2).

Projective geometries and their truncations

507

By the case previously treated,

r(H

f l

F2)= r(Fr)- 1.

Putting all the information together, we obtain the desired result. 0 Let M be a linear space on E. If x and y are two points of M , we shall denote by Fthe line containing x and y , that is, Xy = cl({x, y } ) . A subset S of E is said to be line-closed if x, y E S @ GC S. Clearly, every flat in a linear space is line-closed. In general, a line-closed subset need not be a flat. The situation is different in the case of projective geometries. Theorem 5. A subset S of points of a projective geometry M on E is a flat if and only if S is line-closed. Proof. If S is a flat then, as already observed, S is line-closed. We shall prove the converse by induction on r ( S ) . The verification of the statement when r ( S ) S 2 is trivial. Let S be a line-closed subset of rank 2 3. We may suppose that S C cl(S). Choose a point p in cl(S)\S. Let {x1,x2,.. . ,x,} be a basis of S and let H = cI({x2,x3,. . . ,x,}). Since H and S are line-closed, H fl S is line-closed (more generally, the intersection of any two fine-close subsets of a linear space is line-closed). By induction, H f l S must be a flat. Since H fl S contains a basis of H, H fl S must be the same as H . Now consider the line T p . This line must meet H in a point, say q, different from x,, and p . Since H = H f l S, q E S. And, by choice, x, E S. Now, since S is line-closed by hypothesis, p, which is on r q , must be in S. A contradiction. 0

4. Truncations of projective geometries

All linear spaces on E of rank three or more have identical truncations to rank 2. For this reason truncations to rank 2 are uninteresting. In this section we consider the problem of recognizing a projective geometry from its truncation to rank 3. As an intermediate step, we first look at the truncations of projective geometries to rank 4. Let M be a projective geometry of rank n ( n 3 4), and let N be the truncation of M to rank 4. Then, by Theorem 3, all planes of N are projective planes. The following theorem proves the converse. Theorem 6. Let N be a linear space on E of rank 4 in which each plane is a projective plane. Then there is a unique projective geometry M of which N is the truncation to rank 4. Proof. We shall prove that the lattice of line-closed subsets of E is the lattice of

U.S.R. Muriy

SO8

Rats of a projective geometry M. That N is a truncation of M is easy to see. Uniqueness of M follows from Theorem 5. For any S C E,define cl(S) to be the intersection of all line-closed subsets that contain S. (Line-closed subsets are then precisely those sets S for which S = cl(S).) It is easy to see that the function cl :2F + 2E satisfies conditions Cl-C3. Let F be a line-closed subset and let p be a point not in F. Denote by F * p the set of all points on lines of the form P y where y E F. We wish to prove that F * p is line-closed. To do this, let y , , y 2 be two points in F and let x, E@,, i = 1,2. (We shall prove that x , x 2 F * p . ) Consider the plane 7~ containing the two lines p y , and Pyz. If x3E m ,then x3 E T.By our hypothesis 7~ is a projective plane. Hence pxl must meet the line yly2 in a point, say y3. Since ylysc F, y qE F. I t follows that x 3 EF * p . We conclude that F * p is line-closed. Note that this means that cl(F U { p } ) = F * p . Now let F be a line-closed subset and p and q be two points not in F. Clearly qEF*p

+p EF*q.

From this it follows that the function cl satisfies the condition C3. We may now conclude that the lattice of all line-closed subsets of N is the lattice of flats of a linear space M of which N is a truncation. If H is any hyperplane of M , then E = H * p for any p !Z H. Using arguments similar to the ones already used earlier in the proof, it is easy to see that if L is any line of M, then H fl L # 0. It follows that M is a projective geometry. 0 Now let N be a linear space on E of rank 3. For any three non-collinear points a, b, c, define the block generated by { a ,b, c } as the smallest line-closed subset of E that contains {a, b, c}. If N is the truncation to rank 3 of a projective geometry M then the blocks of N coincide with the planes of M and thus each block of N is a projective plane (that is, induces a projective plane). The converse is also true.

Theorem 7 (Veblen and Young [6]). Let N be a linear space on E of rank 3 in which each block is a projective plane. Then N is the truncation to rank 3 of a projective geometry M on E. Proof. If N is itself a projective plane, then we take M to be the same as N . If not let 9 denote the family of flats of N and 93 be the family of blocks of N . It is easy to see that 9 = 9 U 93 is the family of flats of a linear space M ' of rank 4 in which each plane is a projective plane. Further, N is the truncation to rank 3 of M ' . We now invoke Theorem 6 to complete the proof. 0

Projective geometries and their truncations

so9

5. incidence axioms for geometries An incidence structure is an ordered triple (P, iL, 4 ) where

(i) P and L are non-empty disjoint sets. The elements of P are called points and the elements of L are called lines. (ii) 4, called the incidence relation, is a subset of P x L. If ( p , I) E 4, we say that p and 1 are incident with each other. (We shall use geometrical terms such as intersection of lines and non-collinear points without giving their definitions; they have obvious meanings.) (iii) Not all the points are incident with the same line. It in customary to define a projective geometry as an incidence structure (P, iL, 4 ) which satisfies the following axioms: (Al) Any two points are incident with exactly one line. (A2) If x, y , 2, w are four distinct points, no three collinear, and if Xy intersects ZW, then XZ intersects yW. (A3) Each line is incident with at least three points. This definition is bound to appear a bit curious at first sight. Why is a geometry, which, presumably, has flats of higher ranks, defined merely in terms of points and lines? The reason is that all the higher rank flats are determined and incidence among them implied by what happens between points and lines. We proceed to explain this, and, incidentally, relate the above definition with the one given in Section 2. The first thing to observe is that incidence structures which satisfy axioms Al-A3 are co-extensive with linear spaces of rank 3 in which each block is a projective plane. This is in the sense that if N is such a linear space, then the point-line incidence structure of N satisfies Al-A3, and vice versa. Now N is a projective geometry in the sense that there is a uniquely determined projective geometry M of which N is a truncation. We conclude the paper with a few comments concerning affine geometries. Let M = ( E , 9)be a projective geometry of rank n and let H be a hyperplane of M . Then the restriction of M to E \ H , which can be shown to have rank n, is called an afine geometry. Examples of affine geometries may be obtained as follows: Let V be a vector space of ( n - 1)-dimensions over a finite field K . Then the lattice of affine subspaces of V (0,vector subspaces of V and their translates) is the lattice of flats of an affine geometry of rank n. It is called the afine geometry of rank n ouer K and is denoted by AG(n, K ) . One may define an affine plane in terms of incidence structures quite easily. (See, for example, Welsh [7].) However, in turns out to be quite difficult to define an affine geometry by means of incidence axioms. The difficulty stems from the fact that the affine analogue of Theorem 7 is not valid. Zassenhaus and, independently, Hall have constructed a linear space of rank 3 in which each

510

U.S.R. Murty

block is the affine plane AG(3,3) but which is not the truncation to rank 3 of any affine geometry. Now many such examples are known. See [3] for a description of these examples. In a beautiful paper [l],Buekenhout has shown that if N is a linear space of rank 4 in which each block is an affine plane of order four or more, then N is the truncation of an affine geometry. For a discussion of incidence axioms for affine geometries, see Hall [ 5 ] .

Acknowledgement This paper was written while I was spending a part of my sabbatical leave in the Universidade de SBo Paulo, Brasil. I wish to thank Professor Imre Simon and the Departamento de Matemdtica Aplicada for their hospitality, and Jo20 Baptista Esteves de Oliveira for typing this paper. Some of the ideas came to me during discussions with Denis Higgs about twelve years ago. I wish to record my thanks to him.

References [I] F. Buekenhout, Une caracterization des espaces affins basde sur la notion de droite, Math. 2. 111 (1969) 367-371. (21 H. Crapo, Erecting geometries, in: Proc. the 2nd Chapel Hill Conf. on Combinatorial Mathematics and Its Applications (Chapel Hill, University of North Carolina, 1970) pp. 74-99. [3] M. Deza and N. Singhi, Some properties of perfect matroid designs, in: Proc. Symp. Combinatorial Mathematics and Optimal Design, Ann. Discrete Math. 6 (1980) 57-76. [4] M. Hall, Combinatorial Theory (Blaisdell, Waltham, 1967). [5] M. Hall, Incidence axioms for affine geometry, J. Algebra 21 (1972) 535-547. 16) 0. Veblen and J.W. Young, Projective Geometry, Vol. 1 (Ginn-Blaisdell, Waltham. 1910). [7] D.J.A. Welsh, Matroid Theory (Academic Press, 1976).


TREES AND CUTS Manfred W. PADBERG* and Laurence A . WOLSEY C.O.R.E., Universite' Catholique de Louuain, Belgium We show that the problem of deciding whether or not a point # E W!" can be written as a convex combination of the incidence vectors of the trees of a finite undirected graph G can be solved by computing a minimum cut, i.e., by carrying out at most n - 2 maximum flow calculations where n is the number of nodes of G and IEl the number of edges of G.

I . Introduction

Let G = ( V , E ) be a finite undirected graph with nodes V and edges E. A tree T is a connected acyclic subgraph in G ; a tree T is a spanning free if T has I V I nodes (1 V f means cardinality of V ) .Given edge-numbers d, for all edges e E E, the maximum weighted tree problem is to find a tree T for which the sum C C E T d P is maximum. This problem can be formulated as a linear program as follows, see e.g. [3]:

subject to

x, a 0

V e E E,

where E ( S ) denotes all edges of E which have both ends in the node set S. When we speak of a spanning tree, we tacitly assume that the inequality C I E ES~ I 0 holds for all e E E. We denote

I

E ( S ) = {e E E e has both ends in S }

VS C V.

Proposition 1. If there exist S C C such that ( ( S ) > I S I - 1 holds, then there exist S * , S * contained in a connected component of G,such that ( ( S * ) > 1 S * 1 - 1 holds.

Proof. Since G is finite we can write S as follows:

s = s u s, u I

' ' '

us k ,

(1.3)

where k 3 1, S, n S, = 0 for i# j , and S, is the intersection of S with a connected component of G for i = 1,. .., k. Then there exists at least one i such that [ ( S , ) > 1 S, 1 - 1 holds for otherwise we have

ISIand thus k < 1, a contradiction.

Proposition 2. (i) If there exist S C V such that t ( S ) > 1 S 1 - 1 holds and u E V satisfies b, s 1, then there exist S * C V - u such that ( ( S * ) > I S * I - 1 holds. (ii) Suppose b, - reS 1 and b, - (c s 1 holds for some edge e = [ u, u ] . If there exist S C V such that ( ( S ) > 1 S I - 1 holds, then there exist S* C V such that [ ( S * ) > / S * ) - l and either u , u € S * or u , u k Z S * hold. Proof. (i) Suppose u E S, since otherwise there is nothing to prove. But then S * = S - { u } works. (ii) Suppose to the contrary that u E S and uf? S for S C V satisfying t ( S ) > ( SI - 1. But then S * = S - { u } works. A similar argument applies if u E S and u @ S . 0

514

M.W. Padberg, L.A. Wolsey

While Propositions 1 and 2 permit in general the elimination of certain nodes of the graph G, the repeated application of Propositions 1 and 2 does in general not solve problem (Pl); examples to this effect are easily constructed. To find a violated constrainfof (LP) or to prove that all inequalities of (LP) are satisfied by the point 5 with components le for e E E, it is necessary and sufficient to compute

IVS c V, sz 01.

min{(S1 - [ ( s )

(1 5 )

There exists a violated constraint of (LP) if and only if the minimum value of (1.5) is less than one. The function g ( S ) = 1 S 1 - ( ( S ) is a submodular function [3] and thus to solve (Pl) we have to minimize this special submodular function subject to the (mild) condition that S be nonempty. For S C V we denote

I

{ S : V - S } = { e E E e has exactly one end in S } , ( ( S : v-S)=

2

e a s : v-S)

(1.6)

6..

{ S : V - S } is a cut-set in G and ( ( S : V - S ) its associated capacity, see e.g. [4]. The minimum-cut problem is to find S V, 0 # S # V, such that ( ( S : V - S ) is minimum. { S : V - S } is then a minimum cut in G. If u E S and u E V - S, then { S : V - S } is a cut separating nodes u and u or, simply, a (u,u)-cut. Given u # u, u, u E V, a minimum cut separating u and u is a (u, u)-cut { S : V - S } for which ( ( S : V - S ) is minimum. Minimum (u,u>cuts can be calculated using the maximum flow algorithm, see [4,6]. Remark 3. It has been observed [1,7] in the context of identifying violated subtour-elimination constraints for the symmetric travelling salesman problem that (lS), with the additional restriction S # V, can be solved by solving a minimum-cut problem. This follows because in this case we can assume without loss of generality that 6, = 2 holds for all u E V and thus adding all edges which have at least one end in the node set S we get

2((S)+ ((S:

v - S ) = 2)s1,

and thus problem (1.5) with the additional restriction S # V is a minimum-cut problem. Contrary to what one might expect, the set S for which (1.5) is minimized has no relationship to a minimum-cut in G. To solve the general case of problem (1.5) we define a cupacituted network G * using G = (V, E ) and &, e E E, as follows: (1) Every (undirected) edge e = [u,u] of G is replaced by a pair of directed edges (u, u ) and (u, u ) with capacities cu~= , c,,~= f&. (2) We adjoin a source labelled 0 with edges directed from 0 to u and capacties co., = max{fb, - 1,O) and a sink labelled n + 1 with edges directed from u to n + 1 and capacities c ~ . =~ max{l + ~ - fb,,O} for all u E V. Let S C V

Trees and cuts

s15

and consider the cut set { S U (0): V U { n + 1) - S } in the capacitated network G *. We calculate

c ( S U{O): V U { n +1}-S)= =

max{:b, -1,O}+

= "ES

2 max(1-tb,,O}+c(S:

V-S)

UES

IlEV-S

(max{l - 4bu,0) - max{:b. - 1,0})

+$&(S:V - S ) +

c

max{:b, -1,O)

UEV

= I S l - $ c b,+:((S: D

ES

=ISl-((S)+

V-S)+

2 max{:b,-1,0)

"EV

2 max{tb,-l,O}.

UEV

zoEv

Since the term max{fb, - 1,O) appearing in the above calculation is independent of S we have thus proved the following. Theorem 4. The minimization problem (1.5) is equivalent to the problem of finding a minimum (0, n + 1)-cut in G * satisfying S # 0, i.e., to the problem min{c(S

v u {n -t- 1) - S ) J s c V,S # 01. minimum (0, n + 1)-cut in G * can

u (0):

(1.7)

As noted above, a be found using the maximum flow algorithm [4,6]. To satisfy the condition S # 0 we solve n - 2 maximum flow problems with capacities as defined before except that for the k-th problem we set Cn.k = $ 0 3 and, if k a 2 holds, c,,+, = + c r ~ for u = I , . . . , k - 1 where k assumes the values 1,. . . ,n - 2 . We record the cut-set that is minimum over the n - 2 iterations; the resulting set S solves problem (1.5). The validity of the procedure follows from the simple observation that if S minimizes (1.7) or (1.5) and if 1 S 1 3, then S satisfies k E S, (1,. . . , k - I} C V \ S for some k = 1,. . . , n - 2 . Furthermore, every (0, n + 1)-cut in G * having 1 S 1 = 1 has capacity 1 and any (0, n + 1)-cut having 1 S 1 = 2 has capacity not less than one. Thus n - 2 calculations suffice. We have thus proved the following. Corollary 5. The minimization problem (1.5) can be solved by at most n - 2 maximum flow calculations. Remark 6. There are various possibilities to speed up the calculations of (1.5) and we direct the reader to the appendix of [l] where several techniques to this effect are discussed that remain applicable in the general case discussed here. As

516

M.W. Padberg, L.A. Wolsey

far as the symmetric travelling salesman problem is concerned, the identification of violated subtour elimination constraints can be carried out as above with the following change: We do n - 2 maximum flow calculations in G * as described in the proof of Corollary 5 , but with c,,.+~ = + ~0 in all of the n - 2 subproblems. Verifying the correctness of this procedure is left to the reader. Remark 7. The network G" used in Theorem 4 is a specialization of the network constructed in [ 111 and can be derived by observing that (1.5) is a binary quadratic programming problem with an additional constraint ruling out the trivial solution. Remark 8. While it is possible to calculate a maximum weighted (spanning) tree using any linear programming algorithm and a maximum flow subroutine it is, obviously, not our intent to propose such an algorithm. Rather what we find remarkable is that, as is the case of b-matchings in finite undirected graphs, see [8], an 1.p. based calculation of (spanning) trees is closely related to minimum-cut calculations. The discussion of this paper suggests an algorithm for the minimum-cut problem which is somewhat different from the Gomory-Hu algorithm, see [4], and which we state explicitly for matters of convenience. The Gomory-Hu algorithm uses at most n - 1 maximum flow calculations, where n = I VI is the cardinality of 1 V 1, see [4],to find the maximum flow values between all pairs of the nodes of G and yields a minimum cut as a by-product. The following algorithm - which is designed to find a minimum cut in a connected graph G, but not the maximum flow values between all pairs of nodes - requires n - 1 maximum flow calculations as well and has the advantage of being easier to implement than the Gomory-Hu algorithm. For u € V we let N ( u ) denote the neighbors of u in the graph G. Step 1. Let E = + m, S = 0, k = 1 and choose u I E V. step 2. Choose & + I E U ~ = , ~ ( u , ) -,..., { u , uk}, let u € { u l ,..., u k } be a neighbor of uk+I and calculate (using a maximum flow algorithm) a minimum capacity cut separating u and & + I . Let c ( u , u ~ + be ~ ) the cut-capacity and {Sk: V - S t } be the cut. step 3. If C(U, & + I ) < f holds, Set E = C ( U , Uk+l) and s = s k . Set the edgeweight of edge [ u , u k + l ] equal to + a , and replace k by k + 1. If k < n, go to Step 2 . Else stop: F is the capacity of the minimum cut and { S : V - S } is a minimum-cut set. The validity of the algorithm follows from the following observation: If { S : V - S } is a minimum cut and u E S, u E V - S are any two nodes contained in the opposite sides of the minimum cut, then { S : V - S } is a minimum cut

Trees and cuts

517

separating nodes u and u. Thus if { S * : V - S *} is a minimum cut separating any two given nodes u and u of G, then either { S * : V - S * } is a minimum cut or nodes u and u are on the same side of every minimum cut (which is guaranteed if u and u are connected by an infinite capacity edge). The above algorithm chooses nodes u and u such that they are neighbors in G. (This choice permits one to work with the original list-structures of the graph.) W e ndte furthermore that if a primal maximum flow algorithm is used the maximum flow calculation can be stopped as soon as the flow value is greater than or equal to the current value of C. (The directed version of the minimum-cut problem can be solved by a small variation of the above algorithm: in this case one computes both the maximum flow from u to u and the maximum flow from tl to u and chooses the one which has a smaller capacity, before resetting both respective edgecapacities equal to +M.)

References [ 11 H.P. Crowder and M.W. Padberg, Solving large-scale symmetric travelling salesman problem to optimality, Management Sci. 26 (1980) 495-509. [2] J . Edmonds. Minimum partition of a matroid into independent subsets, J. Res. Nat. Bur. Standards Sect. B 69 (1965) 67-72. [3] J . Edmonds, Submodular functions, matroids and certain polyhedra, in: Combinatorial Structures and Their Applications (Gordon and Breach, London, 1970) pp. 69-87. [4] L.R. Ford and D.R. Fulkerson, Flows in Networks (Princeton University Press, Princeton, NJ 1962). [5] M. Grotschel, L. Lovasz and L. Schrijver, The ellipsoid method and its consequence\ in combinatorial optimization, Combinatorica 1 (1981). [h] E. Lawler, Combinatorial Optimization: Networks and Matroids (Holt, Rinehart and Winston, New York, 1976). [7] M.W. Padberg and S. Hong, On the symmetric travelling salesman problem: A computational study. Math. Programming Studies 12 (1980) 78-107. [8] M.W. Padberg and M.R. Rao, Odd minimum cut sets and b-matchings, Math. Oper. Res. 7 (1982) 67-80. 191 M.W. Padberg and M.R. Rao, The Russian method for linear inequalities, 111: Bounded integer programming, G B A Working Paper (New York University, New York, NY, 1981). [lo] M.W. Padberg and L.A. Wolsey, Fractional covers for trees and matchings, C O R E Discussion Paper No. 8141, Universitk Catholique de Louvain, November 1981. [ l I] J.C. Picard and H.D. Ratcliff, Minimum cuts and related problems, Networks 5 (1975) 357-370.



BINARY SELF DUAL CODES CONSTRUCTION FROM SELF DUAL CODES OVER A GALOIS FIELD F,m G . PASQUIER Centre Universitaire de Toulon, La Garde. France The binary images with respect to a trace orthogonal basis for F,m over F,,or any if m = 2 . are self dual codes. So we construct binary self dual codes.

1. Introduction

Definitions and results used in this paper may be found in (61. A (n, k, d ) code over F = F,m is a subspace of F" with dimension k ; the minimum Hamming distance between its elements is exactly d. The binary subcode of a code (n,k ) over F is the code where qo= (e n (IF,)".

The orthogonal code of a code is constituted by the vector of F" perpendicular to the code words. A self dual code is a code equal to its orthogonal code: then n = 2 k . A doubly even self dual code is a self dual code such that all the word weights are divisible by 4. A n extremal doubly even self dual code is a doubly even code such that its minimal distance reaches the upper bound 4[n/24]+4 (see [7]).

2. Binary images of a code over F

Definition. The binary image of a vector a = (a,,,. . . , a , , . . . , a , - , ) € F" with respect to a basis B = (y,),. . ., ym -,) is the mn-tuple obtained by mapping each x, into the m-tuple of its coordinates. Properties. We can find in [8] the proofs of the following properties: Property 1. The binary images of a code %(n, k, d ) over E m are codes with parameters ( N = mn, K = mk, D 3 d ) . Property 2. Let Go be a generator matrix of a code Ce over F = F , m , 519

G. Pasquier

520

B = (yo,. . . ,ym-J be a basis for F over IFz, and i be the binary images operator. Then the matrix G = ( i ( y o G o ) . . i(ym-,Go))'is a generator matrix of the binary image code. Definition. A trace-orthogonal basis (T.O.B.) of F is a vector space basis for F over Fz such that tr(yi y j ) = 1 if i = j and tr(yi * y j ) = 0 if i# j (tr is the trace operator for F over h). Example. In this paper we use the following T.O.B.'s: B = (a', a h ,a 5 ) ,the only one for IFs --- F,(x)/(x?+ x + 1); B , = (a', a', a " , aI3), B2= ( a h a'), , a l l , a"), the only ones for IFlh = IFz(x)/(x' + x + 1). B , = ( a ' , a " ' , ( ~ ~ ~ '"), , a ~B ,2a= ( a h , a " , a 2a",a"), 1, two T.O.B.'s for F32= lFz(x)/(x' + x z + 1). B = ( a ' , a ' I , a", a4',a'', a " ) a T.O.B. for lFhj -- 6 , ( x ) / ( x h+ x + 1). Theorem (proof in [8, pp. 44,451). The binary images of a self dual code over IFzm with respect to a self orthogonal basis, or any if m = 2, are self dual codes.

3. Binary images of some quadratic residue codes over IF,

From the paper [4], for p a prime of the form p = 8A + 3, A E N , the extended residue codes over IF, are self dual codes and their binary images with respect to any basis for F, over IF2 are self dual doubly even codes. So we obtain the following extremal ones: P

Parameters of extended QR 4. 2, 3 12. 6, 6 20. 10, x 44. 22, 14

3

II 1') 43

For p ((2"+'

=

Parameters of the binary images

x.

4. 4 24. 12. X 40, 20. 8 88. 44, 16

Known equivalent code Hamming Golay ? ?

I 1 we obtain the constacyclic code of length 11 over F, with parameters

+ 1)/3, n - 2s - 1,5), for s = 2(rn[3]).

4. Binary images of an extended Reed-Solomon class Let the Galois field be F = F,m; n = 2"' - 1, I = {1,2,. . . , ( n - 1)/2}. Let a be a primitive root of F. The Reed-Solomon (R-S) code %, generated by g(x) = (x - a ' ) has parameters ( n , ( n + 1)/2, ( n + 1)/2). From [S,

n,,,

Binary self dual codes construction

p . 961, the extended code ( n + 1, ( n + 1)/2, ( n + 3)/2).

@,of

52 1

%, is a self dual code with parameters

Theorem 4.1 [9]. The binary image of code @, over F8 with respect to the T.O.B. of F8 is equivalent to the Golay code (24,12,8)

over FI6 with respect to the Theorem 4.2 [11]. The binary images of code T.O.B.’s of F16 are extremal doubly even codes of length 64.

el

Theorem 4.3. The binary images of code over h2with respect to the two T.O.B.’s introduced above are doubly even self dual codes of length 160; their minimal distance D E {20,24,28}. IS

Proof. n =31; I ={1,2, ..., 15}; g ( x ) = & = , ( x - a ‘ ) ; a! primitive root of IF,(x)/(x’+

x’

+ 1);

F3>=

g ( x ) = g,, = ( l , a 7 ,a 1 7 , a 5 , a ’ 2 , a ” , a ’ 7 , ( Y ’a2”, X,ar, a --,(Y I h , a 15, a 12, a I X , a ’ 7 , 0 1 5 ) ; >,

1

gllthe extended code of g,l is ( g l l a’”). 8, has parameters (32, 16, 17) and their binary images have parameters (160,

80, D

3

17).

We choose the coefficients of g(x) -- gll Let GI,be the generator matrix of as the first row of GI,;the other rows of GI,are the shifts of this first row. If we denote by i the binary image operator, the binary image of %I has a generator matrix such that

G

= (i(Go)i(aGo) i(a2Go)i(a3Go)i(a4Go))’.

Let G be G’s extended. With respect to the T.O.B.3 B1 introduced above G’s rows have respective weights 40, 44, 40, 44, 52. From [6, p. .27], 4 divides all word weights and consequentIy D ; but 20 6 D S 4 ( n/24) + 4 = 28; so D E {20,24,28}. With respect to the T.O.B.’s B2 introduced above G’Srows have respective weights 56, 40, 40, 48, 40 and the binary image of CeI is also doubly even and D E {20,24,28}.

Conjecture. Binary images of all the codes doubly even self dual codes.

with respect to the T.O.B.’s are

G. Pasquier

522

5. Binary images of H-code Let FG, G = (F2r +), be an algebra group with the following operations: If x = C I I E O x R X and R y = &,,y,XS are elements of FG we have

x +y =

(x, REG

+y,)XR

and x y

= REG

(

xhyk) g=h+k

X'.

We call an H-code a principal ideal of FG generated by x = & E G ~ R Xsuch R that x 2 = 0 , and if H is an hyperplane of G considered as a vector space over F,, ZgeHx, = 1. Camion [l] proves that an H-code is a self dual code. We identify the elements of FG with n-tuples, where n is the order of G, with coefficients in F ; these elements of FG are classed according to the following method (Rabizzoni [ 151): Let x,, i E {0,1,. . . , 2 r } , be the elements of F,r; we consider the following encased subspace:

+

El = x o + Eo,. .., E , = x,-i + E,-I,. ..,E,-I = H : En Fzr,

Eo{O};

where x, €5 E , , and x , # x, if i # j . Then an element x = (l,O("'"~', al, . . . , an,:) with ~ ~a, =~ 1 , *generates , an H-code and a generator matrix is G = ( I M ) , where I is the unit matrix. Then the permutation n:(u u)+(u u ) , ( u , u ) E (F"")' is in the automorphism group of (e [8, p. 521. We call a trivial H-code an H-code such that its generator has weight two; then an untrivial H-code has minimal distance d 3. It is easy to prove [S, p. 571 that the binary image of an untrivial H-code (n, k, d ) over FlmG, G = (F2r,+ ), have parameters ( N = m2', K = mk, D 3 d), 3 G D S 2'; we denote FG, G = (h2, +), by F h r .

I

I

I

6. Results

6.1. H-codes over FF4 We choose the generator x = X " + a x " + bX"'= (l,O, a, b ) , a + b = 1, (Y a primitive root of F4= F2(x)/(xz+ x + l), for the untrivial H-code %. We then have the following generator matrix of %: 1 0

Q

6

GO=[ O l b a

]

gi=xX0, g2 = XX".

Then we have the following results (see proof in [8, pp. 60-681).

Binary self dual codes construction

523

Table 1 Extremal doubly even self dual code H-code over F

G

F

Basis

Generator x

Binary image N

D

Equivalent code

IF,

I

0 a

a-'

any

8

4

Hamming A, (not Pless)

F,

1

0 a

a'

B

12

4

B,,

IF,,,

1 0 1 0

Bl

16 16

4 4

F,,

20

4

S?,,

24 24 24 24

4 4 4 4

F,, (not Pless-Slaane)

16 16

3 4

A , t A , (not Pless)

24

8 6 4 4 4 4 4 4

Golay; b(%',) Z,, (not Pless-Sloane)

IF,

I

a ( a!" a a'

0 a

Bi

a'"

Bl

x" x a x a , xa'

IF, IF,

E,

x a ' xa'

1 a 0 0 0 1 a' 1 0 0 0 a' a'

any

1 0 0 0

a1

ad

I

a'

a5

1 a

a' a

1 0 0 0 1 1 0 0 0 a a

a

1 0 0 0 1

1

1 0 0 0 0

0

1 a

a

"

VLd

T2, OL3

B

a'

a' 0 a' 0 a'

F,,

FZ, V2-I 0 2 4

M,, (AR)' (not Pless)

B&B,,

0

1

a"' a ' ' a''

B , , B, 32

8

?

0 0 0

I

a'

B,,

B, 40

8

?

IF,,

1 0 0

IF,,

I

x"+a""'

+

a"

a"

+

t a'x"? a"x"' n"'X"i: + a 1x " + a.t5xe1? +

a

+ +

''

xa5

l 0 0 0 l l l O I D 0 0 0'1 a a

1 0 0 0 1

IF,,

XI

A,+A,

8

B,, B, 80

(not Pless)

or

I2 or extremal if D

=

16

16

F, = F , ( x ) / ( x 2 x 1) F, = F 2 ( x ) / ( x 3 x + I);, T.O.B. B = (a', a', a') F,,=F,(x)/(x'+x 1); T.O.B. B , = ( c ~ ' , a ' , a ' ~ , a IB~2)=; ( a b , a 9 , a " , a ' " ) F 3 2 = F 2 ( ~ ) / ( ~ ' + ~ 1); 2 + T.O.B. B , = ( a S , a ' o , a 3 0 , a 9 , aB ' ',)=;( a b , a y , a " , a 2 4 , a 2 h ) F, = F 2 ( x ) / ( x b + x 1); T.O.B. B = ( a Ja, " , a 4 3 , a s 8 , a 6 1 )

+

+

+

524

G . Pasquier

Theorem 6.1. Binary images of untrivial H-codes over F,F4 with respect to any basis of IF4 are equivalent to the extended Hamming code. Theorem 6.2. Binary images of untrivial H-codes over FxF4with respect to the T.O.B. of IF, are equivalent to the self dual indecomposable code B I 2of the Pless classification [ 121. Theorem 6.3. Binary images of untrivial H-codes over F,F, with respect to T.O.B. B , of IFl6 are equivalent to the codes A g $ A x or F,, of the Pless classification 1121. with respect to T.O.B. B, Theorem 6.4. Binary images of untrivial H-codes of IF32 are equivalent to the code SZ0of the Pless classification [12]. Theorem 6.5. Binary images of untrivial H-codes over FMF4 with respect to the T.O.B. B of 64 are the codes FZ4,VZ4, TZ4,OZ4of the Pless and Sloane classification [ 131. 6.2. H-codes over FF,

We choose the generator x = X " + a l X 1+ o 2 X u ' +a j X o h +a4X"', a l + a2+ a'+ a4 = 1, a a primitive root of Ex = F,(x)/(x3+ x + l), for the untrivial H-code V. Then we have the following generator matrix of V:

gl = x, g2

= XX",

g3 = x x n 2 ,

g., = XX"'. We then have the following results (see proofs in [8, pp. 78-90]).

Theorem 6.6. Binary images of unrrivial H-codes over F$, with respect to any basis of F4 are equivalent to codes Ax+ Axof F,, of the Pless classification [ 121. Theorem 6.7 (Wolfmann [16]). The H-code over IF$, with generator (1, 03,1, a, a2,a') is equivalent to the extended Reed-Solomon code @, over F,; its binary image with respect to the T.O.B. of F8= F2(x)/(x3 -tx + 1) is equivalent to the Golay code (24, 12, 8). Theorem 6.8. Binary images of H-codes over FxFB with respective generators ( I , O', I , a', a ', a'), ( I , O', a ', a ', a', a'), (1, 02,1, I , a', a"), (1,03,a, a, a', a '),

Binary self dual codes consiruciion

525

(1, (Y, a,a,a,a 3 ) , (1,03,a,a 3 , 0,O) with respect to the T.O. B. are the respective codes 2 2 4 , F 2 4 , v 2 4 , OZ4,M241B I 2 @B,, of the Pless and Sloane classification [ 12, 131. Theorem 6.9. Binary images of the H-code FI6F8 with generator (1,03,1, a k 0 , a "a',') with respect to the T.O.B.'s of F16 are extremal doubly even self dual codes of length 32. Theorem 6.10. Binary images of the H-code over h2F8 with generator ( 1 , 03,1 , a', a 14, a") with respect to the T.O.B.'s B1 and B2 of F32 are extremal doubly even self dual codes of length 40. 6.3. H-codes over F F I 6 We choose the generator

x;=l

We have ai = 1, with a a primitive root of Flh F2(x)/(x4+ x untrivial H-code %. -'I

+ 1) for the

Remark. It is easy to prove that if ai E F * and if at least three elements ai, i E (1, S), are different, then the minimum distance of % is equal or superior to 5. Theorem 6.11. The binary image of the H-code over h8[F16, with generator (1, O', 1,1,1, a,a,a', a 3 ,a5)with respect to the T.O.B. of IF8 is a doubly euen self dual code (48, 24, 8). Proof. By the remark, the H-code %? has parameters (16,8), d 2.5,and its binary image with respect to the T.O.B. is a self dual code (48,24), D 5. Let Go be the generator matrix of % and i be the binary image operator. The generator matrix of c is G = ( i ( G o )i(aG0) i(a2Go))f by [8, p. 421. So rows of G have weight 20 or 16. By [6, p. 27],4 also divides code weights and consequently D, or, by Property 2 [8, p. 571, D s 8, so D = 8.

Theorem 6.12 (Wolfmann [16]). The H-code over F16Fj6 with generator (1,07,aI2,a, a 6 ,a', a', a', a 3 )is equivalent to the extended Reed-Solomon code @, over F16; their binary images with respect to the T.O.B.'s of F16, are also equivalent to the extremal doubly even self dual code of length 64.

G. Pasquier

526

Theorem 6.13. Binary images of the H-code over F32F16 with generator (1, O’, a ’, a 4, a ’, a 6 , a ’, a lo, a ”, a ”) with respect to the T.O.B.’s B , and BIof F3z are doubly even self dual codes (80,40, I)), D E {8,12,16}. Proof. By the remark, the H-code V has parameters (16, S), D 3 5 and their binary images C, with respect to T.O.B.3 are self dual codes (80,40), D 3 5. Let Gobe the generator matrix of % and i be the binary image operator. By [S, p. 421 a generator matrix of C, is

G = (i(Go)i(aGo) i(a2Go)i(a3Go))’. So the rows of G with basis B , are 24,28,20 and with basis B2they are 24,20. By [6, p. 271, 4 also divides code weights and consequently D ; or by [7], D s 4[n/24] + 4 = 16. So D E {8,12,16}.

References [ I ] P. Camion. Etude de codes binaires abeliens modulaires autoduaux de petites longueurs, Rev. CETHEDEC (79) 2 (1979) 3-24. 121 J.H. Conway and V. Pless, On the enumeration of self dual codes, J. Comb. Theory, Ser. A 24 (1980) 26-53. [3] 1.I.Dumer and V.A. Zinov’ev, Some new maximal codes over GF(4), Problems of Information Transmission (1979). (41. M. Karlin, V. K. Bhargava and S.E. Tavares, A note on extended quaternary quadratic residue codes and their binary images, Inform. Control 38 (1978) 148-153. [5] A. Lempel, Matrix factorization over GF(2) and trace-orthogonal basis of GF(2”), SIAM J. Comput. 4 (1975). (61 F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes (North-Holland, Amsterdam, 1977). [7] C.L. Mallows and N.J.A. Sloane, An upper bound for self dual codes, Inform. Control 22 (1973) 188-200. [8] G. Pasquier, Etude des codes sur une extension de Fz et de leurs images binaires, Thi.se, UniversitC de Provence (1980). (91 G. Pasquier, The binary Golay code obtained from an extended cyclic code over Fa, Europ. J. Combin. 1 (1980) 369-370. [lo] G. Pasquier, Binary images of some self-dual codes over GF(2”) with respect to trace orthogonal basis, Discrete Math., to appear. [Ill G. Pasquier, A binary extremal doubly even self-dual code (64,32,12) obtained from an extended Reed-Solomon code over F,,, I.E.E.E. Trans. Inform. Theory, to appear November 1981. [12] V. Pless, A classification of self orthogonal codes over GF(2), Discrete Math. 3 (1972) 7-09-246. 1131 V. Pless and N.J.A. Sloane, On the classification and enumeration of self dual codes, J . Comb. Theory, Ser. A 18 (1975) 313-335. (141 D. Slepian, Some further theory of group codes, Bell System Tech. J. 39 (1960) 1219-1252. [ 151 P. Rabizzoni, Private communication. 1161 J. Wolfmann. A new construction of the binary Golay code (24,12.8) using a group algebra over a finite field, Discrete Math. 31 (1980) 337-338. [ 171 J . Wolfmann, Private communication.


SUR UN THEOREME MIN-MAX EN THEORIE DES GRAPHES (D’apris L. Nebesky) C. PAYAN and N.H. XUONG I.M.A.G., B.P. 53X. 38041, Grenoble, Cidex, France Let G = (V(G), E ( G ) )be a connected graph. Denote by fmin(G) the minimum number of faces taken over all 2-cell embeddings of G into orientable surfaces. For every subset A of E ( G ) , let p ( A ) (resp. i ( A ) )be the number of components of G \ A = ( V ( G ) , E ( G ) - A ) . known witheven (resp. odd)Betti number. Set y , , ( G ) = p ( A ) + Z i ( A ) - I An E ( G ) ( Using results on the maximum genus and some matroid theoretical results, L. NebeskL showed that max, y,, ( G ) = f min(G). In this paper, we give another direct proof of this result where only graph-theoretical concepts are used.

1. Definitions et notations

Pour les dtfinitions et notations non prtcistes, on pourra se reporter A [ 11. Soit G = ( V ( G ) , E ( G ) )un graphe, V ( G ) son ensemble de sommets (I V ( G ) J= n ) , E ( G ) son ensemble d’arites ( / E ( G ) I= m ) . Posons P ( G ) = m - n + 1. Pour les graphes connexes, P ( G )est le nombre de Betti (ou nombre cycloma tique). Nous noterons par: G v , le sous-graphe engendrt par Y ( Y V ( G ) ) ; G \ A , le graphe partiel ( V ( G ) E , ( G ) - A); G n F, I’intersection des graphes G et F : ( V ( G )n V(F),E ( G )n E ( F ) ) ; G U F, l’union des graphes G et F : ( V ( G ) U V ( F ) ,E(G)U E ( F ) ) . Soit ( ( G ) le nombre minimum de composantes connexes d’un co-arbre de G, ayant un nombre d’arktes impair. Nous appellerons c-cocycle, un cocycle o(X)tel que Gx et G ” ( G )contien-~ nent chacun un cycle. Un graphe est cycliquement k-ar2res-connexe (not6 c-k-c) si k est inftrieur ou Cgal A la cardinalit6 de tout c-cocyle. Une immersion d’un graphe G dans une surface S est un dessin I G I de G dans S tel que les sommets de G soient des points distincts de S, les arktes des courbes simples qui ne se rencontrent pas en dehors de leurs extrkmitts. Une face de I G I est une composante connexe de S - IG I. Une immersion est dite 2-cellulaire si chaque face est homtomorphe 2 un disque ouvert. 521

C. Payan, N.H. Xuong

528

Remarquons que par surface on peut entendre un ensemble de surfaces et qu’un graphe non connexe ne peut-etre immergk 2-cellulairement que dans plusieurs surfaces. Nous noterons par f m i n ( G ) le nombre minimum de faces d’une immersion 2-cellulaire de G. y max(G), dkfini pour les graphes connexes dksigne le genre maximum d’une surface dans laquelle G est immerge 2-cellulairement. Pour les graphes connexes, la formule d’Euler permet de lier ymax(G) ii fmin(G). On a n - m + f m i n ( G ) = 2(1- y max(G)). Fonction de Nebesky’ VA, soit p ( A ) le nombre de composantes connexes de G \ A ayant un nombre de Betti pair; soit i ( A ) le nombre de celles ayant un nombre de Betti impair. Nebeskjl [2] a dCfini la fonction YA

( G ) = p ( A )+ 2i(A) - IA f l E ( G)I.

Soit y max(G) = maxAyA (G).

I

Remarques. (1) 3A C E ( G ) YA ( G ) = y maxf G). (2) y max(G) L ye(G)L 1 ou 2. (3) D’aprbs les dkfinitions de yA(G), P(G), et la formule d’Euler, on a VA : yA ( G ) = P ( G ) + 1 = f min(G) mod 2.

2. Rappels

Xuong [3] par un thkorkme de type min-min a lit fmin(G) 21 ,$(G). Nebeskjl [2] en utilisant une dkmonstration ‘matroidale’ a par un thiorkme de type min-max relii [(G) ii y max(G). Dans cet article, nous relions directement f min(G) 2I y max(G). Nous donnons ainsi une dimonstration graphique du rksultat de Nebeskjl. Rappelons les rksultats suivants [3].

Lemme 1. Pour tout e E E ( G ) , on a soit

f min(G) - 1,

soit

f min(G) + 1, (ce 4ui est le cas en particulier si e est un isthme).

f min(G \ { e } ) =

Un thior?me min-max en thiorie des graphes

529

Lemme 2. Pour tout e, f, deux ar2tes adjacentes de G, ou a soit

f min(G) + 2,

soit

f min(G).

f rnin(G \ { e , f } ) = Lemme 3. Si G est un graphe non connexe formt de deux composantes connexes G , et Gz,on a f min(G)=fmin(G,)+fmin(G?).

Theorhme. Pour tout graphe G, on a

f min(G) = y max(G). 2. I. Dtrnonstration de f min(G)

y rnax(G )

Soit A C E ( G ) tel que y A (G) = y rnax(G). Soit C une cornposante connexe de G \ A : fmin(C)> 1 si p ( C ) est pair. fmin(C)> 2 si p ( C ) est impair.

On a donc d’aprks le Lemrne 3: f min(G \ A ) 3 p(A ) + 2i(A ),

cornrne f min(G { e } )=sf rnin(G)

+1

(Lemme 1).

On a

f min(G A ) S f m i n ( G ) + l A

1,

c’est A dire f mi n(G)s f r ni n( G \ A ) - I A

1

sp(A)+2i(A)-JA nE(G)( Et donc

f m i n ( G ) s y max(G). 2.2. Dimonstration de f min(G)< y max(G)

La dimonstration se fait par recurrence sur IE(G)(. La proposition est vraie pour l E( G ) ( = 0 ou 1. Soit A C E ( G ) tel que y A (G) = y rnax(G) et tel que ( A 1 soit maximum.

C. Payan, N.H. Xuong

530

ler cas. G \ A est connexe (donc G est connexe). Dans ce cas, nous dirons que G est indkcomposable. YA(G)=(l ou 2 ) - J A I . Comme yA ( G ) s 1 ou 2, on a ( A 1 = 0 ou 1. Si G a au plus une arite f m i n ( G ) = 1 ou 2 et donc f m i n ( G ) C y max(G). Sinon, G a au moins deux arites et, puisqu’il est connexe, il a deux arites adjacentes e et f. Montrons que y max(G\{e,f})C2. En effet, sinon il existerait A ’ C E(G\{e,f}) tel que yA,(G\{e,f})a3 et par suite Y ~ , ~ { . . 2 ~ ~1,( G d’aprks ) la definition de yA. = JJA ( G ) = y max(G). Ce qui est impossible puisque Donc Y~,,,{~.,~(G) ( A ’U {e,f}I 2 2 > ( A 1.

On a donc bien y max(G \ { e , f } ) S 2. Par hypothese de rkcurrence:

f min(G \{e, f } ) y max(G \{e, f)) 2. D’oi~f m i n ( G ) 6 2 d’aprks le Lemme 2. On a donc f m i n ( G ) S y max(G). (Rappelons que f min(G) = y max(G) (mod 2.) U m e cas. G \ A est non connexe. Puisque yA ( G ) = y max(G) et IA 1 maximum, on a en particulier: (i) Si C est une composante connexe de G \ A de nombre de Betti pair: yA (C) = y,(C) = y max(C) = f m i n ( C ) = 1.

(ii) Si C est une composante connexe de G \ A de nombre de Betti impair:

yA (C) = ye(C) = y max( C) = f min(C) = 2, Ve E E ( C ) , f min(C\{e}) = 1. Sinon 3 B ~ E ( C \ { e } ) l y B ( C \ { e } ) 2 3et, par suite y B u ~ . ~ ( C ) S 2 = y ~ ( donc C), yAuBu{el(G)? y A (G), ce qui est impossible puisque / A u B U { e } l > / A I (e5ZA).

(iii) Si X dtsigne I’ensemble des sous-graphes de G engendris par union de composantes connexes de G \ A alors

V H E x: y/i(H)> 1. Nous allons montrer par rkcurrence sur ( E ( G ) l que pour A C E ( G ) vkrifiant (i), (ii), (iii), f min(G)

yA

(G).

Un thiorkme min-max en thiorie des graphes

s31

(a) Si G est non connexe, ou si G possbde un isthme e appartenant I? A, la dkmonstration est immCdiate (d’aprbs la dCfinition de yA et des Lemmes 1 et 3). (b) Sinon / A I 3 nombre de composantes connexes de G \A. Puisque yA > 0, G \ A possttde au moins une composante C de nombre de Betti impair. Cette composante est reliee au reste du graphe G par au moins une arCte de A. Soit 9 I’ensemble des graphes F de X qui contiennent C et une arCte de A ayant une extrimit6 dans C et I’autre hors de C, et qui sont tels que yA (F) soit minimum. Soit Fl, un ClCment de 9 ayant un nombre minimum de sommets. Soit e = {x, y} une arkte de Fo telle que x E C et ye C. Soit f une arCte de C adjacente 6 I’arCte e (f existe puisque C est connexe et posskde au moins une arkte). Soit G ’ = G \{e, f}, A ’ = A - {e}. Pour le graphe G ‘ ,A ’ vCrifie (i), (ii), (iii). En effet, fmin(C\{e})= y max(C\le})= 1. Supposons qu’il existe H E X et H’= H \ { e , f } tel que y A , ( H ’ ) c O . C est contenu dans H et e n’est pas contenu dans H, sinon on aurait: y A ’ ( H ’ ) = y A ( H )3 1.

On a donc Y A ( H ) = 1.

Par suite yA

(FO) = 1 ( Y A (FO) Y A ( H ) ) .

Si C est une composante connexe isolCe du sous-graphe H n F,,, y A ( H n Fo) 3 Y A (c)a 2. Sinon, puisque H fl F,, est strictement contenu dans F,, ( y E F,,, ye H ) YA

( H n F,))> y A (F,,),

et I’on a encore yA

( H n Fo) 2.

D’oh d’aprtts la definition de y A : yA

( H u Fo) = y A ( H )+ y A (

-

~ 0 ) YA

( H n F”)- p,

ou p dCsigne le nombre d’arCtes joignant H\Fo i Fo\H. Donc y A (H U F0)S 1 + 1 - 2 S 0 ce qui est impossible puisque A verifie (iii) dans G. A ‘ = A - {e} virifiant (i), (ii) et (iii) pour G ’ = G \{e,f} on a par hypothttse de recurrence:

(d’aprks la definition de yA)et fmin(G)cfrnin(G - { e , f } ) (Lernrne 2).

On a donc

f rnin(G)

YA

(G),

et par suite f rnin(G) s y rnax(G). Ceci achtve la demonstration du theoreme. 0

Corollaire 4. Si G est un graphe c.4.c, f min(G) 6 2. Demonstration. G contient un sous-graphe G’ sans isthrne, c.4.c, tel que f rnin(G’) = f min(G). Soit A tel que yA (G’) = y max(G’). Toute composante C de nornbre de Betti pair de G’\ A est relie par a u moins deux arites au reste du graphe (G’ est sans isthme). Tonte composante C de nornbre de Betti impair de G ’ \ A contient un cycle. C est donc reliie par a u moins quatre arites au reste du graphe si celui-ci contient un cycle. Sinon elle est soit isolte, soit relike B un arbre. Un calcul immediat donne yA (G’) 2 . Remarque. Determiner y max(G) ou fmin(G) revient B trouver un ensemble d’arites A C E ( G ) telle que yA ( G ) = y max(G) = f min(G). On a vu qu’il suffit pour cela de trouver A vtrifiant (i), (ii) et (iii).

La condition (iii) est facile ii verifier. Restent les conditions (i) et (ii) qui reviennent ii reconnaitre si un sous-graphe est indtcomposable. On a les proprittis suivantes:

P1:

I

G est indCcomposable ($

(VA y A (G) = y max(G) .$ G \ A est connexe) ( 3 A maximal par inclusion yA (G) = y max(G) j G \ A est connexe).

I

Un thiorkme min-max en thiorie des graphes

533

P2 : Ve, f E E ( G ) : f min(G \{e, f}) = 1 ou 2. G est indCcomposable P3 : G est indCcomposable et P ( G ) impair Ve E E ( G ) : f m i n ( G \ { e } ) = 1. P4 : G est indkcomposable et P ( G ) pair 3 Ve(G \{e}) est indCcomposable.

RefCrences [ l ] C. Berge, Graphes et Hypergraphes (Dunod, Paris, 1973). [2] L. Nebeskf, A new characterization of the maximum genus of a graph, Czechoslovak Math. J . 31 (106) (1981). [3] N.H. Xuong, How to determine the maximum genus of a graph, J. Comb. Theory, Ser. B 26 (1979) 217-225. (41 C. Payan and N.H. Xuong, Upper embeddability and connectivity of graphs, Discrete Math. 27 (1979) 71-80.


Annals of Discrete Mathematics 17 (1983)535-548 @ North-Holland Publishing Company

ON TWO COMBINATORIAL PROBLEMS ARISING FROM AUTOMATA THEORY J.E. PIN Uniuersiti Paris VI er CNRS, Laboratoire d’lnformatique Thiorique et de Programmation, 75230 Paris Cedex 05, France We present some partial results on the following conjectures arising from automata theory. The first conjecture is the triangle conjecture due to Perrin and Schutzenberger. Let A = { a , b } be a two-letter alphabet, d a positive integer and let B, ={a‘ba’ 10s i + j s d } . If X C B, is a code, then IX I C d + 1. The second conjecture is due to t e r n 9 and the author. Let d be an automaton with n states. If there exists a word of rank S n - k in d,there exists such a word of length k 2 .

1. Introduction The theory of automata and formal languages provides many beautiful combinatorial results and problems which, I feel, ought to be known. The book recently published: Combinarorics on Words, by Lothaire [8], gives many examples of this. In this paper I present two elegant combinatorial conjectures which are of some importance in automata theory. The first one, recently proposed by Perrin and Schutzenberger [ S] , was originally stated in terms of coding theory. Let A = {a, b } be a two-letter alphabet and let A * be the free monoid generated by A. Recall that a subset C of A * is a code whenever the submonoid of A * generated by C is free with base C; i.e., if the relation c l * c, = c I * * c: where c,, . . . , cl,. . . , c: are elements of C implies p = q and ci = c: for 1 S i S p . Set, for any d > 0, Bd = {a’baj o zz i + j s d ) . One can now state the following.

I

The triangle conjecture. Let d > 0 and X

I I

C Bd. If X is a code, then X S d

+ I.

The term ‘the triangle conjecture’ originates from the following construction: If one represents every word of the form a’ba’ by a point (i,j)E N2,the set B d is represented by the triangle {(i, j ) E N210 S i + j S d } . The second conjecture was orginally stated by t e r n $ (for k = n - 1) [3] and extended by the author. Recall that a finite automaton SB is a triple (Q, A, S ) , where Q is a finite set (called the set of states’),A is a finite set (called the alphabet) and S : Q X A + Q is a map. Thus S defines an action of each letter of A on 0. For simplicity, the action of 535

J.E. Pin

536

the letter a on the state q is usually denoted by qa. This action can be extended to A * (the free monoid on A ) by the associativity rule ( q w ) a = q ( w a ) for all q E Q, w € A * , a € A . Thus each word w in A * defines a map from Q to Q and the rank of w in d is the integer Card{qw q E a}. One can now state the following.

I

Conjecture (C). Let .d be a n automaton with n states and let 0 s k s n - 1. If there exists a word of rank s n - k in d there exists such a word of length s k’.

2. The triangle conjecture

I shall refer to the representation of X as a subset of the triangle {(i,j)E + j S d } to describe some properties of X . For example “ X has at most two columns occupied” means that there exist two integers 0 S i , < i2such that X is contained in a ‘Iba* U a ’2ba*. Only a few partial results are known on the triangle conjecture. First of all the conjecture is true for d S 9; this result has been obtained by a computer, somewhere in Italy. In [ 5 ] , Hansel computed the number t. of words obtained by concatenation of n words of Bd. He deduced from this the following upper bound for 1 X I . N210 zs i

Theorem 2.1. Let X C B d . If X is a code, then IX

1 S ( 1 + (l/V‘?))(d + 1 ) .

Perrin and Schutzenberger proved the following theorem in [9]. Theorem 2.2. Assume that the projections of X on the two components are both e q u a l t o t h e s e t {0,1, ...,r ] f o r s o m e r s d . I f X i s a c o d e , t h e n l X l S r + l .

Two further results have been proved by Simon and the author [lo]. Theorem 2.3. Let X C Bd be a set having at most two rows occupied. If code, then 1x1 z s d + 1.

x is a

Theorem 2.4. Assume there is exactly one column of X C Bd with 2 points or more. If X is a code, then S d + 1.

1x1

Corollary 2.5. Assume that all columns of X are occupied. If X is a code, then IXlSd +l.

Two combinatorial problems arising from automata theory

537

Indeed assume 1 X I > d + 1. Then one of the columns of X has two points or more. Thus one can find a set Y C X such that: (1) all columns but one of Y contain exactly one point; (2) the exceptional column contains two points. Since I Y I > d + 1, Y is a non-code by Theorem 2.4. Thus X is a non-code. O n course statements 2.3, 2.4, 2.5 are also true if one switches ‘row’ and ‘column’. 3. A conjecture on finite automata

We first review some results obtained for Conjecture (C) in the particular case k = n - 1: “Let d be an automaton with n states containing a word of rank 1. Then there exists such a word of length C ( n - l)’.” First of all the bound ( n - 1)* is sharp. In fact, let dn= (Q,(a,b } , a), where Q ={0,1, ..., n - l}, ia = i and ib = i + 1 for i # n -1, and ( n -1)a = ( n - l)b = 0. The the word (ubn-’)’-’uhas rank 1 and length ( n - 1)’ and this is the shortest word of rank 1 (see [3] or [13] for a proof). Moreover, the conjecture has been proved for n = 1,2,3,4 and the following upper bounds have been obtained: 2“-n-1

(tern9 [2], 1964);

i n 3- :nZ+ n + 1

(Starke [16, 171, 1966);

;n”n*++n

(Kohavi [6], 1970);

in’-In?+Pn

-4

( t e r n j , Pirick6 and Rosenauerov6 [4], 1971);

An’ - 2n*+ Y n - 3 (Pin 1151, 1978). For the general case, the bound k 2 is also the best possible (see [13]) and the conjecture has been proved for k = 0,1,2,3 [13]. The best known upper bound was

i k ’ - f k 2 + y k -1

([15]).

We prove here some improvements of these results. We first sketch the idea of the proof. Let d = (0,A, 6 ) be an automaton with n states. For K C Q and w E A *, we shall denote by K w the set {qw q E K } . Assume there exists a word of rank S n - k in d.Since the conjecture is true for k s 3, one can assume that k 2 4. Certainly there exists a letter a of rank # n. (If not, all words define a permutation on Q and therefore have rank n.) Set K , = Qa. Next look for a word m i(of minimal length) such that K 2=Klmlsatisfies I K 21 < I K , I. Then apply the same procedure to K 2 ,etc. until one of the K , ’s satisfies 1 K , 1 S n - k :

I

J.E. Pin

538

-

Then aml * * m,-l has rank S n - k. The crucial step of this procedure consists in solving the following problem: Problem P. Let d = (Q,A, 6) be an automaton with n states, let 2 S m C n and let K be an m-subset of Q. Give an upper bound of the length of the shortest word w (if it exists) such that 1 K w 1 < IK 1.

There exist some connections between Problem P and a purely combinatorial Problem P'. Problem P'. Let Q be an n-set and let s and t be two integers such that s + t S n. Let ( S i ) l r i r pand (T,)lrr.Epbe subsets of Q such that (1) For l s i s p , I S i l = S and IT,I=t. (2) For 1s i =Sp, S, n Ti = 0. (3) For l s j < i s p , S,nTi#0. Find the maximum value p ( s , t ) of p .

We conjecture that p ( s , t) = (':I) = (":'). Note that if (3) is replaced by (37, viz., for 1 S i# j S p , S, n T, = 0, then the conjecture is true (see Berge [l], p. 406). We now state the promised connection between Problems P and P . Proposition 3.1. Let d = (a,A, 6 ) be an automaton with n states, let 0 s s G n - 2, and let K be an (n - s)-subset of Q. If there exists a word w such that 1 K w I < IK I , one can choose w with length S p ( s ,2).

Proof. Let w = a l * . . up be a shortest word such that I K w 1 < 1 K I = n - s and define K I = K,K 2= K l a l , .. . ,Kp = K,-lap-l. Clearly, an inequality of the form 1 K, I = 1 K a , * a, 1 < IK I for some i < p is inconsistent with the definition of w. Therefore IK, 1 = 1 K,(= * = IK, 1 = ( n - s ) . Moreover, since IKpap1 < 1 K , 1, K, contains two elements x, and y p such that x,a, = ypap. Define 2-sets T, = { x , , y , } C K, such that x,a, = x , + ~and y,a, = Y , + ~for 1 S i C p - 1. (The T, are defined from T, = {x,, y p } . ) Finally, set S, = Q \K,. Thus we have (1) For l s i s p , I S , l = s and I T , ) = 2 . (2) F o r l s i s p , S , n T , = 0 .

-

Finally assume that for some 1 S j < i C p , S, f l Ti = 0, i.e., { x i , y t } C K j . Since xiai

* *

a, = yiai

* *

. a,,


539

it follows that

(Ku, ..

*

ui-,uj *

*

up I = JK/Ui. . . up 1 < n - s.

---

But the word a , ui-lui * up is shorter than w, a contradiction. Thus the condition (3), for 1 6 j < i 6 p , Si fl T,# 0, is satisfied, and this concludes the proof.

I shall give two different upper bounds for p ( s ) = p ( 2 , s ) . Proposition 3.2. We have : (1) p ( O ) = (2) p(1)=3, (3) p ( s ) 6 s 2 - s +4 for s 3 2 . 1 9

Proof. First note that the S,’s ( T , ’ s ) are all distinct, because if S,= S, for some j < i, then S,n T, = 0 and S, f l T,# 0, a contradiction. Assertion (1) is clear. To prove (2) assume that p(1)>3. Then, since T4nSI#0, T4nS , # 0 , T4n S, # 0, two of the three 1-sets S , , S 2 , S, are equal, a contradiction. On the other hand, the sequence S , = {x,}, S 2 = {x2}, S3 = {x,}, TI = {xz, x,}, T2= {x,, x,}. T1= {x,, x2}, satisfies the conditions of problem P‘. Thus p(1) = 3. To prove (3) assume at first that SIfl S2 = 0 and consider a 2-set T, with i 2 4. Such a set meets S,,S 2 , and S,. Since SI and S 2 are disjoint sets, T, is compofed as follows: - either an element of S,r l S3 with an element of S2 n S,; - or an element of S,r l S, with an element of S2\S3; - or an element of S , \S,with an element of S2 flS,. Therefore

P ( s ) - 3 s Is, n s,J Is2n s3(+ 1 sIn slI

(s,\s,I + sI \S,I Is2n s31 = Is, n s3J (s21+ Is,1 Is2n s3( - Is,n s, Iszn slJ =s(Jsn I s,l + I s2n s,J)- I sIn s31I s2n s,].

Since S , , S , , S , are all distinct, IS,n S,I S s - 1 and IS, f S,l lS s - 1. Thus if IS, n S,I = 0 or IS2n S,l= 0 it follows that

p ( s ) S S(S

- 1 ) + 3 = S’-S

+ 3.

If I S , n S3I # 0 and I S 2 nS,I # 0, one has

Is, n s,I Is2n s3J3 Is, n s31+Iszn s,J- 1, and therefore:

J.E. Pin

540

We now assume that a

=

1 S, n S , 1 > 0, and

we need some lemmata.

Lemma 3.3. Letx be an element of Q. Then x is contained in at most (s + 1) T,’s. Proof. If not there exist (s +2) indices i , < . - -< is+* such that T, = {x,x,,} for 1 G j d s + 2. Since S,, fl T,,= 0,xbf S,,. On the other hand, S,, meets all T,, for 2 s j s s + 2 and thus the s-set S,, has to contain the s + 1 elements x,,, . . .,x ~,),, a contradiction.

Lemma 3.4. Let R be an r-subset of Q. Then R meets at most (rs + 1) T,’s. Proof. The case r = 1 follows from Lemma 3.3. Assume r 2 2 and let x be an element of R contained in a maximal number N, of T,’s. Note that N, s s + 1 by Lemma 3.3. If N, s s for all x E R, then R meets at most rs T,’s.Assume there exists an x E R such that N, = s + 1. Then x meets (s + 1) T,’s, say T,,= ~ } i , < * * < is+,. {x,x,,}* T,,,, = {x, x , ~ +with We claim that every y # x meets at most s T,’s such that i # il,. . . ,iri2.If not, there exist (s + 1) sets T,, = { y , y,,} * T.+,= {y, y / , , > }with j , < * . * s, a contradiction. This proves the claim and the lemma follows easily.

-

We can now conclude the proof of (3) in the case 1 S , r l S21= a > 0. Consider a 2-set T, with i 2 3 . Since T, meets S , and S2,either T, meets S , n S,, or T, meets S , \Sz and S , \ S , . By Lemma 3.4, there are at most (as + 1) T , ’ s of the first type and at most (s - a)’ T , ’ s of the second type. It follows that p ( s ) - 2 S ( s - a ) 2 +as

+ 1,

and hence p ( s ) S s’ + a’ - as + 3 G s 2 - s + 4, since 1 S a S s - 1. Two different upper bounds were promised for p ( s ) . Here is the second one, which seems to be rather unsatisfying, since it depends on n = 1 Q I. In fact, as will be shown later, this new bound is better than the first one for s > Ln/2J. Proposition 3.5. Let a

=

[ n / ( n- s)J. Then

Two combinarorial problems arising from automata theory

54 1

if n - s divides n, and

if n - s does not divide n.

Denote by Ni the number of 2-sets meeting Si for j < i but not meeting S,. Note that the conditions of problem P‘ just say that Ni > 0 for all i s p ( s ) . The idea of the proof is contained in the following formula

This is clear since the number of 2-subsets of Q is (;). The next lemma provides a lower bound for N i . Lemma 3.6. L e t Z , = r ) , , , S , \ S , a n d ( Z , ( = z , T . henN,a(;~)+z,(n-s-z,). Indeed, any 2-set contained in Zi and any 2-set consisting of an element of Z, and of an element of Q \(Si U Z i )meets all Si for j < i but does not meet S , . We now prove the proposition. First of all we claim that

If not,

Q\W=

n s, Is#sp(s)

is non-empty, and one can select an element x in this set. Let T be a 2-set containing x and S be an s-set such that S n T = 0. Then the two sequences T satisfy the conditions of Problem P‘ in S , , . . . ,&), S and T I , . . , Tp(*), contradiction to the definition of p ( s ) . Thus the claim holds and since all Z,’s are pairwise disjoint:

C zi= n. It now follows from ( 1 ) that

(2)

J.E. Pin

542

Since Ni > 0 for all i, Lemma 3.6 provides the following inequality:

where f ( z ) = (:) + z ( n - s - z ) - 1. when the zi’s Thus, it remains to find the minimum of the expression are submitted to the two conditions (a) zi = n (see (2)) and (b) O < z i 6 n - s (because Z i C Q \ S , ) . Consider a family ( z i )reaching this minimum and which furthermore contains a minimal number a of zi’s different from ( n - s ) . We claim that a 6 1 . Assume to the contrary that there exist 2 elements different from n - s, say z I and z 2 . Then an easy calculation shows that

zf(zi)

z

f(Zl+

if z I + zz S n - s,

22) n - s. z I + z 2 - in the case z1+ z 2S n - s - or by 2,

Thus replacing z I and zz by ( n - s ) a n d z, + z 2 - ( n - s)-in the case z, + z 2 > n - s -leads to a family (2:) such that f(z :) S f ( z i )and containing at most (a- 1) elements z I different from n - s, in contradiction to the definition of the family ( z i ) .Therefore a = 1 and the minimum of f(zi) is obtained for

z

c

and for ZI =

* . .= z, = n - s , ~ , + ~ = rif n = a ( n - s ) + r w i t h O < r < n - s .

It follows from inequality (4)that if n

p ( s )S

(;)

- af(n - s )- f ( r )

=

a ( n -s),

if n = a ( n - s ) + r with 0 < r < n - s,

where f ( z ) = ( ; ) + z ( n - z ) - l . Proposition 3.5 follows by a routine calculation. We now compare the two upper bounds for p ( s ) obtained in Propositions 3.2 and 3.5 for 2 6 s C n - 2. Case 1 . 2 ~ s s ( n / 2 ) - 1 . Then a = 1 and Proposition 3.5 gives p ( s ) 6 s2+ 2 . Clearly s 2 - s + 4 is a better upper bound.


543

Case 2. s = n / 2 . Then a = 2 and Proposition 3.5 gives p ( s ) G s 2 + 2. Again s 2- s + 4 is better. Case 3. ( n + 1)/2 S s c (2n - 1)/3. Then a = 2 and Proposition 3.5 gives p ( s ) 3~s z - 3 n s

+ n 2 + 3= s 2 - s

+ 4 + ( n - s - l ) ( n-2s

+1)

s s z - s +4.

Case 4. 2 n / 3 S s. Then a 3 3 and Proposition 3.5 gives

6 s z- s

+ h ( a - l ) ( n - s ) -~ ((a - I ) ( n - s)-

1)s + a

+ 1.

Since s s ( 1 - a ) ( n - s), a short calculation shows that

p ( s ) ~ s ’ - s + 4 - ~ (- aI ) ( U - ~ ) ( ~ - S ) ~ + ( U - l ) ( n - s ) + ( a -3). Since a

3 3,

-;(a - 1 ) s

p (s) c s

-s

- 1 and thus

+ 4 - ( a - 2 )( n - s)’ + ( a - 1)( n - s ) + ( a - 3),

and it is not difficult to see that for ( n - s ) 2 2, -(a -2)(n -s)’+(a - l ) ( n -s)+(a - 3 ) a

Therefore Proposition 3.5 gives a better bound in this case. The next theorem summarizes the previous results.

Theorem 3.6. Let d = (Q,A, 6) be a n automaton with n states, let 0 S s S n - 2 and let K be a n ( n - s)-subset of Q. If there exists a word w such that I K w 1 < I K 1, one can choose w with length =s cp(n,s ) where a = Ln \ ( n - s)] and c p h s) = 1

if s = 0,

rp (n,s) = 3

ifs=3,

cp(n,s)= s 2 - s + 4

if 3 s s s n / 2 ,

cp(n,s)=( a + l ) s 2 + ( 1 - a 2 ) n s + ( i ) n ? + a = ins

+a

ifn=a(n-s)ands>n/2,

if n - s does not divide n and s > n / 2 .

J.E. Pin

544

We can now prove the main results of this paper.

Theorem 3.7. Let d be a n automaton with n states and let 0 S k s n - 1 . If there exists a word of rank S n - k in d there exists such a word of length s G ( n , k ) where G ( n ,k ) = k'

for k = 0,1,2,3,

G ( n ,k ) = f k ' - k 2 + Y k - 5

for 4 s k s ( n / 2 )+ 1,

G(n,k ) = 9 +

for k a ( n + 3)/2,

3ss Lk -I

q ( n , s)

Observe that in any case

G ( n ,k ) s f k ' - k 2 + Y k - 5 . Table 1 gives values of G(n,k ) for 0 < k s n s 12.

Proof. Assume that there exists a word w of rank G n - k in SP. Since Conjecture (C) has been proved for k S 3, we may assume k a 4 and there exists a word w , of length s 9 such that Q w , = K , satisfies ( K ,1 S n - 3. It suffices now to apply the method described at the beginning of this section which consists of using Theorem 3.6 repetitively. This method shows that one can find a word of

Table 1 Values of G(n,k ) (see Theorem 3.7) ~

h,n

I

1

0

2

3 4

5 6

7 8

9 10

11 12

2

3

4

5

6

7

8

9

1

0

1

~~

1

12

1

4

9

19

34

56

85

125

173

235

310

0

1

4

9

19

35

57

89

128

180

244

0

1

4

9

19

35

59

90

133

186

9 1 9 3 5

0

1 0

4 1

4 0

9

1

59

93

135

9

35

59

Y3

1

4

9

19

35

59

0

1

4

9

19

35

1

4

9

19

0

1

4

9

0

1

4

0

1

0

0

Two combinaiorial problems arising from automata iheory

r a n k s n - k in d of length G 9 + & s s s k - l q ( n , s ) = s ' - s + 4 for s G n/2 and thus

545

q ( n , s ) = G ( n ,k). In particular,

n for4skS-+1. 2

G ( n , k ) = f k 3 - k 2 + ! k- 5

It is interesting to have an estimate of G ( n ,k ) for k = n - 1 .

Theorem 3.8. Let d be an automaton with n states. If there exists a word of rank 1 in d there exists such a word of length G F ( n ) where

Note that this bound is better than the bound in An', since 7/27 = 0.2593 and

(4 - d/36) 0.2258. Proof. Let h ( n , s ) = ( " ; ' ) s 2 + ( l - a 2 ) n + s ( ; ) n 2 + a+ E ( s ) , where &(S) =

I

0 if n = a ( n - s ) ,

1 if n - s does not divide n.

The above calculations have shown that for 3 S s sz-s+4
h,. Further, let k o + k , p + + k,-,p"-l and bo + b l p + - . . + b,-,p"-' be the base p expansion of k and g - 1 (i.e., all k i , g, E p ) , If b, is the last nonzero term, then p z S g - 1 and the minimality of g yields

(t)= (i)= ... );(

I

0

and Lucas' theorem [ 3 ] shows at once that k , = * - * = k, = O . Let go + * . + g m _ , p m - 'be the base p expansion of g. From (i) f 0 again by Lucas' = b, = 0, hence g is divisible by p"' while g - 1 is not, theorem we have b,,= proving g = p"'. Since g > h,, the least possible value for g is p " defined above and we get k 3 g 3 p " . To conclude the proof it suffices to show that for k = p "

-

-

I.G. Rosenberg

578

the equation (2) has a solution. Define y = ( y ] , . .. , Y , , , ) ~ by setting yl(,)=O (i = 1 , . . . ,s), y, = 0 and y, = - N,,, otherwise. Direct check shows that z = (M - Z ) y + N satisfies z I = * . . = z,,,)-, = 0. Noting that (M)," is of the form

dg(H1,. . . , H,-i,O,. . . ,O), where H, are hi x h, matrices (i = 1 , . . .,w - l), it is easy to see that (M),,=> = 0 as required. (2) Let (f)= * * = (L,) = 0. Then ( M ) k is the zero matrix and hence by (2) all cycles o f f have length k. Thus k divides p m ,i.e., k = p " > hl 2 h, where u = n by the minimality of k and part 1 of the proof. 0 We derive a sufficient condition for ( M ) p=uO . Proposition. Let f have a cycle C of length p " and afine dimension m. Then p " is the length of the shortest cycle off.

Proof. Let {a,,...,a,} C C be a basis for C. Since m is the affine dimension of C, there exists c = y,a, in C such that y , f 1. Define the permutation h by setting h ( x ) = f ( x + c ) - c for a l I x E V , . Since h ' ( O ) = f ' ( c ) - c ( i = 1 , 2 , ...), clearly 0 is on the cycle C - c of length p " of h. The claim is that al - c, . . .,a, c form a basis for V,. Indeed, let Er-, xt (a- c ) = 0. From c = Er=,yta, and the independence of a , , .. . , a m we see that this system is equivalent to x, (x, + . . . + x m ) y , = 0 ( i = 1 , . . , , m ) whose matrix is A = [a,,] with a,, = - y , - 68,. By elementary calculations, det A = 1 - y l - . . - y,f 0, proving x, = . . = x, = 0. Thus we may assume that 0 is on a cycle of h of length p " spanning V,". By (1) this cycle is formed by

zy=,

zy=l

9

0, (M)lN,. . . ,(M)," IN

(3)

and (M),.N = 0. Here the polynomial matrices ( M ) , (i = 1,. . .,p" - 1) commute with (M)," and therefore each ( M ) , N belongs to the null space of (M),P. By assumption, the vectors (3) span V, ; therefore (M)pn is the zero matrix and we may apply the lemma. 0

Now we have the following. Theorem 1. A cyclic afine permutation exists if and only if m = 1 o r p = m

=2

Proof. Necessity. The assumptions of the proposition are satisfied. Thus p"' is the least power of p exceeding h,, i.e., h, 3 p m - ' , Combining this with m 3 h, we get m 2 p m - ' . Using the binomial expansion of (1 + ( p - l))"-' we easily obtain that either m = 1 or p 2 m = 2. Sufficiency. The full discussion of the exceptional cases has been given in Section 2. 0

Cycle structure of afine transformations

s 79

Theorem 2. A n afine permutation with cycles C,of length p m - ' (i = 0,. . . , p -- 1) which are not of the form Ci= {x E p m : aTx = i (mod p ) } for some a E p'" exists if and only if p = 2 , 2 < m S 4 or p = m = 3 . Proof. Necessity. We claim that there exists at least one cycle satisfying the conditions of the proposition. Firstly, for cardinality reasons, each cycle not spanning V,,, is an ( m - 1)-dimensional subspace of V,,,. The cycles being disjoint, they do not share the zero vector and therefore there is at most one cycle not spanning V,,,. Suppose the p - 1 cycles C,spanning V,,, are of the form {x E p m : a:x = i,} (j= 1,. . . , p - 1). If a,. and a,- are not collinear again there is a common point. Thus the cycles are of the form {x Ep"' : aTx = i,} ( i = 1 , . . . ,p - 1) with all i, f 0. Rearranging, we obtain that C, = {x E p"' : a T x = j } . Clearly the remaining cycle is C,,= {x E p"' : a T x = 0). Applying the proposition, we obtain h, 2 p m - 2 ,hence m 3 pm-'. This holds only for the values indicated in the theorem. Sufficiency: Let M be the zero-one simple Jordan block and let N = (1,0,. . . ,O)T. A direct but rather tedious check of the four cases shows that the corresponding f has p cycles of length p " - ' . 0 Acknowledgment I would like to thank R.W. Moody for many helpful suggestions. References [I] B. Csikany. Homogeneous algebras are functionally complete, Algebra Universalis 1 1 (lclXO) 149-1 58. (21 C . Cullen, Matrices and Linear Transformations (Addison-Wesley, Reading, MA, 1966). [3] N.J. Fine, Binomial coefficients modulo a prime, Amer. Math. Monthly 14 (1947) 580-502. [4] H.P. Gumm, Algebras in congruence permutable varieties: Geometrical properties of affine algebras, Algebra Universalis 9 (1979) 8-34. [5] A. Hoffmann, Cyclic affine lines, Canad. J. Math. 4 (1952) 295-301, MR 14-196. [6] D. Hughes, Personal communication. [7] N. Jacobson, Lectures in Abstract Algebra I1 (Van Nostrand, New York, 1953). ( X I R. McKenzie. On minimal locally finite varieties with permuting congruence relations, preprint (1976). [9] R.C. Mullin, Personal communication. [lo] J.C. Muzio and I.G. Rosenberg, Large classes of functionally complete operations I, in: Proc. 10th Intern. Syrnp. Multiple-Valued Logic, Evanston, 1980; IEEE, 94-101. [ 111 I.G. Rosenberg, Functional completeness of single generated or surjective algebras, preprint CRM-856, Montreal, 1979; to appear in: Proc. Coll. Finite Algebra and Multiple-Valued Logic (North-Holland, Amsterdam, 1981) pp. 635-652. (121 L. Szab6 and A. Szendrei, Almost all algebras with triply transitive automorphism groups are functionally complete, Acta Sci. Math. 41 (1979) 391-402. [13] A. Szendrei, A new proof of the McKenzie-Gumm theorem, Algebra Universalis 13 (1981) 133- 135.



CONSTRUCTION OF COLOUR-CRITICAL GRAPHS WITH GIVEN MAJOR-VERTEX SUBGRAPH Horst SACHS and Michael STIEBITZ Technische Hochschule Ilmenau, D D R -6300 Ilmenau PSF327, German Democratic Republic

Dedicated to Dr. Tibor Gallai on the occasion of his 70th birthday T. Gallai (1963) characterized the class L, of all subgraphs spanned by the low vertices for the class of k-critical graphs (k 4) having ‘low’ (minor) vertices (i.e., vertices of valency k - 1). In this paper, the class Hkof all subgraphs spanned by the ‘high’ (major) vertices (i.e., vertices of valencies, k ) is characterized. A general principle is given which, for given H E Hk, enables arbitrarily many k-critical graphs having H as their high-vertex subgraph to be constructed. The main results are contained in Theorems 3.1 and 7.1.

1. Introduction

This paper continues research work carried out by the first author together with G.A. Dirac at Aarhus University, June 1980.

I . 1. Notation ‘ All graphs considered are finite, undirected, and have neither loops nor multiple edges. A graph G = ( V , E ) consists of a finite set V = V ( G )of vertices and a finite set E = E ( G ) of 2-subsets (i.e., edges) of V ( G ) .If e = {P, Q } E E ( G ) then P and Q (the end vertices of e ) are adjacent in G. For P E V ( G ) ,N(P : G ) denotes the set of all vertices of G which are adjacent to P. The valency of P with respect to G is val(P : G )= IN(P : G ) f . Let V ’ C V ( G ) ,E’ C E ( G ) .The subgraph spanned by V ’ ,denoted by G[ V ’ ] , is defined by V ( G [ V ’ ]= ) V‘,

1

E ( G [V ’ ] )= { e e € E ( G ) and e

V’},

further,

G - V ’ = G [ V ( G ) -V’],

G-E‘=(V(G),E(G)-E‘).

Most of the concepts used in this paper can be found in [lo, pp. 528-5401 (see also [9]). 581

H. Sachs, M. Stiebirz

582

The join G = GIV G2 of the disjoint graphs G I , G2 is defined by

V ( G ) = V(Gi) U V(Gz),

I

E ( G )= E(GI)U E(G2)U {{P,Q) P E ~ ( G I Q ) , E V(G2)}. A maximal subgraph B of G such that any two edges of B are contained in a circuit of G is called a block (or member) of G. The set of all blocks of G will be denoted by B ( G ) .A vertex of G which is contained in more than one block of G is a cut vertex (i.e., a separating vertex). An end block of G is a block which contains at most one cut vertex of G. Two blocks which have a vertex in common (they cannot have more than one vertex in common) are called adjucent. The block hypergraph HG of a graph G is defined by

~ ( H G=) V(G),

E(&) = { v ( B )

I

EB(G)};

note that HG has no (non-trivial) circuits, i.e., it has a ‘tree-like’ structure. A colouring c of the graph G is a mapping of V(G) into the colour set {c1,c2,. . . } where c ( P ) # c ( Q ) for any two adjacent vertices P, Q E V(G). Let C ( G ) denote the set of all colourings of G. If, for c E C ( G ) , Ic(V(G))l d k, then c is called a k-colouring of G. Let ck (G) denote the set of all k-colourings of G. For c E c k (G), we shall always suppose that c(V(G))C {cl,c 2 , .. . , ck}. The chromatic number x ( G ) of a non-empty graph G is the smallest integer k with Ck( G ) # 0. A graph G is called critical if x(G’) < x ( G ) for every proper subgraph G’ of G, it is called edge-critical if x ( G - {e}) < x ( G )for every edge e E E ( G ) ,and it is called k-(edge-)critical ( k 2 1) if it is (edge-)critical with x ( G ) = k . Obviously, a connected graph is k-critical if and only if it is k-edge-critical. 1.2. Some fundamental observations and theorems

The important notion of criticity has been introduced by Dirac [2], [3]; for major contributions to the theory of critical graphs see [2]-[6], [lo]-[14].

Observation 1.1 (immediate). If G is k-critical then val(P : G ) 3 k - 1 for all P E V(G). The vertices whose valencies equal k - 1 are usually called minor and the others are called major. In order to have a more distinctive notation, we shall follow a proposal of Dirac and use the terms low and high vertices (i.e., vertices of low or high valency) instead of minor and major vertices. The subgraphs of a k-critical graph G spanned by the low or high vertices will be denoted by L(G) and H(G), respectively. Let, as usual, K, and C, denote the complete graph and the circuit on n vertices, respectively.

Construction of colour-critical graphs

5x3

The problem of determining all critical graphs without high vertices is settled by Observation 1.2 (an immediate consequence of Brooks’ theorem [l]). A k critical graph G has no high vertices if and only if either G = Kk (k 3 1) or k = 3 and G = Gq+, (4 3 1; note that C,= K3). Furthermore, for k < 4 these are the only critical graphs. Therefore, in what follows we will always assume that k 3 4 . This problem being settled, the ‘complementary’ problem of determining all critical graphs that have no low vertices is still open and seems to be a very difficult one. Gallai [6] was the first to show that such graphs exist for every k 2 4 (see Theorem 1.2). In this paper, however, we shall confine our investigations to critical graphs that do have low vertices. Let us return to the trivial cases mentioned in Observation 1.2. Following Dirac, for reasons soon to become visible, we formulate the following. Definition 1.1. A k-critical graph without high vertices is called a k-brick. If B is a k-brick we shall say that B has size k and write k = s(B). Note that the exceptional graphs in Brooks’ theorem [I] are precisely the bricks. Thus ‘bricks’ alludes to ‘Brooks’ as well as to ‘cliques’ (a k-clique is a complete subgraph on k vertices). Definition 1.2. A connected graph all of whose blocks are bricks is called a Gallai tree (see Fig. 1.1); a Gallai forest is a graph all of whose components are Gallai trees. A k-Gallai-tree (forest) is a Gallai tree (forest) all of whose vertices have valencies 6 k - 1. Definition 1.3. Let T be a Gallai tree. A subset B , of B ( T ) is called a matching of T if every vertex of T is contained in at most one block of B , . A matching B , is calted perfect if every vertex of T is contained in exactly one block of B,.

Fig. 1.1.

583

H.Sachs, M . Sriebitz

Let T be a Gallai tree which has a perfect matching B , . Note that, because of the tree-like structure of its block hypergraph HT, BI is uniquely determined, therefore we can write BI = B 1 ( T ) .For the Gallai tree T of Fig. 1.1, B I ( T )= {Bl,B4,Bs,B,,Bs}is its unique perfect matching.

Definition 1.4. For k 3 4,let T i denote the set of all k-Gallai-trees T having the following properties: (1) B ( T ) = B‘U B” (B’Z0, B ” # O), where all blocks of Blare 2-bricks and all blocks of B” are ( k - 1)-bricks; (2) B’ is a matching; (3) B” is a perfect matching. The following three theorems are fundamental.

Theorem 1.1 (Gallai [6, Satz (E. l)]). If G is a k-critical graph then L(G) is a k Gallai-forest (possibly empty). Theorem 1.2 (Gallai [6, Satz (E. 2)]). Let k 3 4 and let L* be a k-Gallaiforest which does not contain a k -clique as a component. Then there is a k -critical graph G with L(G) = L *. This is particularly true if L * is the empty graph. Theorem 1.3 (Dirac [4];Gallai [6, Satz (3.3)]). (1) Let k 8 4 and let G be u k-critical graph which has exactly one high vertex. Then L(G) E T ! . (2) Let T E T i (k 3 4) and connect every vertex of T which is not a cut vertex with an additional vertex Q. Then the resulting graph G is k-critical, L ( G ) = T, and H(G) consists of the single vertex Q. Definition 1.5. Let S, denote the set of all graphs F having the following properties: (1) x ( F ) S k - 1; (2) for each edge e = {P, Q ) E E ( F ) ,the graph F -{e) has a (k - 1)-colouring c with c ( P ) = c ( Q ) .

Observation 1.3 (immediate). Let k 3 1. If G‘ is a proper subgraph of a k-critical graph G then G ‘ E S,. The converse of this proposition is also true.

Theorem 1.4 (Greenwell and Lovisz [ S ] ; Miiller [ l l ] ; see also [lo, Problem 9.19(b), p. 611). Let G I € & , k 2 1. Then there is a k-critical graph G which contains G’ as a proper subgraph.


585

1.3. Problem and results The problem : For the class of k-critical graphs (k 4) having low vertices, characterize the class Hkof all subgraphs spanned by the high vertices. The main results: According to Observation 1.3, Hk C sk. By constructive methods (not using Theorem 1.4) it is shown that, in fact, Hk= S,. In particular, this means that a graph that can be embedded in a k-critical graph (k 3 4) as a proper subgraph can also be made the high-vertex subgraph of some k-critical graph. The constructions used are obtained in a very intuitive form: they turn out to be a far-reaching generalization of the classic Dirac-Haj6s construction applied to joins of bricks.

2. Extensions of colourings In this section, using a well-known simple colouring principle, we shall prove a general colouring theorem needed for the investigations to follow.

Definition 2.1. Let G be a graph with n vertices. The sequence (Pi,P2,. . . ,P,) is called a feasible labelling of (the vertices of) G if V ( G )= {PI,P2,.. . ,P,} and, for all i E {1,2,. . . , n}, G[{Pi,Pi+,, . . . ,P,}] is connected. Then, obviously, for i E {1,2,. . . , n - l}, Pi is adjacent to a vertex Pi with j > i.

Definition 2.2. The set X C V(G) is called a k-set of the graph G if it satisfies (1) for all P EX, val(P : G) s k - 1; (2) G[X]is connected; (3) x(G - X ) S k - 1. Let X be a k-set of the graph G with 1x1 = m, let 1 = (Pi,P2,. . . ,P,) be a feasible labelling of G[XJ, and let c E Ck-,(G- X ) where c(V(G - X))C {cl,c 2 , .. . ,c & - ~ }Construct , a colouring c ' of G, depending on 1 and extending c from G - X to G, in the following way: (i) For P E V(G -X), put c ' ( P ) = c(P). (ii) Suppose that i = 1 or i > 1 (i S m )and c ' ( P j )has already been defined for j = 1 , 2 ,..., i - 1 . Let Xi = N(Pi : G)-{Pi+i, P,+z,. . . ,Pm},

H . Sachs, M. Stiebitz

586

and put C'(Pi) = c,,

where r is the smallest among all integers s with csE c'(X). Lemma 2.1. (a) c' E c k ( G ) ; (b) c' E Ck-t(G)if and only if (c'(N(P,,,: G ) ) I < k - 1.

Proof. 1 being a feasible labelling of G [XI, 1 Xi1 < k - 1 for i E {1,2,. . . , m - 1); therefore, Ic'(Xi)l < k - 1 and hence c' ( P i ) € ( c l ,c2,. . . ,ck-1) for i < m. This means that the only vertex that can possibly have colour c k is P,,,. This occurs if and only if (c'(N(P,,,: G ) ) l = k - 1. Lemma 2.2. For every connected graph G with n vertices and for every vertex Q E V ( G ) , there is a feasible labelling (PI,Pz,.. , ,P,) of G with P,, = Q. Lemma 2.2 follows by a simple induction argument from the fact that every non-empty connected graph has a non-separating vertex. Lemmata 2.1 and 2.2 immediately imply the following theorem. Theorem 2.1. Let G be a graph and let X be a k-set of G. Then we have: (1) x ( G ) < k. (2) ZfthereisavertexQEXwith v a l ( Q : G ) s k - l , t h e n X ( G ) s k - l . (3) Let Q E X be a vertex adjacent to Q,, Q2E V(G) - X . If there is a colouring c E c k - I (G - X ) with c ( Q , ) = c(Q2), then x ( G )6 k - 1. (4) For all edges e E E ( G ) - E ( G - X ) , x ( G - { e ) ) S k -1.

3. The construction K

In this section, a procedure will be described which, starting from a suitable Gallai tree T and a vertex set Q disjoint from V ( T ) ,enables arbitrarily many k edge-critical graphs to be constructed. (A) Let q 2 1 and k 3 3 be given integers, let r = k - q and put Q = {1,2,...,ql. If a Gallai tree T has a perfect matching then this matching will always be denoted by B I ( T ) .We put Bz( T ) = B ( T ) - Bi( T ) .

( B ) Denote by Tk (4) the set of all k-Gallai-trees having a perfect matching BI = BI(T)and satisfying:


5%7

(Bl) for each B E B , ( T ) ,s ( B ) a r ; ( B 2 ) for each B E B2(T),s ( B ) S q + 1. Note that a Gallai tree which has a perfect matching belongs to infinitely many sets T k ( q ) ,e.g., the Gallai tree T of Fig. 1 . 1 belongs to all T k ( q ) with 5 s k s q + 2 ; in particular, 7 E T5(3),T E T46). ( C ) Let T E T k ( 9 ) and define U k ( T ; q )to be the set of all mappings u of B ( T ) into the power set of Q satisfying: ( C l ) for each B E BI(T),lu(B)I = s(B)-r ; (C2) for each B E B 2 ( T ) ,l u ( B ) ( = s ( B ) - 1 ; (C3) for any pair of adjacent blocks B , , B z E B ( T ) , u(BI)r l u(B,)= 0. It is easy to see that for each T E Tk(4)there exists a mapping u E u k ( T ;4 ) ; in general, there are many such mappings which can all be listed without a n y difficulty in a straightforward manner. For example, in Table 3.1, two special mapping u, u ’ are given for the Gallai tree T of Fig. 1.1, corresponding to q = 3 and q = 6 , respectively. Table 3.1

(D) For T E Tk( q ) , u E U,( T ;q ) , and P E V ( T ) ,put

u ( P )=

u u(B),

where B runs through all blocks of T containing the vertex P. (E) For a given pair T E T k ( q ) , u E uk ( T ; q ) , define the graph H = H ( T , u, Q ) by (El) V ( H ) = Q ; ( E 2 ) E ( H ) = { { i , j ) l i# j and there is a B € B I ( T ) such that {i, j } C Q - u(B)}. For our examples (Fig. 1.1 and Table 3.1), H ( T, u, { 1 , 2 , 3 } ) K j ,

H ( T , u’,{I, 2,3,4,5,6})is the graph depicted in Fig. 3.1.

6

5

Fig. 3.1.

H. Sacks, M . Stiebifz

588

(F) For a given pair T E Tk( q ) , u E Uk(T; q ) and an arbitrary graph H with V ( H )= Q, define the graph G = G(T,u, Q; H) by (FI)V(G) = V ( T )u Q ( v ( T )n Q = 0); (F2) E ( G ) = E ( T ) U E ( H ) U {{P, i } P E V ( T ) , i E Q - u(P)}. (GI Put G(T,u, Q ) = G(T,u, 9 ;H(T,u, 9)).

I

Remark. Despite the somewhat lengthy description of the construction K given above, the procedure for constructing the graphs G ( T , u , Q ) for given T, Q = {1,2,. . . ,q } , and k is, in fact, quite simple and in all its details straightforward. The following theorem may be considered the main result of this paper. Theorem 3.1. Let T E Tk(q), u E U k ( T ; q ) .IfH(T,u,Q)ESk then G ( T , u , Q ) is a k -edge -critical graph.

4. Proof of Theorem 3.1 Let T € Tk ( q ) , u E u k ( T ; q ) , and let H E & be an arbitrary graph with V ( H )= Q containing H(T, u, Q ) as a subgraph (i.e., E ( H ( T ,u, Q ) ) C E ( H ) ) . Let G represent G (T, u, Q ; H ) . We shall prove in order the following four lemmata from which Theorem 3.1 immediately follows. Lemma 4.1. For all P E V ( T ) ,val(P : G ) = k Lemma 4.2.

- 1.

x( G ) G k and x (G - { e } )S k - 1 for all edges e E E ( G )- E ( H ) .

Lemma 4.3. x ( G )= k. Lemma 4.4. x ( G - { e } ) S k - 1 for all edges e E E ( H ( T ,u, Q ) ) .

Proof of Lemma 4.1. P E V ( T ) is contained in exactly one block (brick) B E B I ( T ) ;let B I ,B2,. . . ,B, ( m 2 0) be those blocks of &(T) which contain P. Then val(P: T)= s(B)-I +s(B,)-1

+ . a

- + s(B,)-

1,

I Q - u ( ~ ) l =I Q - (u(B)u u(B,)u . . . u u(B,))( = q - (S(B) - r + s(B1)- 1 + . + s(B, ) - 1) = k - 1 + ( s ( B )- 1 + s(B1)- 1 + + s(B,) - 1); *

*

* * *


5x9

there fore, val(P : G )= val(P : T ) +I Q - u(P)J= k - 1. Proof of Lemma 4.2. Lemma 4.2 is an immediate consequence of Lemma 3.1 and Theorem 2.1.

The proofs of Lemmata 4.3 and 4.4 need some preparation 4.1. Let B E B ( T ) be a brick which is adjacent to at most one other block of T. Then B E BI(T), B is an end block. Put Q I = Q - u(B);then IQI1 = q -(s(B)-r ) = k - s(B)(see (A) and (Cl)) and GIQl.]= Kk-,(B,(see (E2)).

4.1.1. Case 1: B * ( T ) = O . Then B I ( T ) = { B } and T = B. Hence, for all P E V ( B ) , u(P)= u(B);according to (F2), P is adjacent to all vertices of QI, i.e., E ( B V G [ Q , ] ) c E ( G )implying ,

x ( G )3 x ( B v G[QlI)= x ( B )+ x ( G [ Q I I ) = s ( B ) + ( k -s(B))=k. By Lemma 4.2, x ( G )= k. This proves Lemma 4.3 under the assumption Bz( T ) = 0. Let e E E ( H ( T , u , Q ) ) . Then e CQt, i.e., both the end vertices of e are adjacent to all vertices of T = B . Theorem 2.1 now immediately implies x ( G - { e } )S k - 1 (note that E ( H ( 7,u, Q)) C E ( H ) and H f Sk). This proves Lemma 4.4 under the assumption B , ( T ) = 0.

4.1.2. Case 2: B 2 ( T ) #0. Then there is exactly one block B * E B,(T) to which B is adjacent; let V ( B )n V ( B * )= { P } . According to (C3), u(B)n u(B*) = 0, consequently, u ( B *) C QI= Q - u ( B ) , hence G [u(B*)] is a complete graph (Fig. 4.1).

Fig. 4.1.

H. Sachs, M. Stiebirz

5 0

Every vertex P‘ E V ( B )- { P } is adjacent to all vertices of QI (see (F2)), vertex P is adjacent to all vertices of Q1- u(B*).Therefore, G* = (B -{P})V G [ Q , ] is a subgraph of G. Note that x ( G * )= k - 1. If c is any colouring of G with Ic(V(G*))l= k - 1, then either c ( P ) E c(V(G*)) (i.e., P has a kth colour) or c ( P ) E c ( u ( B * ) ) . 4.2. Lemmata 4.3 and 4.4 will be proved by induction over 1B2(T)l. For 1 Bz(T)l = 0, the validity of these lemmata has already been established under (4.1.1). Assume 1B2(T)l= s + 1 5 1. The graph T having a perfect matching, there is at least one block (brick) in B ( T ) - say, B* with I V ( B * ) l =t - having the following properties: The vertices of B * can be labelled so that V ( B * )= {PI,Pr, . . . ,P,} and, for i = 1,2,. . . ,t - 1, P, is contained in exactly one block other than B * - say, P,E V ( B , ) , B,# B * - and all of the B, are end blocks. Then, necessarily, B * E B , ( T ) a n d B , E B , ( T ) ,i = 1 , 2 ,..., t - 1 (Fig.4.2).Thus ,-I

U

T’= T -

V(B,)E Tk(q),

,=I

and T’ has the perfect matching

B i ( T ’ ) = B i ( T ) - { B i , B z , ..,B,-I}; . further, B > ( T ’ ) =B , ( T ) - { B * } . Denote the mapping u restricted to B ( T ’ ) by u’. Put (i) G ’ =G ( I ’ , u ‘ , Q ; H ) ; (ii) G I = G [V ( T ’ )U Q]; (iii) E , = {{P,,i} i E u (B*I}. By (F2), E ( G )r\ El 0.

1

Fig. 4.2.

Construction of colour-critica/graphs

s9 I

Let u ’ ( P ) ( P E V(T‘)) be analogously defined for T’ and u’ as is u(P) ( P E V ( T ) )for T and u (see (D)). Then u ’ ( P l )= u ( P l ) - u(B*)and, by virtue of

W), G ’ =(V(GI),E(GI)UEI).

Proof of Lemma 4.3 (the induction). Suppose ,y(G)< k. Then, by (4.1.2), , y ( G ) = k - 1 and, for C E C ~ - ~ ( Gc)( ,P i ) E c ( u ( B * ) )( i = 1 , 2 , ..., t - 1 ) . For colouring the vertices P I ,Pz,. . .,P,-l of B* E Bz(T), at least s(B*)- 1 = x ( B * )- 1 colours are needed, thus IC({Pl,

Pz,. . . ,P1-1})1* s(B*)-1.

Further, G[u(B*)] being a complete graph on s(B*)-1 vertices, ( c ( u ( B * ) )=( s(B*)-1. As an immediate consequence, we obtain c ( P , ) P c ( u ( B * ) ) ,i.e, c induces a ( k - 1)-colouring of G ‘ = ( V(GI),E ( G I ) U El), which contradicts the induction hypothesis (see (4.2)). Lemma 4.3 is now proved. Proof of Lemma 4.4 (the induction). Let e = {a, b } E E(H(T, u, Q)) C E ( H ) and let B denote the set of all those blocks B E Bl(T) for which e C Q - u(B); because of (E2), Bf 0. Case I. B n {BI,Bzr. . . , B I - J# 0. Assume, w.l.o.g., B1E B. Since B I is a brick, V(BI)-{PI}# 0. Let P E V(Bl)-{P,}. Then, by (F2), P is adjacent to a and to b. Because of H E Sk, there is a colouring c E C k - I ( H- { e ) ) such that c ( a )= cfb). Now Theorem 2.1 implies x ( G - { e } ) s k - 1. Case 11. B n {BI,B z ,. . . , BI-,}= 0. Then B C B1(T’)= B1(T )- {B1,B z ,. . . ,B,-l}, therefore,

e EE(H(T’,u’,Q))CE(H(T,u,Q))CE(H) and, consequently, by the induction hypothesis,

x (G ‘ - {e}) G k - 1. Let C ’ E C ~ - ~ ( G ’ - { ~ Because }). G 1 - { e } = G ’ - { e } - E l , we also have Note that c’(P,)!Zc ’ ( u ( B * ) ) .The colouring c’ can be extended to a ( k - 1)-colouring c of G -{e} in the following way. First, the vertices p E V ( B i )- {Pi}( i = 1,2,. ..,t - 1) are coloured. This can be done in such a way that, during the colouring procedure, each vertex of Bi -{Pi} (note that B, - { P i } is a connected graph) not yet coloured is adjacent to at most k - 2 vertices already coloured, thus, in this step, k - 1 colours suffice; P being adjacent to all vertices of u(B*), the colour of P is not contained in c’(u(B*)). c ’ ECk-I(GI-{e}).

H.Sachs, M.Stiebitz

592

Eventually, the vertices PI,Pz,. . . ,P,-lare coloured: the colours of c ‘ ( u ( B*)) can be used, and since exactly s ( B *) - 1 colours are needed it suffices to show that I c ’ ( u ( B * ) ) l > s ( B * ) - l .To do this, note that in H ( T , u , Q ) C H ,any two vertices of u(B*) are adjacent. The same holds for H - {e} and, therefore, also for G’ - {e}, since not both the end vertices a, b of e can belong to u ( B*) for, otherwise, because u ( B *) fl u (Bi) = 0, we would have a P u (Bi) and b E u ( B i ) , i.e., Bi E B, contradicting the definition of B ; hence, Ic ’ ( u ( B*))I = s(B *) - 1. Thus we have proved ,y(G - { e } ) S k - 1 . In both cases, x ( G - { e } ) S k - 1, proving Lemma 4.4.

5. Some remarks concerning the construction K 5. I. Note that isolated vertices may occur in G (IT, u, Q ) .

5.1.1. It is easy to see that precisely those vertices of Q remain isolated which are contained in each of the sets u(B)where B E B l ( T ) .Let Z denote the set of all isolated vertices of G( T, u, Q).Then,

5.1.2. Let I Z l = d and suppose, w.l.o.g., Z = { q - d + l , q - d + 2 ,..., q } . Put q - d = q *, Q - i = { 1 , 2 , ...,q *} = Q *, r + d = r *, and u(B)- Z = u * ( B) for all B E B ( T ) . For B E Bl(T), we have Z C u(B)implying Iu*(B)I = l u ( B ) - Z l = s(B)-r - d = s ( B ) - r * ; for B E B2(T),we have Z

n u(B)= 0 (see (C3)) implying

J u * ( B )= J Iu(B)I = s ( B ) - 1 .

Thus, T E Tk( q *), u * E Uk(T,q *), and clearly, according to the construction K ,

G ( T, u *, Q *) = G ( T, U, Q ) - Z. 5.1.3. If I = 0 then H ( T , I(, Q )and, therefore, also G ( T ,u, Q ) are connected.

The proof is left to the reader.

5.2. For the construction K , only such Gallai trees were admitted which have a perfect matching. In what follows we describe a possible extension of K that can be applied to arbitrary Gallai trees. Let T be a Gallai tree and let B ’ C B ( T ) be a maximal matching of T ;put

5Y3

The graphs Bp E B" are "1-bricks'' (but no longer blocks) of T, and every vertex of T is contained in exactly one brick of B I . We can now proceed quite analogously to the way described in Section 3. For the resulting graphs G(T,u , Q ; H ) ,Lemmata 4.1, 4.2 and 4.3 remain valid (the proofs need not be altered), but in the proof of Lemma 4.4, essential use must be made of the hypothesis s(B)> 1. However, in many cases the statement of Theorem 3.1 is still true. Therefore, the question as to under what conditions 1-bricks can be admitted deserves to be investigated.

6. The construction K

- a generalization of

the Dirac-Haj6s construction

The following construction due to Dirac (see, e.g., [4]), who used it in some special cases, was described explicitly by Hajbs [8]; it is a powerful tool in the theory of k-critical graphs.

The Dirac-Hajos construction (DH) (slightly generalized). Let GI and G' be disjoint k-critical graphs (k 2 3 ) . For i = 1,2, let C' be an h-clique of G ' (1 S h S k - 2) and {I",Q ' } E E ( G ' ) ,where P' E V(C'), Q ' E V ( G ' ) - V(C'). Delete the edges {I",Q ' } and {P2, Q2},identify C' and Cz and connect Q ' and Q' by an edge. Then the resulting graph G is k-critical.

The construction K described in Section 3 is based on a set-assignment formalism and, at a first glance, seems to have little to do with the classical methods of constructing k-critical graphs. But that is not true. On the contrary, K is a very general form of the Dirac-Haj6s construction. The generalization lies in the fact that many k-critical graphs simultaneously act together in creating a new k-critical graph; K is more special than the Dirac-Hajbs construction in as much as the graphs participating all have the form Kk or K k - 3 7 C r r + , . Let T E Tk ( q ) , u E Uk (T,q ) , and Z = u(B)= 0. Then G = G ( T, u, Q) has no isolated vertices and H ( T , u, Q) is connected. According to the construction K (see (E2)), to every block B E BI(T)there corresponds in H ( T , u, Q ) a clique on k - s(B)vertices, namely, the clique spanned by Q - u(B)which we shall denote by C(B)(the 'complement' of B ; note that C ( B )= G [ Q - u(B)]).Clearly, by our construction rules,

n,,

H. Sachs. M. Stiebitz

594

Now consider the graphs

B =B V C(B),

B E Bi(T);

B is s(B)-critical, and C ( B )is a (k - s(B))-clique, thus B is a k-critical graph of (if s(B)= 3). V the form Kk (if s(B)# 3) or L3 Now consider the union G=

u

B.

BEBi(T)

In order to obtain G from G, we have, on the one hand, to add certain edges interconnecting the bricks B E BI(T)(how to do this is prescribed by the shape of T ) and, o n the other hand, to omit certain edges which connect, UBEBItl)B with UBEBI(T) C(B)(this is regulated by the sets u ( B *), B * E B2(T ) ) . This corresponds precisely to the steps of the Dirac-Haj6s construction; in particular, we observe that the number of edges to be omitted is just twice the number of edges to be added. Consider the simple case IBI(T)I= 2. Put Bl(T)= { B , ,B2), 6, = B , V C(BI), B2= B2V C(B2) (where Bi, B2 are k-critical). The only possibility of obtaining a Gallai tree T from B i U B , which has {Bi,B2)as a perfect matching is to add an edge {Pi,P2}(Pi E V ( B I )P2 , E V ( B 2 ) which, ) together with its end vertices, is the single brick B * (a 2-brick) contained in B2(T ) . Then, necessarily, u ( B *) = { i} (i E V(C(BI))nV ( C ( B 2 ) ) )and , the rules of the construction K say: Consider G = B, Q B2,add the edge {Pi,PJ and omit the edges {i, PI}and {i, P?). Since, trivially, the graph H = C ( B i ) U C ( B J is in sk, by Theorem 3.1 the graph G obtained by the above construction is k-critical. This is just the classical Dirac-Haj6s construction.

7. Construction of critical graphs with given high-vertex graphs

In this section, we shall use the construction K to prove the following theorem. Theorem 7.1. A graph H * i s the high-oertex graph of a k-critical graph which has low vertices (i) if and (ii) only if H * E s k (k 2 4).

Observation 1.3 immediately implies that for each k-critical graph G with low vertices, H ( G )E sk,proving assertion (ii) of Theorem 7.1. Next we shall establish assertion (i) of Theorem 7.1 for connected graphs H * . If H * = K , then (i) follows from Theorem 1.3: thus we may assume that H * has an edge. We shall prove the following theorem.


595

Theorem 7.2. Let k 3 4 and w L 1 be given integers and let H * E sk be a connected graph with 1 V ( H * ) J> 1. Then there is a k-critical graph G satisfying

(I) H ( G ) = H * ; (11) L(G) is a non-empty Gallai tree having a perfect matching; (111) each vertex o f H(G) is adjacent to at least w vertices of L(G). If k 3 5 , then, in particular, G can be chosen so that L(G) E T i - , (see Definition

1.4). Proof of Theorem 7.2. Let V ( H * )= Q = {1,2,. . . ,q } , q > 1. The underlying idea. H * is considered as resulting from the overlapping of sufficiently many k-cliques B = B V C ( B )with B = Kk-2 and C ( B )= K 2 ,where the 2-cliques C ( B )cover the 2-cliques (i.e., edges) of H * (each edge of H * may be covered by many C ( B ) )and the ( k - 2)-bricks B are the elements of a perfect matching B , ( T ) of a Gallai tree T E Tk(4). The bricks B in T are interconnected by 2- or 3-bricks (if k 3 5, 2-bricks suffice). The mapping u E uk ( T ,q ) can be so defined that H ( T , u, Q )= H * ; thus Theorem 3.1 becomes applicable. Using sufficiently many B, the valencies of the vertices of H ( T , u, Q ) with respect to G = G ( T , u, Q )can be made 3 k, thus H ( G ) = H ( T , u, Q )= H * . The details of the proof. We shall now simultaneously describe two different constructions C', C" where C' is possible for all k 2 4 and C" is possible for k 3 5 only. Let L * denote the line graph of H ( V ( L * )= E ( H * ) , E ( L *) = {{e,f})e,f E E ( H * ) & J en f l = l } ) and let D* be a spanning tree of L * . The constructions C', C" (' and " always refer to C', C" respectively). PART I. Construction of Gallai trees T ' , T", Corresponding to each e E E ( H * ) = V ( D * ) ,let FL(e, m ) ( m = 21 + 1, 1 2 1) and Fi(e, m ) ( m L 1 ) denote special Gallai trees which are defined as follows: (a') F;(e, m ) consists of m = 21 1 blocks B ; ,B;,. . . ,BA which are ( k - 2)cliques and one block B' = C, ; every vertex of B' is contained in exactly one of BI, B i , . . . ,B h (Fig. 7.1). (a") F'L(e, m )consists of m blocks B'i, B;,. . . , B L which are ( k - 2)-bricks and m - 1 blocks I?';, By,. . . ,B;-, which are 2-bricks interconnecting the BY,BY,. . . ,BL in such a way that Fll(e, m ) E T i - , (Fig. 7.2).

+

Fig. 7.1.

596

H. Sachs, M . Sriebitz

Fig. 7.2.

Gallai trees assigned to distinct edges of H * are disjoint. Let us call a vertex of a Gallai tree which is not a cut vertex, a boundary vertex. Clearly, F;(e, m) has m (k - 3) and F”(e,m ) has m(k - 4) + 2 boundary vertices’ (these are precisely the vertices of valency k -3). Thus, for large enough m, the number of boundary vertices of F;(e, m ) (k 2 4) and F‘L(e, rn) (k 3 5), resp., will exceed any prescribed bound. Now the Gallai trees F;(e, m ) or F;(e, m ) are interconnected by additional 2-bricks (i.e., edges) &’; or &, resp., according to the following conditions: (A) A 2-brick BLf or B% (exactly one) between F;(e, m ) and F;(f, m),or between FIl(e,m) and F ; ( f , m ) , resp., is introduced if and only if e and f, considered as vertices of L * , are adjacent in D*. (B) BLf (or B:i) connects a boundary vertex of F;(e, m ) (or F;(e, m))with a boundary vertex of FLU, m ) (or of F;U, m),resp.). (C) The I?:, are pairwise vertex disjoint, and so are the &. It is easy to see that, for sufficiently large m, these conditions can always be satisfied. The graphs G & Gb’ resulting from this construction are Gallai trees since, if each of the FL (or F ; , resp.) is contracted to a single vertex, then G,!,as well as G ; is transformed into a graph isomorphic to D * (which is a tree). We shall therefore write T’, T” rather than GA, Gbr, resp. Clearly, the set of all of the (k -2)-bricks B : (or BL)of all of the FL(e, m ) (or F;(e, m)) is a perfect matching B 1 ( T ’ )of T’ (or B,(T”) of T”, resp.). B,(T’) consists of the 3-bricks B’ of all of the F;(e, m ) and of the 2-bricks BLf ; Br(T”) consists of 2-bricks only. Thus T’E Tk(9) and T”E Tk(q)n T i - , . PART11. Definition of u‘ E Uk(T‘,4) and u’’€vk (T”,4). Put b = k - 2, r = k - 4 ; then b - r = q - 2 2 0 . (1) To each of the b-bricks BA(h = 1,2,. . .,m )of F;(e, m) which is contained in B , ( T ’ ) the (b - r)-set u’(BL)= Q - e is assigned; likewise, to each of the b-bricks B : (h = 1,2, * ., m ) of F;(e, m ) the (b - r)-set u”(B:)= Q - e is assigned.

‘

Or more,


597

(2) To the 3-brick B' of FL(e, i n ) (which belongs to B,(T'))the 2-set u'(B') = e is assigned; to each of the 2-bricks B" of F'((e,rn) an arbitrary one of the two 1-subsets of e is assigned (i.e., for e = {i,j}, u"(B")= {i} or = G}). (3) To a 2-brick BLf or B:i the 1-set u'(B$) = u"(B2) = e n f is assigned. Now it is easy to check that u' E uk ( T ' ,q), u" E uk (T", q ) . Thus T ' , Q and u ' , or T", Q and u", resp., have all the properties required for executing the construction K which now yields the graphs G'= G ( T ' ,u ' , Q) and G" = G ( T " , u " , Q )where, due to the definitions of u ' and u", H ( T , u ' , Q ) = H ( T " ,u", Q ) = H*. Since, by hypothesis, H * E sk,we conclude from Theorem 3.1 that G' and G" are k-edge-critical. Further, because H * is connected, neither of G',G" has isolated vertices; thus G' and G" are k-critical. All vertices of T' (or T")in G'(or in G")have valency k - 1, i.e., they are low vertices. If in is chosen large enough, then every vertex of G ' [ Q ] = H * is connected with edges to more than w vertices of T', and the analogue holds for G". In particular, this implies that for w k - 1 the high-vertex graphs of G' and G" are G ' [ Q ]and G " [ Q ] resp., , both isomorphic to H * . Thus

This proves Theorem 7.2.

In order to finish the proof of Theorem 7.1 we shall now show by induction over the number c of components of H * assertion (i) of Theorem 7.1 holds for disconnected graphs, too. Clearly, by Theorem 7.2, Theorem 7.1 is true if c = 1. Suppose that Theorem 7.l(i) is true for all graphs H * E Sk with less than c0 components (co3 2). Let fi* E sk have precisely co components, say, fi?,fi?,. . . ,ti?(,. Denote the subgraph of fi* consisting of the components fi?,H : , . . . , fi?"-,by H : and put fi;(,= HT;then, clearly, HT, HT E Sk. By the induction hypothesis, there are k-critical graphs G I , G 2 having low vertices and satisfying H(GI)= HT, H(G2)= HT. We may assume that G2has a low vertex P2 which is adjacent to no high vertex. For HT = Kl this follows from Theorem 1.3(2), and for graphs HT that have more than one vertex it follows from the proof of Theorem 7.2, construction C' (any vertex P of T' with val(P : T ' ) = k - 1 may be taken for P2).Let PI be a high vertex of G I and PI a low vertex of GI adjacent to PI,and let P ; be any vertex of G2 adjacent to P 2 . Now apply a simple Dirac-Haj6s construction to G I and G2.Identify P I and Pz, omit the edges {PI,PI} and { P 2 ,P;}, and add the edge { P j , P;}. Clearly, this results in a k-critical graph G with H ( G ) = H * . Theorem 7.1 is thus proved.

H. Sachs, M. Stiebitz

SO8

Example (Theorem 7.2, Construction C'): H* = 7.3).

, k = 4, w = 3 (see Fig.

}

Q -

T =

-

H(G)

T'

G = G ' i s 4-critical

Fig. 7.3.

References [I] R.L. Brooks, On colouring the nodes of a network, Proc. Cambridge Phil. SOC.37 (1941) 194-1 97. [2] G.A. Dirac, A property of 4-chromatic graphs and some remarks on critical graphs, J. London Math. SOC.27 (1952) 85-92. (31 G.A. Dirac, Some theorems on abstract graphs, Proc. London Math. SOC.2 (1952) 69-81. [4] G.A. Dirac, A theorem of R.L. Brooks and a conjecture of H. Hadwiger, Proc. London Math. SOC.7 (1957) 161-195. [5] D. Greenwell and L. LovBsz, Applications of product colouring, Acta Math. Acad. Sci. Hungar. 25 (1974) 335-340. [6] T. Gallai, Kritische Graphen I, Magyar Tud. Akad. Mat. Kutatb Int. Kozl. 8 (1963) 165-192. [7] T. Gallai, Kritische Graphen 11, Magyar Tud. Akad. Mat. Kutatd Int. KO& 8 (1963) 373-395. [8] G. Hajbs, Uber eine Konstruktion nicht n-farbbarer Graphen, Wiss. 2. Univ. Halle, Math.Nat. X (1961) 113-114. (91 F. Harary, Graph Theory (Addison-Wesley, Reading, MA, 1969). [ 101 L. LovBsz, Combinatorial Problems and Exercises (Akad. Kiadb, Budapest, 1979). [ 111 V. Miiller, On colorable critical and uniquely colorable critical graphs, in: Recent Advances in Graph Theory (Proc. 2nd Czechoslovak Syrnpos., Prague, 1974) pp. 385-386. [12] M. Simonovits, On colour-critical graphs, Studia Sci. Math. Hungar. 7 (1972) 67-81. [t3] B. Toft, Some contributions to the theory of colour-critical graphs, Doct. Diss., Univ. of London, 1971; Var. Publ. Ser. 14. [ 141 H.J. Voss, Graphs with prescribed maximal subgraphs and critical chromatic graphs, Comment. Math. Univ. Carolinae 18 (1977) 129-142.


GRAPHS AND INVERSE SEMIGROUPS S.C. SHEE and H.H. TEH National University of Singapore, Singapore

1. Introduction

Numerous papers have been published on graphs and groups and a new topic, group graphs, has also been studied. However, extremely few papers have dealt with graphs and semigroups. We present here some results involving graphs and inverse semigroups in the hope that it will stimulate a wider study of graphs and semigroups. In this paper, a graph D I- ( G , E ) means a digraph such that the vertex-set V ( D )= G and the arc-family A ( D )= E C G X G. An inverse semigroup is an ordered pair ( S , . ), where S is a non-empty set and ’ is a binary operation such that: (i) ‘ - ’ is associative, and (ii) for every a E S, there exists uniquely an a * E S such that aa * a = a and ‘ a

a*aa* = a * .

Obviously a group is an inverse semigroup. Note that the elements aa * and a * a are idempotent, and an inverse semigroup is a group iff it has only one idempotent element. A very important class of inverse semigroups is obtained as follows. Take a non-empty set X. Let A, B C X such that I A 1 = IB I. Then any one-one mapping LY from A onto B is called a partial mapping of X with d o m ( a ) = Aand rank(a) =B. mapping (Y is a special relation on X, i.e., (Y C X X X . It is obvious that the collection

rx= { a I a

is a partial mapping of XI

forms an inverse semigroup under multiplication of relations. In 1954, Preston showed the following.

Every inverse semigroup (S;) is embeddable into the inverse semigroup (Ts, - ). This is a very important theorem, since, because of it, we need only study for a set X to be able to study inverse inverse subsemigroups of (G;) semigroups. To relate Tx to graphs, let D = (G, E ) be a graph. 599

S.C. Shee, H.H. Teh

600

An a of f, is called a partial automorphism of the graph D iff for all a , b E dom(a),

(a,b)EEe(aa,ba)EE. Then it is easily verified that the collection

I

Q = { a a is a partial automorphism of D }

is an inverse subsemigroup of I', and hence itself an inverse semigroup. We observe that for every K C G, the partial identity mapping IK E Q, and that for all a, p E if a E Q and p C a, then p E Q. In general, the inverse semigroup Q is rather large, and its structure is difficult to analyse. We shall explore some inverse subsemigroup of Q, which possesses interesting properties. To do this, we introduce the concept of an ideal of a graph.

r,

2. The ideal inverse semigroup of a graph

A subset K (possibly empty) of G is called an ideal of the graph D = (G, E ) iff for every u, u E G, if u E K and there is a dipath from u to u, then u E K. It is clear from this definition that the collection of all ideals of a graph forms a complete distributive lattice under the inclusion relation. It can also be shown that the collection

I

I(G.E)= {a E Q dom(a) and rank(a) are ideals} is an inverse semigroup, called the ideal inverse semigroup of the graph D. We have the following: Theorem 1. Every inuerse semigroup is embeddable into the ideal inverse semigroup of some non -empty graph. Sketch of proof. A sketch of the proof is as follows: Let ( S ; ) be an inverse semigroup. By Preston's result we can identify S as a subset of f,,that is, elements of S are identified as subsets of S x S. We then construct a graph D = ( S , E ) , where, for a, p E S, (a,/3) E E iff a C p. For each a E S, consider ideals of the form S Q * a -' and S a - l . a, where a-' is the inverse relation of a and the operation ' * ' is the multiplication of relations. Define 4u : S . ( ~ * a - ~ + S * abysettingy4, ~'.a = y . a f 0 r e v e r y r E S . a . a '.Then 4" is a partial automorphism of D, whose domain and range are ideals. Define 4 : S R = {r& a E S } by setting a4 = &. The mapping 4 turns out to be an isomorphism from S onto 0. Hence the result.

I

Graphs and inuerse semigroups

60 I

Let D = (G, E ) be a graph. Regard N = {1,2,3,. . . } as a set of colours. Then any mapping t+h from E into N is.called an arc-colouring map of the graph D. The graph D together with an arc-colouring map is called an arc-coloured graph and will be denoted by D * = (G, E, $). Now let us consider an arc-coloured graph D * = (G, E, 4). An a E Q is called a colour partial automorphism of D* iff for every a, b E dom(a) and (a,b ) E E, (a, b ) $ = (aa, ba)$. We notice that the collection

I

I(G.E.IL) = {a E I(G.E) a is a colour partial automorphism of D *}

is an inverse semigroup.

Theorem 2. Let D = (G, E ) be a graph. Then for every arc-colouring map $, I(G.E.JI)is an inverse semigroup whose idempotent elements form a complete distributive lattice iosmorphic to the complete distributive lattice of the ideals of D. In particular, the idempotent elements of I(G,E,IL, form a Boolean algebra i f D is locally connected. In view of Theorem 2 we raise the question: Given an inverse semigroup ( S , - ) can we find an arc-coloured graph D * = (G, E, +) such that I(G.E.*)

= S?

We have so far been unsuccessful in constructing such an arc-coloured graph for the general case. But if S satisfies some additional conditions, then we can say something, as indicated in the following theorem. Theorem 3. Let ( S , . ) be a finite inverse semigroup such that each element of S is idempotent and that the idempotent elements form a distributive lattice under the usual partial ordering el s e2 ($ el .e2= e l . Then there exists an arc-coloured graph D * = (G,E, t+h) such that I ( G ~ E . * )= S.

To construct D * = (G, E, t+h), we first note that S, satisfying the prescribed C), where 2’= { S 6 I S € A } is a condition, is isomorphic to a lattice (3, collection of sets, each containing at least two elements. Let G = u(s6 E A ) . For every x E G, let S, be the smallest member of 3’ containing x. For every a, b E G, define (a,b ) E E ($ Sb C S.. Then I(G,E,$L) = S, where t+h is a one-one mapping.

IS

3. The principal ideal inverse semigroup of a graph

An ideal K of a graph D = (G, E ) is called a principal ideal iff either K is

S. C. Shee. H. H. Teh

602

empty or there exists a u E K such that for every w E K,there is a dipath from u to w. Let

I

P(c.E)= { a E Q dom(a) and rank(a) are principal ideals}. In general, P(G.E)is not an inverse semigroup. But we have the following. Theorem 4. P(G.E) is an inverse semigroup iff the intersection of every two principal ideals of D is again a principal ideal. In this case, for every arc-colouring map IJ,

is an inverse subsemigroup of P(G.Ej, containing all the idempotent elemerits of P,G.€). Furthermore if the intersection of every two non-empty principal ideals is again a non-empty principal ideal, then

P&) = PG.E \{empty map1 is an inverse semigroup. If (G, E ) satisfies some conditions, then P;",.,, can be quite interesting, as illustrated in the following theorems. Theorem 5. Let ( G ,E ) be a tree such that the out-degree of every vertex is at most one. Then for every arc-colouring map q?, P;CG.E,*) is a full inverse subsemigroup of P&.E).Conversely, for every full inverse subsemigroup S of P&.E), there exists an arc-colouring map q? such that S = P(&.J). Theorem 6. Let ( G ,E) be any given graph such that the intersection of every two non -empty principal ideals is again a non-empty principal ideal. Let S be any full inverse subsemigroup of P&) such that S contains CY whenever a .(Y = (Y -' a. Then there exists an arc-colouring map q? such that P&.E.+)= S. A graphical faithful representation of an inverse semigroup ( S , - ) is a graph D = (G, E ) such that (i) P& = S; (ii) for every full inverse subsemigroup H of S, there exists an arc-colouring map q? such that P&.*)= H, and conversely. To end this paper we state the following.

Theorem 7. Every finitely generated Clifford inverse semigroup has a graphical faithful representation.

Graphs and inverse semigroups

References [ I ] A.H. Clifford and G.B. Preston, The algebraic theory of semigroups, Amer. Math. SOC.(1961). [2] G.B. Preston, Representations of inverse semigroups, J. London Math. SOC.29 (1954). 131 C. Berge, Graphs and Hypergraphs (North-Holland, Amsterdam, 1973). [J] H.H. Teh and S.C. Shee, Algebraic Theory of Graphs (Lee Kong Chian Institute of Mathematics and Computer Science, Nanyang University, Singapore, 1976). [S] H.H. Teh and S.C. Shee, Colour graphs and inverse semigroups, SEA Bull. Math. 3 (1979).



COLORATIONS EXTREMES DANS LES HYPERGRAPHES F. STERBOUL and D. WERTHEIMER Universite‘ de Lille 1, France

Conditions are given on the number of edges needed for a r-uniform hypergraph to have a &-coloring, where all edges are either r-colored or unicolored.

1. Introduction 1.I. Dkfinitions

Nous appelons r-graphe un hypergraphe H = ( X , 8)oG X est I’ensemble des sommets et 8 I’ensemble des arCtes. Les arCtes sont des sous-ensembles de cardinal r de X ([l]). Le cardinal de X est l’ordre de H, et le cardinal de 8 la taille de H. Une k-coloration est une application g de X sur un ensemble de cardinal k (ensemble des couleurs). Une partie Y de X est dite h-colorCe si I g ( Y)l = h. 1.2. Position du problime dans la litte‘rature

Considtrons un 3-graphe H . Si pour toute 2-coloration il existe au moins une arCte de H 1-colorCe, alors H posskde au moins 7 aretes. En gCnCral, si H est un r-graphe et si, pour toute k-coloration, il existe au moins une ar$te 1-colorCe, on peut chercher le nombre minimal d’arCtes de H. Ce problttme est trks classique, et liC B la propriCtC B et au nombre chromatique ((41). ConsidCrons B nouveau un 3-graphe H. Si, pour toute 3-coloration, il existe au moins une arkte de H 3-colorCe, alors H posskde au moins n ( n - 2)/3 arCtes, oc n est I’ordre de H. Ce type de problkme, plus recent que le prCcCdent, a CtC 6tudiC dans plusieurs articles par I’un des auteurs (une bibliographie est donnte dans [6]). En ce qui concerne les 3-graphes, il restait donc A considher le problkme suivant. Soit H un 3-graphe d’ordre n. Si, pour toute 3-coloration, il existe dans H une ar2te 2-colorCe, quel est le nombre minimal d’arites de H ? De faGon Cquivalente, quel est le plus grand entier rn = f ( n ) tel que tout 3-graphe d’ordre n et de taille m possttde une 3- coloration dans laquelle toute artte est ou bien 3-colorie ou bien 1-colorCe? 605

606

F.Sterboul. D.Werrheimer

Ce problkme admet plusieurs gtntralisations possibles; nous choisissons d’ttudier ici une des plus naturelles. 1.3. La fonction R

Definitions. Soit H un r-graphe, on dira qu’une arete E de H est ma1 colore‘e dans une k-coloration si E est h-colorCe, avec 1< h < r. On dira qu’une k-coloration est extrtme si aucune arete de H n’y est ma1 colorCe (donc toute argte est soit r-colorCe, soit 1-colorte). O n verra dans la Section 2 qu’on ne perd pas en gkntralitt en supposant de plus que H est connexe (quand k = r). Dhfinition. R (n, r, k ) est le plus grand entier m tel que tout r-graphe H connexe d’ordre n et de taille m posskde une k-coloration extrgme. On prendra toujours n a k

3 r 3 3.

2. Influence de la connexite (k = r ) Proposition 2.1. Pour que le k-graphe H n’aitpas de k-coloration extreme, il faut et il sufit que le nombre de ses composantes connexes soit infirieur Ci k et que chaque composante n ’ait pas de k -coloration extreme. Demonstration. Soit H sans k-coloration extreme. Alors le nombre de ses composantes est infCrieur 1 k car sinon une k-coloration oh chaque composante est unicolore fournirait une contradiction. D’autre part, si une k -coloration extreme existait dans une composante, elle pourrait s’ttendre 1 tout H en prenant les autres composantes unicolores. RCciproquement, supposons que les composantes d’un hypergraphe H vtrifie les hypoth2ses de la proposition et considCrons une k-coloration quelconque. Comme le nombre des composantes est infCrieur ti k, au moins une des composantes reGoit h couleurs, avec h 5 2. Si h = k, cette composante contient au moins une arete ma1 colorCe d’aprks I’hypothcse. Si h < k,cette composante contient au moins une arete ma1 colorCe, ceci rCsulte de la connexitC de cette composante. D’ob H n’a pas de k-coloration extreme.

3. Le cas k = r Proposition 3.1. O n a

Colorations exrrimes d a m les hypergraphes

R ( n , k , k ) S nL-m 2]

607

s i n = 2 o u 3(modk-2), sauf n = k

+ 1s 5,

et

R (n, k, k ) S

1-

n -2

+1

dans les autres cas.

Demonstration. Posons n - 2 = s(k - 2) + t, avec 0 S t S k - 3. Si s = 1 et k > 3, il est facile de construire un k-graphe connexe d’ordre n et de taille 3 sans k-coloration extreme. On peut donc supposer s 2 2. Soit X union disjointe des ensemblesA,B,,..., B,,D a v e c ( A ) = 2 , ( B i I = k - 2 ( i = 1,..., s ) , l D l = t . Soit 9 = { A U Bi l i = 1,..., s}. Soit % = 9 U { E } avec E CX, / E l = k, E n A = 0 , IE n D I = m i n { t ; l } . Le k-graphe H = (X, ‘8) ne poss&depas de k-coloration extreme. Supposons, en effet, qu’une telle coloration existe. Si A est 1-colorC, alms toutes les ar2tes de 9 sont 1-calories, et X est au plus ( t + l)-colort: contradiction. Si A est 2-colorC, par exemple par les couleurs 1 et 2, les ensembles Bi sont (k -2) colorCs et les couleurs 1 et 2 n’y figurent pas, donc une des deux couleurs 1 et 2 ne figure pas dans I’arete E qui est alors h-colorCe avec h = k - 2 ou k - 1: contradiction. Si t = 0 ou 1, H est connexe d’oh la premihre inCgalitC; dans les autres cas, il faut ajouter une arkte B H pour assurer sa connexitk. 0 Notations. M ( L , K ) dCsigne une matrice M dont I’ensemble des indices de lignes est L et l’ensemble des indices de colonnes est K. Si L ’ C L et M ’ C M , M ( L ’ , K ’ ) est la sous-matrice de M correspondant B ces sous-ensembles d’indices. De meme, V ( L )dCsigne une matrice colonne ayant pour ensemble d’indices de lignes L, et si L ’ C L, V ( L ’ )est la sous-matrice correspondante. ThCorkme 3.2. Soit k = p a , puissance de nombre premier. Soit H un k-graphe connexe d’ordre n et de taille m avec m S (n - 2)/(k - 2), alors H posstde une k -coloration extrime. DCmonstration. Soit H = ( X , . % ) vCrifiant les hypothhes. Soit M ( L , K ) une matrice B valeurs dans le corps GF(k), avec I L I = k - 2 et IK I = k. On suppose de plus que

I

Ker M = { V V E (GF(k))k,MV = 0}, est de dimension 2 et est engendrC par les deux vecteurs V , et V2,oh V , a toutes ses composantes Cgales B 1 et oh V z a pour composantes les k ClCments distincts de GF(k) dans un ordre arbitraire. Notons que M existe. Tout vecteur de Ker M a ou bien toutes ses composantes Cgales ou bien toutes ses composantes

F. Sterboul, D. Wertheimer

608

distinctes. Soit A (8 X L, X) la matrice ayant comme ensemble d’indices de lignes le produit cartesien %‘ X L et comme ensemble d’indices de colonnes X , definie ainsi: pour toute ar6te E E %‘, soit f E une bijection quelconque de E sur K ; on prend A de fagon que, pour tout E E %‘, A ({Elx L, {Y 1) = M ( L ,{fE (y)))

pour tout y E E,

A ( { E )x L, {Y )) = 0

pour tout y E X - E.

A a donc m (k - 2) lignes et n colonnes. D’oii rang A s m (k - 2) S n - 2 et dimKerA 2 2 . Soit V ( X ) E K e r A . D’aprks la dtfinition de A, on a V ( E ) E Ker M, pour toute artte E de H. Donc, en considkrant V ( { x ) )comme la couleur associee au sommet x, V(X) est une coloration de H oh toute artte E est k-colorCe ou I-colorte. Comme H est connexe, V ( X )est soit une k-coloration soit une 1-coloration. Comme dim Ker A 3 2, et les 1-colorations forment un sous-espace de dimension 1, il existe bien un vecteur V ( X ) , donnant une k-coloration extreme de H. 0 Compte tenu de la Proposition 3.1, on en deduit la suivante.

Corollaire 3.3. Si k = pa, puissance de premier, alors: R(n, k, k)=

n-2

s i n = 2 ou 3 (mud k - 2 ) , saufn = k

+135,

Corollaire 3.4 R(n,3,3)= n - 2 = 2, R (5,4,4)

pour tout n 3 3 , R (n, 4,4)=

1 7 1 n -2

pour tout n 2 6 .

4. Le cas n - k petit Le problkme semble plus difficile quand le nombre des couleurs est suptrieur B la cardinalite des aretes. Ntanmoins, on peut obtenir encore des valeurs exactes de R quand la difference n - k est petite.

Definition. Soit H = ( X , %‘)un r-graphe d’ordre n. On dit que H est un recouvrement des paires de X si toute paire de sommets distincts est contenue dam au moins une ar6te de H.

Colorations extremes dans les hypergraphes

609

On note cov(2, r, n ) la taille minimale d’un tel recouvrement. Notons qu’un systgme de Steiner S(2, r, n ) est nicessairement un recouvrement minimal (nombreuses rCfCrences dans [3] et [5]). Proposition 4.1. Soit H un r-graphe d’ordre n = k + 1, H ne posstde pas de k-coloration extrime si et seulement si H est un recouvrement des paires. Demonstration. I1 s u s t de faire correspondre ? toute i paire de sommets la k-coloration oh ces sommets sont de mCme couleur. 0 Corollaire 4.2. R (k

+ 1, r, k ) = cov(2, r, k + 1) - 1.

Lemme 4.3. Soit H u n r-graphe d ’ordre n = k + 2, avec r > 3. Pour que H soit sans k-coloration extrime, il faut et il sufit qu’il existe un sommet c tel que les arites de H recouvrent les paires de l’ensemble X -{c}, ou X est l’ensemble des sommets de H. Demonstration. La condition est suffisante. En efiet, pour toute k-coloration, il existe au moins 2 (et au plus 3) sommets de mCme couleur dans X - { c } et une arCte de H contenant ces 2 sommets ne sera ni k-colorie ni 1-colorie (car r > 3). Riciproquement, supposons que H n’ait pas de k-coloration extrCme. Si H est un recouvrement des paires de X , le risultat est dkmontri. Sinon, soit {a, b } une paire non recouverte. E n considbrant les colorations oa a et b sont de couleur 1, et x . et y sont de couleur 2 avec { x ,y } C X -{a, b } , on voit que les arftes de H recouvrent toute paire { x , y } de ce type. Supposons qu’une paire {b,d } , d E X - {a, b } , ne soit pas recouverte par une arCte de H (sinon, on peut prendre c = b ) . Alors, en considirant la k-coloration oh a, b, d sont de couleur 1, on voit que la paire { a , d } est recouverte. D’autre part, en considirant la k-coloration oh b et d sont de couleur 1, et a et z sont de couleur 2, on obtient que toute paire {a, z } , z E X - { a , b, d } est recouverte. On peut donc prendre c=a. 0 Corollaire 4.4. Le plus grand entier m tel que tout r-graphe H d’ordre n = k + 2, avec r > 3 , et de taille m posstde une k-coloration extrime est m = C O V (T,~k, + 1) - 1. DCmonstration. I1 suffit d’utiliser le lemme pricident, en remarquant qu’un r-graphe sur X dont les arCtes recouvrent les paires de X -{c} a au moins autant d’arCtes qu’un r-graphe sur X - {c} qui est un recouvrement des paires de X - { c } (pour toute arCte qui contient c, il suffit de remplacer c par un sommet de X - { c } ) .

F. Srerboul, D.Werrheirner

610

Corollaire 4.5. Pour r > 3, R ( k +2,r, k ) = C O V (r,~k,

+ 1)

ou C O V (r,~k,

+ 1)-

1.

Dimonstration. La difference avec le corollaire prickdent rCside dans l’hypothkse de connexitC. 0 Lemme 4.6. Soit H un 3-graphe d’ordre n = k + 2. Pour que H soit sans k-coloration extr2me, il faut et il sufit qu’il existe urt sommet c tel que les arktes de H recouvrent les paires de X - { c }et que, pour toute ar2te E de H, il existe une arkte F de H telle que 1 E f l F 1 = 2 . Demonstration. Toute la deuxitme partie de la dkmonstration du Lemme 4.3 reste valable. Le reste vient de la consideration des k-colorations oh 3 sommets sont de couleur 1. Corollaire 4.7. Soit H un 3-graphe d’ordre n = k + 2 et de taille m. Si m < k (k + 1)/5, alors H posstde une k-coloration extr2me. Cette borne est la meilleure possible pour une infinite‘ de valeurs de k. Dhonstration. Soit H = ( X , $) un 3-graphe d’ordre n = k + 2 sans k coloration extreme. Soit G le graphe biparti dont une classe, notie Y, est l’ensemble des paires de X , et dont I’autre classe est 8, les arites de G correspondant Q la relation d’inclusion. Le Lemme 4.6 implique que Y a au plus k + 1 sommets isolCs et que les autres composantes connexes de G ont au moins 6 arCtes chacune. (Soit p le nombre de ces composantes non rCduites Q un point.) Soit M le nombre d’arttes de G, on a donc

1 Y I + 18 1 - p - (k + 1)a18 I - p + k(k + 1)/2, et 3 )27 1 = M 3 6p. I1 en risulte 127 13 k ( k + 1)/5. D’autre part, si k + I = 0 ou 1 (mod 5 ) et k # 4, on peut construire un 3-graphe H dTordre k + 2 et de taille M

2

k(k + 1)/5 sans k-coloration. En effet, Bermond et Schonheim [2] ont montri qu’il existe dam ce cas une partition des arktes du graphe complet d’ordre k + 1 en graphes Gi (1 S i C k(k + 1)/10)isomorphes Q Q, qui est le graphe d’ordre 4 constituk de deux triangles ayant une arkte commune. On obtient alors un 3-graphe H vCrifiant les conditions du Lemme 4.6 en remplacant dans chaque graphe Gi les deux triangles par les deux triplets correspondants, et en ajoutant un sommet c isolC. Corollaire 4.8. R (k + 2,3, k ) < k (k + 1)/5; cette borne est la meilleure possible a une unite‘ pr2s pour une infinite‘ de valeurs de k.

Colorations exrrimes dans les hypergraphes

On peut obtenir des rCsultats analogues pour n - k mCthodes deviennent rapidement fastidieuses.

61 1 = 3,

etc.; mais ces

5. Le cas general Proposition 5.1. R(n,r, k ) S ( n / ( r -2))+c(r, k ) . Demonstration. Supposons pour simplifier que n - k - 1 = s ( r - 2), s entier. Soit X union disjointe de A , B , ,. . . , B , , D avec ( A I = 2 , IBi 1 = r - 2 (1 s i s s), ID I = k - 1. Soit 9 = { A U B,11 S i S s} et H le r-graphe dont I’ensemble des aretes est

9U { E IIE

1 = r, E C A

U D}.

On vCrifie aisCment que H n’a pas de k-coloration extreme. Plusieurs variantes sont possibles, et la valeur de c ( r , k ) que fournit la construction prCcCdente peut Ctre amCliorCe. Mais il est inutile de prCciser davantage tant que la limite de R (n, r, k ) / n , quand n croit, n’est pas connue. En effet, la seule borne infCrieure gCnCrale pour R que nous obtenions est la suivante, qui rCsulte de la seule hypothbse de connexitk. Proposition 5.2. R (n, r, k ) 2 ( n - l)/(r - 1).

6. Problkmes ouverts et conjectures

Le problbme gtntral paraissant hors d’atteinte, nous donnons quelques Ctapes possibles. 6.1. Determiner la valeur exacte de R (n, k, k ) pour k puissance de premier (voir Corollaire 3.3).

6.2. Peut-on amknager la mCthode algCbrique du ThCorbme 3.2 quand k n’est pas une puissance de premier? Conjecture 6.3. R (n,k, k ) 3 [(n- 2)/(k - 2)J. Conjecture 6.4. R (n,r, k ) 3 n / ( r - 2)

+ d ( r ,k ) .

612

F. Sterboul, D. Wertheimer

References [ I ] C. Berge, Graphes et Hypergraphes (Dunod, Paris, 1970). (21 J.C. Bermond and J. Schonheim, G-decomposition of K. where G has four vertices or less, Discrete Math. 19 (1977) 113-120. 131 J. Doyen and A. Rosa, A bibiography and survey of Steiner systems, Bollet. U.M.I. 7 (1973) 392-41 9. 141 P. Erdos, On a combinatorial problem, Nordisk Math. Tidskr. 11 (1963) 5-10. 151 A. Schrijver (ed.), Packing and covering in combinatorics, Math. Centre Tracts 106, Mathematisch Centrum, Amsterdam (1979). 161 F. Sterboul, Un problkme de coloration aux aspects varies, Colt. Franco-Canadien de Combinatoire, Montreal, 1979, Ann. Discrete Math. (North-Holland, Amsterdam) h paraitre.


EVERY t-IRREDUCIBLE PARTIAL ORDER IS A SUBORDER OF A t 1-IRREDUCIBLE PARTIAL ORDER

+

William 'T. TROTTER, Jr.* and Jeffrey A. ROSS*' Department of Mathematics and Statistics, University of South Carolina, Columbia. South Carolina ZYZOX, USA The dimension of a partial order (X, ) is the least integer t for which there exist linear extensions X i , X>,...,Xc of X so that xi < x 2 in X if and only if x , < x , in X , for each i = 1.2,. . . , t. For an integer t 3 2 , a partial order is said to be t-irreducible if it has dimention I and every proper nonempty subpartial order has dimension less than t. We answer a natural question concerning dimension by proving that for each t 2. every f-irreducible partial order is a subpartial order of a t + I-irreducible partial order.

1. Introduction

I n this paper, we answer one of the most natural questions that can be asked concerning the dimension of partially ordered sets. Utilizing a construction whose origins lie in chromatic graph theory, we prove that for each t 2 2 , every t-irreducible partial order can be embedded in a t + I-irreducible partial order. The construction also relies on two fundamental concepts in dimension theory: the structure of nonforced pairs and realizers of irreducible partial orders. Nevertheless, for the reader who is familiar with little more than the most basic concepts concerning partial orders, the paper is entirely self contained, and it is only necessary to present a few definitions and preliminary lemmas before proceeding to the principal result. The reader who desires additional background material on the dimensional theory of posets is referred to the survey article [4] which also contains an extensive bibliography of papers on this subject. A partially ordered set (poset) is a set X equipped with a reflexive antisymmetric and transitive binary relation < . If x I , x 2 € X , x I sf x, and x. $ x I , then x I and x. are incomparable and we write x I IIx?. For each point x I E X , we let D,, ( x l ) = {xz E X : x2 < xl}, V,y(xl)= {x2E X : x I < xJ, and I , ( x , ) = {xzE X : x , Ilx2}.We let l x = {(xl, x,) : x , Ilx2}.We say X is a linear order if I, = (rl. * Research supported in part by NSF Grant ISP-8011351. ' Research of this author supported in part by a Grant from the USC Research and Productive Scholarship Fund. 613

W. T. Trotter,Jr., J.A. Ross

614

If XI and X z are partial orders on the same set and xI< x2 in X z whenever x1 < x2 in XI, we say X 2 is an extension of X I ;if X 2 is a linear order and an extension of XI, X 2 is called a linear extension of X I . Dushnik and Miller [ l ] defined the dimension of a poset X , denoted dim(X), as the least positive integer t for which there exist f linear extensions XI,X2,. . .,X , of X such that x , s x2 in X if and only if x1S x2 in X i for each i = 1,2,. . . ,t. If XI and X , are posets and the point set of X I is a subset of the point set of Xz, the poset X I is called a subposet of X 2 when xId x 2 in X I if and only if x Is xz in X z for all xl, x2E X , . For each point x E X,we let X - {x} denote the subposet of X whose point set contains all points in X except x. Of course, dim(X - {x}) G d i m X for each x EX. For an integer t 2 2 , a poset X is t-irreducible if dim(X) = t and dim(X - {x}) < t for each x E X.A poset has dimension one if and only if it is a linear order (a chain) so the only 2-irreducible poset is a two point antichain. There are infinitely many 3-irreducible posets, and a complete listing of these posets has been made by Trotter and Moore [7] and by Kelly [ 3 ] . These posets can be conveniently grouped into 9 infinite families with 18 odd examples left over. An incomparable pair (xl,xz) E Zx is called a nonforced pair if x3 < xI implies x3 < x 2 and x2< x 4 implies x1 < x4 for all x3,x4E X . We let Nx denote the set of all nonforced pairs. For the poset X shown in Fig. 1, Nx = {(2,3), ( 3 , 2 ) , (6, l), (5,6), (2,4), (3,4)). It is customary to consider Nx as a directed graph whose vertex set is the point set of X . When (xl,x2) E N,, we draw an edge from x2 to xl. For the poset X in Fig. 1, we have the digraph shown in Fig. 2. The properties of the digraph Nx are central to the theory of rank for partial orders and we refer the reader to [5] and [6] for additional material on this subject. In this paper we will need only a few basic facts concerning Nx.We state these elementary results without proof. The reader may enjoy providing the arguments, although full details are given in [5]. Lemma 1. As a binary relation X U N , is transitiue, that is, if

subsefof X a n d f o r e a c h i = 1,2,..., m -1, eitherxi then either xl < x, in X or (xl, x,) E Nx.

{xi : 1 C i G m } is a i n X o r (x,,x,+,)€ N,,

1

X

Fig. 1 .

Fig. 2.

Every t-irreducible partial order is a suborder of a t

+ 1-irreducible partial

order

615

Lemma 2. If A = {a ,,a2,. . . , a, ) is a subset of X and Nxcontains a directed cycle { ( a i a, i + ] )1: S i < n } U {(a",a,)}, then the set A is an antichain in X , Furthermore, if x E X - A, then x > ai if and only if x > a, for each i, j with 1 s i < j =z n. Dually, x < ai if and only if x < ai for each i, j with 1 S i < j 6 n.

If t 2 3, a t-irreducible partial order is indecomposable with respect to ordinal sum [ 2 ] so in particular, it never contains an antichain satisfying the conclusion of the preceding lemma. A 2-irreducible poset (a two point antichain) is itself such an antichain and has a directed cycle of length two for its digraph of nonforced pairs. However, when t 3 the digraph of nonforced pairs of a t-irreducible poset contains no directed cycles. In this case, we abuse terminology somewhat and write X U N x to denote the set X equipped with the binary relation defined by x Is xz in X U Nx if and only if x Izs x2 in X or (xl,xz) E N,. Lemma 3. I f t 3 3 and X is a t-irreducible partial order, then X U Nx is also a partial order.

We illustrate the preceding lemma for a 3-irreducible poset (Fig. 3). A set R = {XI,X2,. .. ,XI}of linear extensions of X is called a realizer of X when x 1s x2 in X if and only if x 1S x2 in Xifor i = 1,2,. . . ,t. Lemma 4. A set R = {XI,X z , . . . , X I } of linear extensions of a poset X is a realizer of X if and only if for each nonforced pair ( x , ,x2) E N,, there exists some i s t for which x2 < x 1 in Xi.

Note in the preceding lemma that the emphasis is on a linear extension X,with xz < x , in Xi, so it is natural to say that Xireuerses the nonforced pair (x,, x2). The dimension of a partial order X is then the minimum number of linear extensions of X required to reverse the nonforced pairs of X. It is therefore

x6

x3

X UNX

X Fig. 3

.'

W.T. Trotter,Jr., J.A. Ross

616

natural to associate with a partial order X a hypergraph Hx whose vertices are the nonforced pairs in N x . A subset N C Nx is an edge in the hypergraph Hx when there is no linear extension of X which reverses all the nonforced pairs in N, but if N' is a nonempty proper subset of N, then there is a linear extension of X reversing the nonforced pairs in N'. It follows immediately that the dimension of X is the chromatic number of the hypergraph H x , that is, the least number of colors required to color the vertices of Hx so that no edge of Hx has all of its vertices assigned the same color. For the posets in Figs. 1 and 3, the associated hypergraphs are illustrated in Figs. 4a and 4b, respectively. Note that the graph in Fig. 4a is 2-colorable and that the graph in Fig. 4b is 3-colorable as it contains an odd cycle on seven points. Example 5. For the poset X shown in Fig. 3, the following three linear extensions realize X :

XI

= {XZ

< X I < x4 < x5 < x3 < x6 < x7},

xz= {x,< x,= { X I
1 for which such a design does exist, and, in general, the spectrum of SKC3(u)’s.

References [ I ] R.D. Baker, Partitioning the planes of AG,, (2) into 2-designs, Discrete Math. 15 (1976) 205-21 1. 121 J. Dinitz and D.R. Stinson, The spectrum of Room cubes, Europ. J. Comb., to appear. [3] A. Rosa, Room squares generalized, Ann. Discrete Math. 8 (1980) 45-57. [4] A. Rosa and S.A. Vanstone, Kirkman Cubes, 2nd Internat. Conf. on Combinatorial Geometry, Rome (1981) to appear.


ON THE BANDWIDTH OF RANDOM GRAPHS W. Fernandez de la VEGA Laboratoire de Recherche en Informatique, Universite'de Paris-Sud, B i t . 490, 91405 O R S A Y, France It is proved that almost all graphs on n vertices with cn edges have bandwidth 3 b,n, where b, is strictly positive for c > 1 and tends to 1 as c goes to infinity. Similar results are proved for the bandwidth of random r-regular graphs, r b 3, and for the profile of random sparse binary matrices.

1. Introduction

Let G be a graph on n vertices, with vertex set V(G) and edge set E ( G ) .The bandwidth of G, denoted by B ( G ) , is defined as

B ( G )= min B,(G), f

where

B , ( G ) = max{lf(u)-f(v)l: (4U)EE(Gh and f ranges over the set of all 1-1 mappings from V(G) to {1,2,. . . ,n}. The bandwidth of a real n X n symmetric matrix A = ( a j )is defined analogously by replacing the non-zero entries by ones and interpreting the resulting matrix as the adjacency matrix of a graph. Another related parameter of symmetric matrices is the profile which, for given A = ( a j )is, defined as the minimum value of the sum

2 (i

- min(j:

bij>011,

$=I

where B = ( b , j )ranges over the set of all symmetric permutations of A. In this last definition it is assumed that aii > 0, i = 1,. . .,n. In this paper we investigate the asymptotic behaviour of the bandwith of random graphs with constant average degree, when the number of vertices goes to infinity. It turns out that 'almost all' these graphs have 'large' bandwidth, greater than a constant times the number of vertices, if only the mean degree exceeds 2. We also obtain similar results for the bandwidth of random r-regular graphs, r 5 3, and for the profile of random sparse binary matrices. 633

W.F.dc la Viga

634

We refer the reader interested in bandwidth problems to the forthcoming review of Chinn et al. [3]. 2. The bandwidth of random graphs We denote by % ( n , m ) the set of all labelled graphs on n vertices, with m edges. Clearly, the number of these graphs is equal to the binomial coefficient (n(n2 2 ) . Let G be a graph on n vertices and suppose B ( G ) S bn - 1 for some constant b E]O,l[. Let f be a 1-1 mapping from V ( G ) to { 1 , 2 , ..., n} such that B ( G ) = B,(G). Then, clearly, there cannot be any edge joining a vertex x with f(x)< [ n ( l - b)/21 to a vertex y with f ( y ) a n - rn(1- b ) / 2 ] + 1. In other words, setting k = [ n ( l - b)/21, there exist two disjoint sets of vertices S and T, with I S 1 = I 71 = k, such that there is no edge going from S to T. Hence, denoting by Nn.m.bthe number of graphs in %(n,m )with bandwidth S bn - 1, we have n )(n(n - 2 / 2 - k ' 2Nn.m.b < ( k , n - 2 k , k where the first right-hand factor is twice the number of ways of choosing the sets S and T and the second factor is the number of graphs in %(n, m )such that there is no edge between two given subsets S and T. Dividing by #%(n, m ) , (1) gives

(

Nn.m.b n! 2k2 # % ( n , n 1 ) ~ 2 ( k ! ) ~- 2( kn) ! 1 - n7 n"(1-2kZ/n2)" k Z k ( n-2k),-" ' where the last inequality, obtained using Stirling's formula, is valid for sufficiently large k and n - 2k. The last factor is a decreasing function of k regarded as a real variable. Hence one can replace k in it by n(1- b ) / 2 . Taking, moreover, m = LcnJ, we obtain

where g ( c , b ) = ( l - ( 1 - b)'

) b - b ( - ) '2- ' b I-b

We have, for c > 1, taking b = 1 - ( 1 / 2 c ) ,

The bandwidth of random graphs

635

and g(c,O) = 2l-‘. Hence, for c > 1, we have g ( c , 0) < 1 and g(c, 1 - ( 1 / 2 c ) )> 1 and the equation g ( c , b) = 1 has a root b, which verifies 0 < b, < 1- ( 1 / 2 c ) . By (2), the proportion of graphs in Y ( n , LcnJ)which have a bandwidth 3 b,n tends to 1 when c +03. Moreover, it is easily checked that b, tends to 1 when c + 00. Hence we have proved the following theorem.

Theorem 1. Almost all graphs with n vertices and Lcn J edges have bandwidth 3 b,n, where b, is strictly positive for c > 1 and b, tends to 1 as c -+ w. We do not know if this theorem is best possible.

Problem. What is the smallest co such that c > co implies that almost all graphs on n vertices with cn edges have ‘large’ bandwidth? Coupling our theorem with the know fact (see Erdos and RCnyi [ 4 ] )that for c c 1/2, Y(n, [cn J) has only ‘small’ components, it is seen that co E [ 1 / 2 , 1 ] .

3. The case of random regular graphs

Regular graphs of degree 2 are disjoint unions of cycles and have bandwidth equal to 2. For random r-regular graphs, r 3 4 , one can apply the same method as above, using the asymptotic formula of Bender and Canfield [ l ] (see also Bollobis [2]) for the number of these graphs, and using for the number of r-regular graphs on n vertices which exclude all edges between two disjoint sets of vertices S and T the trivial bound

n ( n - 1 ) / 2 - I S I IT1 nr/2

(

1

9

obtained by dropping the regularity constraint. We omit the easy calculations and just state the theorem obtained.

Theorem 2. Almost all r-regular graphs on n vertices have bandwidth where b, tends to 1 as r + ~ .

3

b,n,

Actually this theorem holds with strictly positive b,’s for each r 3 because cubic graphs also have large bandwidth as we proceed to prove using a modified method. Let k be any integer>.. If the bandwidth of a graph G verifies B ( G ) S [I V ( G ) l / k J , then there exists a partition of V ( G ) into k disjoint subsets V1, V2,. . . , V,, with 1 V , I = 11 V ( G ) J / k J or 1 V , 1 = 11 V(G)l/kJ + 1 and such

W.F. de la Viga

636

that, for all i, j verifying I i - j 13 2, G contains no edge going from V, to V, (these V,'s are just inverse images of intervals of {1,2,. . . , n } for a mapping f such that B ( G )= B,(G)). Suppose I V(G)I = 2n = kh (the treatment of the values of n such that k does not divide 2n requires only trivial modifications) and B ( G ) S h. Note that, for fixed V1,V2,.. ., Vk with I V, I = h, the number of edges not forbidden for G is equal to

w + 2( k - l ) h 2 < ( y - l ) h Z . Then

2 # {G: G cubic on 2n vertices, B (G ) S h } S 2n [ ( 3 k / 2- l ) h 2 ] s(h,...,h)( 3n 4,,2 3n 3n -3" 6 k2" 1) F ]

{ (T-

(--)

{ (y) k-4/3ne}3"

(3)

The number of cubic graphs on 2n vertices is asymptotic to (see [6, p. 1751) (6n)!e-' -fie-'(-) 6Wn 3n .

288" (3n)!

e

Comparing with (3) we see that if we choose k = ko verifying the inequality

then almost all cubic graphs with 2n vertices have bandwidth > 2 n / k o . This inequality is verified for ko = 536. Hence we have proved the following theorem. Theorem 3. Almost all cubic graphs on 2n vertices have bandwidth greater than n /268. Surely there is room left for improving the constant involved in this theorem. 4. The profile of a random sparse symmetric matrix

Let A = ( a i j )be a symmetric real n X n matrix with aii > 0, i = 1 , . . . ,n. We define the profile (see [ 5 ] ) P ( A ) of A as the minimum ialue of the sum

2 ( i - minfi: bij > 011,

i-1

The bandwidth of random graphs

637

where the matrix B = (bij)ranges over all symmetric permutations of A. Since in the definition of the profile it is only the pattern of the non-zero entries which matters rather than their actual values, we shall confine ourselves to the case of (0, 1)-valued symmetric matrices. We shall prove the following theorem.

Theorem 4. Almost all symmetric binary matrices with 2 Lcn J off -diagonal ones have profile greater than pcn2,where pc is strictly positive for c > 1 and pc tends to 1/2 as c goes to infinity. Proof. Let A be an n X n symmetric binary matrix and suppose P ( A )d n2b2/2 for some b E]O,l[. We claim that there then exist two disjoint sets of indices I and J, with 1 I )= IJI = Ln(1- b )/ 2 J ,such that all entries aij with i E I and j E J are zeros. Indeed, put h = Ln ( 1 - b)/2] and suppose, without loss of generality, that A is such that the sum n

2 ( i - minG: aij> 0))

i=l

actually achieves the value P ( A ) . Put J = {1,2,. . .,h } and let rn denote the number of values of i greater than h such that aij > 0 for at least one j E J. Then m

P ( A ) ai - 1 i 3 m ( m + 1)/2, this bound being obtained by choosing the values of i equal to h + 1, h 2,. . . ,h + m. Hence we have m ( m + 1 ) s n2b2, which entails m < nb, and there remain at least n - nb - h 2 h values of i greater than h such that aii = 0 for every j E J, concluding the proof of our claim. Denoting by rn.c.b the proportion of matrices with 2 LcnJ off-diagonal positive entries and profile S n2b2/2we obtain, as in Section 2,

+

rn.c.b

= o(g(c,b y >,

which implies Theorem 4, where, for each c > 1, the value of pc can be taken equal to b f and the b,’s are those of Theorem 1. Theorem 4 is perhaps best possible.

Conjecture. Almost every binary symmetric n X n matrix with 2n of-diagonal ones has profile o(n). Acknowledgement Our thanks are due to Jean-Claude Bermond who pointed out an error in a

638

W.F. de la Viga

previous version of this paper and made several suggestions improving readability.

References [l] E.A. Bender and E.R. Canfield, The asymptotic number of labelled graphs with given degree sequences, J. Comb. Theory, Ser. A 24 (1978) 296307. [2] B. Bollob~s,A probabilistic proof of an asymptotic formula for the number of labelled regular graphs, Europ. J. Combin. 1 (1980). [3] P.Z. Chinn, J. Chvitalovi, A.K. Dewdney and N.E. Gibbs, Graph bandwidth: A survey of theory and applications, to appear. [4] P. Erdos and A. RCnyi, On the evolution of random graphs, Publ. Math. Inst. Hungar. Acad. Sci. 5 (1960) 17-61. [5] N.E. Gibbs, W.G. Poole and P.K.Stockmeyer, An algorithm for reducing the bandwidth and profile of a sparse symmetric matrix, SIAM J. Num. Analysis 13 (1976) 236250. [6] F. Harary and E.M. Palmer, Graphical Enumeration (Academic Press, New York-London, 1973).


ON THE USE OF BICHROMATIC INTERCHANGES D. de WERRA DCparrement de MarhCmariques, Ecole Polyrechnique Fidirale de Lausanne, Switzerland

1. Introduction The purpose of this note is to review some results related to the partitions of the node sets of hypergraphs by systematically applying simple coloring techniques based on bichromatic interchanges. We will not consider here the powerful method (fan-recoloring technique) developed by Fournier [3] for coloring the edges of multigraphs. We will use the graph-theoretical definitions of Berge [l]. Most results given here will be concerned with regular colorings. Given a hypergraph H = ( X , a ) and a positive integer k, a k-coloring of the nodes of H will he a . . . ,xk) of the node set X. Assume that for each edge E we partition % = (X,, are given 2 integers a ( E ) , p ( E ) such that 0 G a ( E )S IE Ilk G p ( E ) ;for each color i and each edge E let Ci( E )= IXi n E I. Then a k-coloring % such that a ( E )s Ci( E )s p ( E ) holds for each color i and each edge E will be called regular [6]. In the next section we will sketch the recoloring procedure and use it for characterizing a class of hypergraphs having regular k -colorings. The case where H is the dual of a multigraph will be considered and some refinements will be described for this particular case.

2. General properties

Given a k-coloring % = ( X I , . . , x k ) of a hypergraph H = (X, 8 ) we can define the deficiency a(%) of % as follows: Let 6, ( E , %) = max(0, a ( E )- C, ( E ) ,Ci( E )- p ( E ) )5 0 for each color i and 6 ( E , %). each edge E. Also let S(E, %) = 6, ( E , %) and finally S (%) = % will be regular if and only if S (%) = 0. A simple recoloring procedure would be the following. If % is such that S(Eo,%) > 0 for some edge Eo, then there are 2 colors a, b with either c b (Eo)< a ( E o )< C,(E,) or cb (Eo)< p (Eo)< C,(Eo). Then if the subhypergraph Hob generated by X . U Xb can be recolored in such a way that the new coloring %' satisfies

x,

c,,,

639

I>. de Werra

640

60(Eo, g‘)

6b

(Eo, %’) < 60( E o%) ~ -I- a b (Eo, q)

(2.1)

and 6, (E, 5%”) + Sb (E, %’) 4 6, (E, %) + ab (E, %)

for E # Eo,

(2.2)

then a(%‘)< a(%) and, by repeating this, one can get a regular k-coloring. Consider the subhypergraph H’ = Ha*of H generated by X, U X b . If H = (X’, 8’)can be recolored in such a way that the new coloring %’ = ( X i ,XL) satisfies, for every nonempty edge E ’ = E n X ’ ,

6, (E, %’)+Sb (E, %’) = max{2a(E) - IE’I ,IE’I - 2 P ( E ) , 0 ) ,

(2.3)

then (2.1)and (2.2) will be satisfied, an improved coloring will be obtained, and one will finally get a regular k-coloring. Hence we have the following. Proposition 2.1. If every subhypergraph H’ of H = ( X , 8 ) can be bicolored in such a way that (2.3) holds, then H has a regular k-coloring. Notice that for each edge E ’ of H’ such that 2a ( E )C I E’I S 2 P ( E ) , we have 6, ( E , %’) = 8 b (E, %’) = 0 in the new coloring V‘. Given a hypergraph H = (X, 8 ) , a positive k and integers a ( E ) , P ( E ) with 0 s a ( E )S I E Ilk 6 P ( E ) ( E E $), we shall say that H has property X ( k ) if any subhypergraph H’ = ( X ‘ ,8’) of H for which a ( E ) S I E n X ’ ( / pS P ( E ) ( E E 8 ) holds for some p S k has a regular p-coloring defined by a ( E ’ )= a ( E ) and P ( E ’ )= @ ( E ) for each edge E’ = E fl X ’ of %’. Clearly if H satisfies the hypothesis of Proposition 2.1, H has property X(k). However, if H has X(k), it may not satisfy (2.3) (this is the case for conormal hypergraphs for instance, as can be seen by considering a bicoloring of the hypergraph defined by X = { 1 , 2 , 3 ) , 8 = ({1,2),{ 2 , 3 ) ,{3,1),{ 1 , 2 , 3 ) ) ) . Now a regular k-coloring % of H will be called canonical if the following holds: Whenever a node x is in Xi,there is an edge E 3 x such that either C , ( E ) = @ ( E ) (j= 1,. ..,i - 1) or C i ( E ) = a ( E ) . By using induction on k, one can show the following (see [ 6 ] ) . Proposition 2.2. Let H, k, LY ( E ) , P ( E ) be given as above and suppose H has the property X(k). Then H has a regular k-coloring which is canonical.

As a consequence, if H = ( X , 8) is the hypergraph of maximal cliques of a perfect graph G ( H is conormal [5] or its dual H * is normal [ 2 ] ) and if

The use of bichrornatic interchanges

64 I

k = max IE 1, a ( E )= 0, P ( E ) = 1 ( E E 8), then H has property X ( k ) and there exists a regular k-coloring of H which is canonical: it corresponds to a node coloring of G where any node with color i is in a maximal clique where colors 1 , 2 , . . . ,i - 1 also appear [6]. Another consequence is the existence in balanced hypergraphs of good k-colorings such that whenever a node x gets color i, there is an edge E containing x, where colors 1 , 2 , . . . ,i - 1 also appear [8]. 3. The case of multigraphs

More refined results can be derived in the particular case of edge colorings in multigraphs. Given a multigraph G = ( X ,E ) , a positive integer k and integers a @ ) , p ( x ) such that 0 s a ( x ) S dG (X)/k S p ( x ) (x E X ) where dc ( x ) is the degree of node x in G, a regular k-coloring of edges is a partition % = (El,..., & ) o f E w i t h a ( x ) s C i ( x ) 6 p ( x ) ( i S k ; x E X Here ). C,(x)is the degree of node x in the partial graph ( X , E , ) with loops counted twice. U * )of G with An obstruction in G is a connected partial subgraph G * = (X*, d G . ( x )= 2 a ( x ) or 2 p ( x ) (x E X*) and with an odd number of edges [7] (’i.e., an odd cycle with degrees 2 a ( x ) or 2 p ( x ) ) . With each regular k-coloring % = ( E , ,. . . ,E k ) of G (with given a ( x ) , p ( x ) ) one can associate a unique sequence F = ( f l , . . . ,fk ) with f l 3 * 3 fk and fi = I E, I ( i = 1,. . . , k). Let 5 k be the set of all sequences corresponding to a regular k-coloring of G. We can define an order relation on s k as follows: 1

F = (fl,.

. .,

fk)>

-

F’ =

(fr,.

.., f L )

if f l + , . . + f i z-fl+...+ft’( i = I , ..., k -1) (obviously we have f l + * * * + f k = * * * +fL = I E I). By repeatedly using alternating chain methods, one easily obtains the following.

fr +

Proposition 3.1 [ 6 ] . Given a multigraph G = ( X , E ) , a positive integer k, integers ~ ( x ) ,p ( x ) with O S Q ( X ) S d G ( X ) / kC p ( X ) (X EX), the set p k is partially ordered. There is a unique ‘minimal’ sequence F* = (f:, . . . ,f t ) in g k . F* satisfies f: 2 .af: L f 7 - 1. Furthermore the analogue of Proposition 2.1 also holds: If G contains no obstruction, there exists a regular k -coloring. Canonical colorings are defined as before and one can show that there exist such colorings when G contains no obstructions. In fact, one can state the following. Proposition 3.2. Given a multigraph G = ( X , E ) , integers k, Q (x), p ( x ) (x E X )

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as before, if G contains no obstruction, all maximal sequences in .?& correspond to canonical regular k -colorings. Here F is a maximal sequence in %k if there is no F' in s k with F' # F and F' > F. The proof technique follows exactly the same lines as the proof given for a special type of regular coloring [8]. Let us assume now that we are given a multigraph G = ( X ,E ) , an integer k, and integers (Y (x), p (x) (x E X ) . Furthermore suppose (Y (x) < dG ( x ) / p < p ( x ) is satisfied at each node x for p = k as well as for p = k + 1. Then if G contains no obstruction, there exist regular k-colorings of G and also regular (k + 1)colorings with the same (Y (x), p (x). Again by performing bichromatic interchanges, one can obtain Proposition 3.3. Let G = ( X ,E ) , k, a ( x ) , p ( x ) (x E X ) be given with a ( x ) S dG ( x ) / p < p ( x ) (x E X ) for p = k and k + 1. If G contains no obstruction, then for any sequence F = ( f l , . . . ,f k + l ) in 5 k + l there exists a sequence F' = . .,f ; ) in Sksuch that (fr,. . . ,f ; , O ) > (fl, . . . ,f k + l ) .

(fr,.

In particular, if a ( x ) = 0 (x E X ) and k = max, [dG( x ) / p ( x ) ] and if G contains no odd cycle C with degrees d c ( x ) = 2 p ( x ) , all maximal sequences corresponding to regular colorings have k positive members (here as before, in order to be able to compare 2 sequences of different length we may add zeroes at the end of the shorter one). If p ( x ) = 1 (x E X ) we have usual edge colorings (adjacent edges have different colors) and all maximal sequences corresponding to usual edge colorings of a bipartite multigraph with maximum degree d have exactly d positive members [4]. Finally, there are some more chromatic properties which can be derived by applying a simple recoloring procedure based on bichromatic interchanges. Proposition 3.4. Let G = ( X , E ) be a multigraph, k 2 3 an integer, cu(x), p ( x ) integers such that a ( x ) < dG (x)/k 6 p ( x ) for all x E X . G has a regular k coloring if any one of the following conditions hold : (a) in each obstruction G * there is a node x with either

d c ( x ) a k ( d c * ( x ) / 2 +l ) - l , or d G (X /

)

k ( d c *(x)/2 - 1) + 1;

(b) in each obstruction G * there are 2 consecutive nodes such that none of them is joined to some node of G * by an odd chain C whose intermediate nodes z have degree d c ( z ) = 2 a ( z ) or 2 p ( z ) and are not in G*.

The use of bichromatic interchanges

643

4. A non-regular coloring In this section we will describe a type of coloring which does not have all nice properties of regular colorings. Consider a multigraph G = (X, E ) with a partition '9 = (8,0) of the node set X. Let k be a positive integer. A partition % = ( E l , .. . ,Ek)of the edge set E will be called a parity k-coloring if for each node x in 8 (resp. in 0) all C,(x) except at most one are even (resp. odd) and maxi, 1 Ci(x) - C, (x) I C 2. It is easy to see that for any x the values of C,(x), . . . , Ck(x) are completely determined (up to a permutation of indices) when k, 8 and O are given. Let us call a graph of type F a connected multigraph G = ( X ,E ) together with a partition 9 = ( 8 , O ) of the node set which has the following properties: (i) all degrees are even (so X = No U N 2 where, for i = 0,2, Nj = {x x E X , dc ( x ) = i (mod 4)); (ii) I E I + I N o n O ( + ( N z 8 n ) is odd.

I

Proposition 4.1. Given a connected multigraph G = ( X , E ) with a partition 9 ' = ( 8 , O ) of X , there exists a parity bicoloring of G iff G is not of type F. Proof. In any parity bicoloring % = ( E a ,E b ) of a multigraph G, we have 1 C, ( x ) - Cb( x ) l = 2 iff x E ( N on 0 )U (N2f l 8). For all remaining nodes I C, ( x ) - cb ( x ) l = 0 or 1. Hence a multigraph G has a parity bicoloring iff the multigraph Go obtained by introducing a loop at each node in (Nofl 0) U ( N zf l 8 ) has a bicoloring %" = (EO,,E!) satisfying I C. (x) - c b (x)/ S 1 for each node x. Such colorings are known to exist iff G ois not an odd cycle, i.e., iff G is not of type E 0 According to Proposition 4.1, given a multigraph G = ( X , E ) , a positive integer k, and a partition '9 = ( 8 , O ) of X, we will say that a partial subgraph of G is an obstruction if it is of type E Proposition 4.2. Let G = ( X , E ) be a multigraph, k a positive integer, and P = (8,0 ) a partiton of X . If G contains no obstruction, there exists a parity k -coloring. Proof. We use a recoloring procedure which will reduce the deficiency in a way similar to the proof of Proposition 2.1. We will here construct the coloring in two steps. For a k-coloring % = ( E l , .. . ,Ek) we define the deficiency of node x by 6 (x, %) = number of colors i for which Ci(x) does not have the correct parity; 6(x, %) is always even. We will first reduce 6(x, %) to 0 for each node x. Let xo be

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a node such that both Co(xo) and c b ( x o ) have the wrong parity. Consider the connected component C o b (xo) containing xo of (X,Ea U Eb).Define No (resp. N , ) as the set of nodes x in c a b ( x o ) for which (x)+ c b ( x ) = 0 (mod 4) (resp. 2 (mod 4)). We add a loop I, at each node x of (Nono) U ( N z n 8). Since G contains no obstruction,

c.

Cab

I

( x 0 ) u {L x E ( N , n 0) u ( N n ZP)I

is not an odd cycle and it can be recolored in such a way that I C:(x)- CL(x) 1 s 1 for each node x in C o b ( x 0 ) . After removal of the loops, the new coloring %' satisfies S(x,, %")C S(xo, %), since Ca( x ) and c b ( x ) now have the correct parity. Furthermore, for all nodes x # xo, we also have S (x, %') S S ( x , %). By repeating this procedure one finally gets a k-coloring 0 = (El,.. . ,E k ) , where all Ci( x ) have the correct parity for each color i and for each node x ( 6 ( x , @)= 0 ( x E X)). We now have to modify the in order to get IC,(x)- C , ( x ) ( < 2 (i s k ;x E X) while keeping for each node x all Ci( x ) (except at most one) even (resp. odd) if x is in 8 (resp. 6). Assume that for some node xo, there are 2 colors a, b such that (xo)c b ( x o ) > 2. As before we consider the connected component C a b (xo) and we define No and N2.We add loops at the nodes x of (Non 0)U ( N 2n $) as before. Since G contains no obstruction, we obtain again (after removal of the loops) a bicoloring where for each node x both Ci(x) and C6(x)have either exchanged their parities or kept their previous parities. Furthermore, for node xo,

c(x)

ca

1 c i ( x 0 ) - CL(x0) 1 S 2
2, I

we will obtain a parity k-coloring of G. 0

Corollary 1 [9].Let G = (X,E ) be a bipartite multigraph with all degrees euen and let k be a positive integer. There exists a partition of E into subsets E l , ...,Ek such that each Ei is a collection of elementary cycles and I C, ( x ) - C, ( x ) 1 = 0 or 2 for each node x and for any 2 colors i, j .

The use of bichromatic interchanges

645

In other words, the edge set E is partitioned into k collections of elementary cycles. The number of cycles going through a node x is almost the same for all collections C,. Proof of Corollary 1. One takes 0 = 0 and 8 = X.G contains no obstructions, since for each cycle H = (X’,E ’ )we have E’I even and IX‘n N21 is also even.

Remark 4.1. For the parity k-colorings, it is not true in general that if G has such a coloring then there exists also a parity k-coloring % ’ = ( E l , .. .,Ek)with - 1 s ( E iI - ( E ,1s 1 ( i , j < k). Remark 4.2. An extension of regular colorings would be to associate with each node x four integers a ( x ) s p ( x ) < y ( x ) S 6 ( x ) and to require that either a ( x ) s Ci( x ) s p ( x ) or y ( x ) c Ci ( x ) s S ( x ) . In the parity k-colorings we have a ( x ) = p ( x ) = y ( x ) - 2 = S ( x ) - 2 when dc ( x ) is the sum of k factors with same parity. Remark 4.3. Another type of coloring related to parity colorings would be d,, d2)of the node set. For a node x in dpall C,( x ) defined by a partition (do, (except at most one) should satisfy C i ( x ) = p (mod 3) and furthermore 1 Cj( x ) - C,( x ) ( s 3 (i, j s k). Unfortunately, graphs which do not have bicolorings of this type (i.e., obstructions) cannot be described as easily as for parity colorings. For instance, in Fig. 1, G, has a bicoloring with do= X,d 1= d z= 0 and G2has no such bicoloring with do= X, .dI= d 2= 0;but GI and Gzhave the same degrees and the same number of edges.

Fig. 1.

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D. de Werra

References [I] C. Berge, Graphs and Hypergraphs (North-Holland, Amsterdam, 1973). [2] C. Berge, RCsultats rCcents sur certaines classes d’hypergraphes qui gtnkralisent les graphes bipartis, Cahiers du C.E.R.O. 20 (1978) 243-278. 131 J.C.Fournier, MCthode et thCoritme gCn&ralde coloration des arttes d’un multigraphe, J. Math. Pures Appl. 56 (1977) 437-453. [4] J. Folkman and D.R. Fulkerson, Edge colorings in bipartite graphs, in: R.C. Bose and R.A. Dowling, eds., Combinatorial Mathematics and its Applications (Univ. of North Carolina Press, Chapel Hill, 1969). [ 5 ] A. Schrijver, Fractional packing and covering, in: A. Schrijver, ed., Packing and Covering in Combinatorics (Mathematical Centre Tracts 106, Amsterdam, 1979) pp. 201-274. [6] D. de Werra, Regular and canonical colorings, Discrete Math. 27 (1979) 309-316. [7] D. de Werra, Obstructions for regular colorings, J. Comb. Theory, to appear. [8] D. de Werra, On the existence of generalized good and equitable edge colorings, J. Graph Theory 5 (1981) 247-258. [9] D. de Werra, Partial compactness in edge chromatic scheduling, Oper. Res. Verfahren 32 (1979) 207-21 9.


EXTREME COLORING OF THE EDGES OF A GRAPH D. WERTHEIMER Universite‘ de Lille 1, France A cubic graph has an extreme 3-coloring of its edges. Partial results are given on Sterboul’s conjecture (1983): “Every simple 4-regular connected graph has an extreme 4-coloring of its edges.”

1. Extreme coloring of the vertices of a k-uniform hypergraph Let H = ( X , S)be a k-uniform hypergraph of order / XI = n and size I 8 f = m. For every E E 8, we have 1 E 1 = k. A k-coloring of the vertex-set of H is a mapping g from X onto a set of cardinal k (color-set). An extreme k-coloring is such that for every edge E, either I g ( E ) l = 1 or Ig(E)l = k. The following was proved in [l]. Theorem 1. If k is a prime-power, every connected k-uniform hypergraph with m s ( n - 2)/(k - 2) has an extreme k -coloring.

2. Extreme coloring of the edges of a graph Let G = ( X , U ) be a k-regular graph. The dual hypergraph of G is the k-uniform hypergraph whose vertex-set is U and edge-set is X , the incidence relations in H being deduced in an obvious way from those of G. An extreme k-coloring of the edges of G is an extreme k-coloring of its dual hypergraph. Hence, in such a coloring, either all edges incident to a vertex have the same color (in this case this vertex is called 1-colored), or all edges incident to a vertex have a different color (the vertex is called k-colored). A vertex x is called equitably 2-colored in some coloring g of the edges of a 4-regular graph if, calling e l , .. . ,e, the edges incident to x , we have g(el) = g(e2)Z g ( e z )= g b ) . A simple graph has no multiple edges. 647

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Theorem 2. Every connected 3-regular graph has an extreme 3-coloriiig of its edges. Proof. Apply Theorem 1 to the dual hypergraph of the graph. Sterboul made the following conjecture.

Conjecture 1. Every simple connected 4-regular graph has an extreme 4-coloring of its edges. In connection with this conjecture we make the following remarks: (1) Every 4-regular graph with chromatic index 4 has an extreme 4-coloring of its edges. (2) If a 4-regular graph has a partial cubic subgraph with chromatic index 3, then it has an extreme 4-coloring of its edges (it suffices to color the edges outside the subgraph with a fourth color). In connection with the latter remark, let us recall the following.

Conjecture 2 (Berge). Every 4-regular simple graph has a partial cubic subgraph. Conjectures 1 and 2 would be a common consequence of the following stronger conjecture.

Conjecture 3. Every 4-regular simple graph has a partial cubic subgraph with chromatic index 3. We now prove a result somewhat weaker than Conjecture 1

Theorem 3. Every 4-regular connected graph has a 4-coloring of its edges where every vertex is either 1-colored or 4-cOfOred or, for a t most one vertex, equitably 2-colored. Proof. Let G = (X,U ) be a 4-regular graph, and x a vertex which is not a cut-vertex. Let H be the dual hypergraph of G, and H’ the hypergraph having the same vertex-set as H and whose edge-set is the edge-set H minus the edge corresponding to x. By Theorem 1, H’ has an extreme 4-coloring. Hence, G has a 4-coloring g of its edges where every vertex except x is either 1-colored or 4-colored. Let us assume that the color-set is { 1,2,3,4}, and let el, e2,e3, e4 be the edges of G incident to x. In order to finish the proof, we have to prove that the following two cases are impossible:

Extreme coloring of the edges of a graph

g(eJ = g(e2)= g ( e 3 )= 1

g(eJ = g(e,) = 1,

649

and g ( e , ) = 2;

g ( e 3 )= 2 and g(e,) = 3.

Calling G ’ the subgraph whose edges are the edges of G colored with color 2 or 4, G ’ would in those two cases have a unique vertex of odd degree, namely x, which is impossible.

Reference [l] F. Sterboul and D. Wertheimer, Colorations extrtmes dans les hypergraphes, in: C. Berge et al., Combinatorial Mathematics, Proc. Internat. Coll. on Graph Theory and Combinatorics, Annals of Discrete Mathematics 17 (North-Holland, Amsterdam, 1983) pp. 605-612 (this volume).



NEW DECODING METHODS OF THE GOLAY CODE (24,12,8) J. WOLFMANN UER de Sciences et Techniques, Unioersite' de Toulon, 83130 La Garde. France

1. Introduction

In this paper we give two new decoding methods of the Golay code (24,12,8). We use a new construction of this code as binary image,of a principal ideal in a certain group algebra over the Galois field of order eight [6]. This ideal is also an extended Reed-Solomon code [ 5 , 21. This construction is a particular case of a general theory on binary images of certain self-dual codes [2]. The techniques use computation in the Galois field of order eight, and proofs are just sketched; for further details the reader is referred to [7]. IF, denotes the Galois field of order n, and other notations and definitions concerning coding theory are standard (see [4]).

2. Construction of the Golay code

We consider the following Reed-Solomon code: The principal ideal of

1 = IF8(X)/X7- 1 generated by the polynomial

g(X)= (X- P ) ( X

- P')(X

- P3),

where P is a primitive root of Fs. As usual, each polynomial a,X' is identified with the word (ao,a,, . . . ,a s ) and this ideal is a cyclic code of length 7 over IF8. Let C be the extended code of the previous code, that is the code whose words are of the kind a = (urn,uo,a , , . . . ,as)obtained by adding, to each word, the new component urn= ZY=oai. Each word of C is identified with the element a,X" + u,X"' of the group algebra $33 = IFs(%), where % is the additive group of Ex. $33 is the set of all formal linear combinations a ( g ) X 8 , a ( g ) E IFx. The laws in $33 are classically defined:

zy=,,

zy=o

/

c,,,

65 1

J. Wolfrnann

652

One can show [6] that C is the principal ideal of $33 generated by

+

+ ax"+ b X b + c X " ) ,

x = X o X'(X"

with a = a , b = a * , c = a 4 and /I2= a. Further, one can also show, as a particular case of a general theory [ 2 ] , that C is a self dual code. Let us consider now the basis ( a ', a h ,a ') for IFx which is a n F,-vectorial space and the binary image of C relative to that basis. That binary image is obtained by replacing, in each word ( u , , u 2 , .. . , uR)in C, each component u, by the 3-uple of the component of u, with respect to that basis. It is shown in [6] and [7] that this image is the Golay code (24,12,8). For practical reasons we will write the elements of IFx in the order 0, a, a', a', 1, a 3 ,a', a', and then each element a , X n + ~ ? = o a , X in " ' C will be identified with ( a z ,a , , a', a4,ao,u3,ah,a5). In that way a generator matrix of C is G = (Z4,M), where Z4 is the identity matrix of order 4, and

M=l I 1, a, a ? , a 4 a, 1, a4,a 2 a ? , a d ,1, a a d , a ? , a, 1

We note that H = (0,a , a', a d }is a hyperplane of IFx (2-dimensional subspace) and this fact is very important ( H is associated with the four first coordinates of the word of C). In all the following the Golay code will be the binary image, relative to the basis ( a 3 , a h , a a )of, the code C over F8 with generator matrix G. 3. Decoding methods

We define the binary distance of two words of (IFx)' as the Hamming distance of their binary images. In the same way, we define the binary weight of a word over FS. The words of (IF2)*' are described by words in (IFx)' using the basis ( a 3a", , a5), and, in particular, the words of the Golay code are described by those of C. Therefore, decoding consists of finding, for each word y of (F$ describing a received word, the unique word z of C whose binary distance is at most 3 from y .

New decoding methods of the Golay code (24,12,8)

653

3.1. First decoding method Notation. Each word u in (IF8)8 is denoted by u = (ul, u2)with u 1and u2,4-weight words. dz(u,u ) denotes the binary distance, while w2(u) denotes the binary weight of u. r is the permutation of (IF8)' defined by r ( a l , a2,. . . as) = ( & ( I ) , 7

ar(Z),. . . as@)), 3

with s = ( 1,27374,596,778 173,678, 2,47597> .

M is the matrix defined above. Theorem 1. Let y be a word of (IFR)' such that there is a word z in C with d2(y,2 ) s3. y ; and y; are such that ~ ' ( y=) (y;, y;). There exists i E {0,1,.. . ,6} such that: (i) w2(y;+y;M)G3 and then .rr'(z)=(y;M,y;), or (ii) ~ 2 ( ~ ; + y i M and ) ~ 3then r ' ( z ) = ( y ; , ~ ; M ) . Sketch of the proof. We use the property of an extended Reed-Solomon code to be invariant under the action of a certain affine group (see [4]). More precisely, C is preserved by the transforms of the kind:

The permutation s corresponds to the case u = a,u = 0. The decoding method is a modified version of the Mac William's permutation decoding [3]. The case (ii) is the one with no error in the first four components in ~ ' ( u )The . generator matrix G = (Z4, M) is used to complete the word r ( z ) . In the case (ii) there is no error in the last four components. We check that M 2 = I4and we now use the generator matrix M X (I4,M ) = ( M , L). To check that there always exists an integer i such that r ' ( y ) satisfies one of the case (i) or (ii), we must just see that, for each 3-subset E of h?,there always exists an affine transform mapping E onto H or onto the complementary set of H. Example. The received word is (1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0). The corresponding word in C is y = (1, a,a3,O,O, 0, a4,a6). We find ~ ~ (= y(1, )a4,0, a 3 ,a 6 , 0 , 0 )= (Y:, Y:), Y:M

= (1, a 3 , a 6 , 0 ) ,

y:+ Y:M

= (0, a 6 ,a 6 ,a 3 ) ,

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654

and therefore

w2(y: + y:M) = 3. According to Theorem 1,

a6,a,o,o),

zI = r s ( 2 ) = ( y : ~ y:) , = (1, a 3 , a 6 , 0 ,

and z =T

- ~ ( Z ~=) (1, a , o , o , ( ~ ~ , Oa6). ,(~~,

The transmitted word is thus (111011000000010000100010). 3.2. Second decoding method Notation is as above. Theorem 2. Let y be in (IFg)'; y = ( y l , y2) such that there exists z in C with d 2 ( y , z ) G 3 .y l and y; are such that y : = y l + y 2 M , y ; = y 2 + y l M . Then (i) z = (y2M, yz) if w 2 ( y ; ) c 3, and z = (y,, y l M ) if wz(y;)c 3; (ii) if wz(y;)>3 and w 2 ( y 9 > 3 , then there exist j E{l,2} and u E (IF8)' such thar w2(u) = 1 or 2, w ( u M + y ; ) = 1, and the only non-zero componenr of u M + y ; not equal to 1 is z = ( u M , u ) + ( y 2 M , y 2 ) if j = 1 , and z = (u, uM)+ (yl, ylM) if i = 2. Sketch of the proof. Case (i). See Theorem 1 with. i = 0. Case (ii). We use the fact that, in that case, the error on the word on IF8 has weight at most 3 and, therefore, necessarily there is only one error either in the first four components or in the last four components.

Use of Table 1 Table 1 gives u and uM for each word such that w2(u) = 1 or 2. To decode the received word y using Theorem 2 in the case (ii), we look for j E {1,2} and u in the table such that uM and yi are different only by one component and the only non-zero component of uM + y:, not equal to 1. The table gives immediately the or ( u , u M ) used to the correction. In the table, each a i in IFx is word (uM,u) represented by the integer i and 0 by an empty cell. Example 1. y =(O,O,l,O,O,aJ,l,a), y I M = ( a 2 , a 4 , 1 , a ) , a n dy : = y 2 + y l M = (a2,0,0,0,1). Therefore, w2(y;) = 2 and z = (yl, y l M ) = (O,O, 1,0, a', a', 1, a ) . Example 2. y = (1, a, a 3 , 0 , 0 , 0a', , a6), y 2 M = (a4,0,as,a ) , y , + y 2 M = (a', a, a', a ) . The exponents of y I are 5 , 1, 2, 1.

and

yl =

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655

Table 1

uM 0 1 2 2 3 4 0 0 0 0 0 0 0 0 0

0 1 2 4 4 5 6 6

1 1 1 1 1 1 1 1 1 1 1 1

0 1 1 1 2 3 3 4 4 5 6

2 6 1 4 0 4

5 0 1 2 4 1

6 5 6 3 5 3 3 6 5 3 6 5

2 2 2 2 2 2

2 2 0 6 3 0 4 4 3 1 3 1

6

5 6 6 3 3 5 5 3 5 5 3 3 6 6

2 2 2 2 2 2 3 3 3

3 3 4 5 5 6

6 0 1 2 2 6 6 3 5 2 3 6 0 4 5 5 1 5 5 4

6 3 3

3 3 3 3 3 3 4 4 4 4 4 4

2 2 2 4 4 5

5 2 1 5

0

uM

U

3 3 5

3 5 5 1 3 6

5

6 5 6 5 3 6 6 4

0 1 2 3

1 2 6 5 0 1

1 4 5 3 0 3

3 2 4 6 2 0 2 5 6 4 4 0 1 0 6 1 3 2

2 3 6 4 5 1 1 0 3 0 4 1 1 5 2

uM

U

3 1 4 3 0 2 4 0 4 5 2

5 3 3 6

5 1 6 6 3 6 5

2 6 5 3 6

5

3 6 3 5 3 5 6

6 3 5

5 6 3

2 6 6 3 1

5 6 3 5 5 5 6 5 5 3 6 3

U

4 4 4 4 4 4

3 3 4 5 6 6

0 3 6 6 2 6

5 4 6 1 3 2

5 5 5 5 5 5 5 5 5

0 1 2 2 3 4 5 5 6

3 5 0 5 2 1 1 4 0

4 1 2 1 6 4 1 2

6 6 6 6 6 6 6 6 6

0 0 1 2 4 4 5 6 6

1 4 4 2 3 2 1 4

1 3 5 6 6 2 0 6 4

3 6

6 3 3

4 2

5 6

6

3 6 5 6 3 3 1

4

5

3 3 5 5

3

6

6 4

6 5

5 6 2

5 5

5

3 3

Galois field of order 8: a3=a+l a‘=a’+a a’=a’+a+l a6=a2+1

a = f f 6 + a 5(0 1 1 ) a 2 = a 3 + a s(1 0 1) a 4 = a 3 + a 6 (1 1 0 ) 1=a3+a6+a5 (1 1 1 )

Table 1 gives 5, 1,5,1 for u = a 6 , 0, a 6 , 0 and, therefore, uM + y I = (0, 0, a ’,0) because a’+ a’ = o 3 and z = (uM, u ) + (y2M,y2) = (1, a,O,O, a6,0,a’, a‘). 4. Conclusion

The property used in [2] for a certain extended Reed-Solomon code to be also a principal ideal in a group algebra can be generalized to a full class of codes. We can then hope for a generalization of the methods and properties of that work to other interesting codes.

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J. Wolfmann

References [ l ] P. Camion, Etude de codes binaires abiliens modulaires autoduaux de petites longueurs, Rev. CETHEDEC, 79-2 (1979) 3-24. [Z] G. Pasquier and J. Wolfmann, A class of binary self dual codes, submitted for publication. [3] F.J. MacWilliams, Permutation decoding of systematic codes, Bell System Tech. J. 43 (1964) 485-505. [4] F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes (North-Holland, Amsterdam, 1977). f5] G. Pasquier, The binary Golay code obtained from an extended cyclic code over F, Europ. J. Comb. 1 (1980) 369-370. [6] J. Wolfmann, A new construction of the binary Golay code (24, 12,8) using a group algebra over a finite field, Discrete Math. 31 (1980) 337-338. [7] J. Wolfmann, Nouvelles mithodes de dicodage du code de Golay (24,12,8), Rev. CETHEDEC, 81 (2) (1981) 79-88.


UNIFORM DISTRIBUTION OF A SUBGRAPH IN A GRAPH Thomas ZASLAVSKY * Deparimeni of Mathematics, The Ohio Siate University, Columbus, Ohio 43210, USA

1. Introduction Given a graph M of order r (perhaps having multiple edges but not loops), we discuss the property of a graph G that M is uniformly distributed in it, that is every induced subgraph of G of order r contains the same number of subgraphs isomorphic to M. It turns out that the order of G is bounded unless M = E,, or M is not contained in G at all, or G = pKn (K,, with multiplicity p on every edge), and quite strictly so if M and G are simple, Indeed in the latter case, if G 2 A4 and Gf Kn then n = I V(G)IG 2r - 2, and I suspect the true bound is n s r + 2. But the exact bound when G or M may be a multigraph is very much an unsolved problem.

Background I became interested in uniform distribution of a subgraph because of the problem of complementation identities for generalized matchings. A generalized I-matching with model M is a disjoint union of 1 copies of M; it is an ordinary matching if M is an edge. For the generalized matching polynomial of a connected model, that is ( - I)'m, (M; G)x"-'',

a M ( G;x ) = '30

where ml (M; G ) is the number of generalized 1-matchings in G, there is the complementation identity a M ( G ; x ) =rz.0 * i ( M ; G ) a M ( p X n -;x), rj

(1)

where G p K . and m, (M; G ) is the number of generalized i-matchings in pKn of which no constituent M lies in G. What is remarkable about (1) is that the Research supported by the National Science Foundation 657

Z Zaslavsky

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coefficients in the expansion of a ’ ( G ; x ) in terms of the a M ( p K N; x ) have a simple combinatorial interpretation. (The proof is also simple.) What properties explain this? (Identity (1) was observed first by Godsil[3] for matchings and then generalized in [5].) Looking for an answer led me to consider the Appell property, D p N ( x )= N p N - , ( x and ) deg(pN(x)) = N, which the sequence Q “ ( p K N;x ) has. It is an open question whether or not an Appell sequence of matching polynomials must have an analog of (1); there is no obvious connection, so the answer probably depends on which such sequences exist. But there is an obvious combinatorial property of Appell sequences [5]. The associated graphs GN have uniform expected distribution of generalized matchings modelled on M: for each I 3 0, the average number of generalized I-matchings in induced subgraphs of G, of order rl is a function of 1, independent of N (provided N b rl). Therefore one asks: Which graph sequences have uniform expected distribution? That seemed quite difficult, so I took the strengthened condition of uniform distribution of generalized matchings: the number of generalized l-matchings in GN on any set of rl vertices should be the same. This is equivalent to uniform distribution of M alone. Thus by Theorem 1 of this paper the only solutions for E ( M )# 0 are GN = pKN and sequences of graphs not containing M. In the sequel we will always have the following situation: M is a graph of order r, not g,. G is a graph of order n b r in which M is uniformly distributed (except in Coroliary 9), but not a trivial example, which means we assume n > r and G 2 M but G # p K n .(Thus r b 3.) m ( M , G ) means the number of copies of M on each r-set X C V(G). We write g, for the graph without edges. For X C V(G), X‘ is its cornplement. X will also denote the subgraph of G induced by X , as in m (M; X). Adjacency is indicated by . The graph u + H is H with an extra vertex u adjacent to every vertex.

-

2. The Ramsey bound We begin by observing that Ramsey’s theorem implies an upper bound on the order of G. Theorem 1. The order of a nontrivial example G of any multiplicity is bounded by a function of r.

Proof. Let m = m ( M , G ) and choose p o so that

mo = m (MIp o K , ) m < m (M, (P O+ 1)K).

Uniform distribution of a subgrapk in a graph

659

Let p(x, y ) = the multiplicity of the edge xy in G.Color the pairs of vertices of G in three colors: black if p(xy) > po, red if p ( x y ) = po and mo = m, white otherwise. By Ramsey's theorem, if n > f ( r ) there is a homogeneous set of r vertices for white or black or of 2r - 3 vertices for red. But a homogeneous white or black r-set has the wrong number of copies of M (too few for white, too many for black), so there must be a homogeneous red (2r - 3)-set. That entails m = mo. Now let n > f ( r ) , let X be a largest homogeneous red set in V(G), and let y E x'if X # V ( G ) .We can certainly find W C X of size r - 1 such that either all p ( w y ) s poand some p ( w y ) < po,or all p ( w y ) s po and some p ( w y ) > po. In the former case m ( M , W U y ) < mo = m, a contradiction. Similarly for the other case. So G is homogeneous for red; that is, G = poK,.

3. Isolated vertices

If M has isolated vertices, we can in general neglect them, thereby making r smaller and improving the bound on n. The justification is the following result. (The example M = K2 U K , and G = K 2 U K 2 shows that one cannot always discard all the isolated nodes.) Proposition 2. Let M have s isolated vertices. If s 3 2r - n, and in particular if n 2 21 - 1, then M less its isolated ueitices is uniformly distributed in G.

I

Proof. We write M = M'U Rs. Define f ( S T ) for disjoint S, T c V(G) to be the number of M'in G using all the vertices of S and none from T. First we show f(S 7')= q ( i , j ) ,where IS I = i and IT1 = j, provided i + j S n - r. That is true for i = 0. (For j = n - r we use the hypothesis and for j < n - r an easy summation argument.) Since

I

I

I

1

f ( u~x T ) = f ( ~7)- f ( ~T

u x ) = q ( i , j )- q ( i , j + 1)

for xk2 S U T and i + j < n - r, we reach the conclusion by induction on i . N o w l e t j = O a n d i = r - s , i f r - s s n - r . Wehave f(SIO)=constantfor I S I = r - s. But this means precisely that M' is uniformly distributed. If n 5 2r - 1, then n 3 2 r - s for any s > 0. So we are entitled to assume henceforth that the number of isolated vertices in M is 0 if n 2 2r - 1, at most 2r - n - 1 otherwise. In particular, if n 3 2r - 1, there can be no r vertices with an isolated member in G. Hence we have the following.

T. Zaslavsky

660

Corollary 3. If n s 2 r - 1, then each y E V(G) has a t least n neighbors.

-r

+1

We can deduce an easy bound for simple G.

CorollaEy 4. If M and G are simple, then n c (r - 1)'. Proof. The idea is to take a vertex x1 of degree 3 n - r + 1, in its neighborhood an xz of degree>[n - ( r -l)]- r + 1, and so on, thus building a clique xI,. . . ,x , - ~ .If n 3 r(r - 2 ) + 2, there will be at least r more vertices adjacent to all of xl,. . . ,x ~ - in ~ , particular an adjacent pair x , - ~ ,n,. This gives an r-clique, contradicting the nontriviality of G. 0

4. Two small examples The first example has r = 4 and M = a two-edge matching. Assuming G is simple, one can prove that n 6 6 = r + 2, the only G with order n = 6 is the octahedral graph CP6, and there are only three possible G with n = 5 (the circuit C,,the bowtie z M, and the pyramid z + C4).The arguments are omitted because they are rather tedious. A second example in which n = r + 2 (suggested by Lovasz) has r = 8, M = a 4-edge matching, and n = 10, G = the Petersen graph. One can check uniform distribution easily because G and its complement are edge-transitive, so there are only two kinds of induced r-subgraphs.

+

5. Centers A center in a vertex set X is a vertex adjacent to every other in X . An important example is that in which M has a center. For instance, M may be the star S,-,. This example introduces the technique we use later to get our best bound for simple graphs. Let CP, denote the cocktail party graph on m vertices ( K , less a perfect matching). Proposition 5. Suppose M is simple and has a center u, and suppose G is simple. Then n = r + 1, r is odd, M \ u C CP,-I, and G = CP,+I.

Proof. Say M = u + MI.We write X i for a set of i vertices of G. Take an X , and a center y in it; let X,-, = X , \ y . Now let y ' E X , and consider X : = X , - , U y r . Any copy of M on X , (similarly on X : ) arises in one of the following ways:

Uniform distribution of a subgraph in a graph

661

(i) MI lies in X,-l and u corresponds to y ; (ii) MI contains y and u goes to w E X , - , . A copy of M in X:of either type corresponds to one copy on X, by exchanging y for y ' , because y is a center. (We rely on the simplicity of M and G.)But if y ' is not a center in X i , a copy of type (i) on X, has no corresponding copy on X:. This, if it occurred, would violate uniform distribution. But type (i) does occur, since each w in type (ii) is a center for X, and can therefore be interchanged with y. We conclude that every y ' E X L is adjacent to all X,-l. In fact, X L is a coclique. Suppose not, say y y ' . Let w be a noncenter in X,-l. (It exists because X , cannot be a clique, since G is simple.) M has no copy on X , with u corresponding to w ; but if we replace w by y then it does. Since that contradicts uniform distribution of M, y and y f could not have been adjacent. Since n > r, we have I X:-l I 3 2. We show X,-, has no centers. Any such center z is also a center of X,, whence XfU z is a coclique; but that contradicts the adjacency of every y f E XC to every member of X,-l. (This argument demonstrates that the center in every X : is unique.) Finally let X:= (X,-l U yy')\x for x E X,-, and let z be the center of X:.We see that X = (X:\z)c is a coclique adjacent to all X:\z, in particular to y . If Xfr l Z # 0, there is a contradiction. So n = r + 1. Moreover by choosing x to be each member of X,-l in turn we find that G consists of cocliques of size 2 and all points in different cocliques are adjacent. 0

-

6. Complete models If M is complete with constant multiplicity we have an easy strong bound on n.

Proposition 6. If M = AK,, then n = r + 1.

Proof. Since G is nontrivial, all edge multiplicities p ( x y ) are at least A. Uniform distribution means that

(rn is a constant), for every X , C V ( G ) . Multiplying the expressions for rn(M,X,) over all X, in a fixed X,+lwe get rn'"

=

fl (p ( AI ) w )) '-'= f(X,+,Y-',

uw s x, + I

where f ( Y ) is the number of copies of M in Y C V ( G ) .Thus, for any X , and

vex,,

662

7.Zaslavsky

a constant. If n 3 r + 2, X,-, 5 V(G), and u, w, ye X ,-l, we can conclude by comparing X , = X,-, U w and X,-, U y that

whence p ( u w ) = p ( u y ) for immediate. 0

any

u, w, y E V(G).

The

proposition

is

There are no nontrivial examples of order n = r + 1 if r = 2, but for r 3 3 we U p ‘ H , where p 2 A and H is any regular can take, for instance, G = simple graph of order r + 1 besides K,+,and K,+l.One can describe all possible G in a fairly effective but complicated way.

7. Our best bound

We can improve greatly on Corollary 4 by an argument developed jointly with Paul Seymour.

Theorem 7 (Seymour and Zaslavsky). If M and G are simple, then n C 2r - 2. This is surely not the best bound. I conjecture n d r + 2. The two small examples given earlier show that n $ r + 1 in general, but they may be exceptional. Examples with n = r + 1, however, are plentiful. Lovisz pointed out to me that one can take G to be vertex transitive and A4 any spanning subgraph of its vertex deletion (so generalizing the example of Proposition 5). The theorem follows directly from Corollary 3 and the following lemma.

Lemma 8. If M and G are simple, then G has maximum degree at most r, at most r - 1 if n > 2 r - 2 . Proof. Again let Xistand for any i-set in V(G). Suppose z is a center in X,+l.Let X,= Xr+t\z. Call an edge p q ‘real’ if there is a copy of M in X,+, using it. We will show that every edge zy is ‘real’. If not, say zy is not ‘real’. Each copy of M \ u (where u E V ( M ) )in X,\ y extends to a copy of M in Xr+,\ y, therefore (by uniform distribution) to a copy in X,.So if zx is ‘real’, say with the correspondences u -+ z and u‘-+ x (where u, U‘ E V ( M ) ) ,

Uniform distribution of a subgraph in a graph

663

then so is y x with v + y and v ’ + x . But now we have acopyof M \ u ’ o n X,\x, which extends to a copy of M on X I + \, x by v ’ + z. So zy is ‘real’ after all. Next suppose w $Z Xr+land x E X,. Since zx is ‘real’, there is a copy Mo of M using zx, on vertices X , + ,\ y say. Let X : = (Xr\ y ) U w. Any copy of M on X : (say with v -+ w ) yields a copy on X,+,\y (by shifting v to z), so by uniform distribution there are no other copies of M on X,+,\y. That means M , corresponds to a copy on X:using wx. Since x was arbitrary in X,, we conclude w x for all W EX,+, and x E X,. Now suppose z can be chosen with dG( z )3 r + 1 and let X,+2consist of z and r + 1 neighbors. By the preceding, X,+*\z is a clique; thus G = K,. But this we excluded as trivial. So every vertex in G must have d e g r e e s r. Let us return to the case where z is a center for X , + ] .We showed that every w P XI+] is adjacent to all of X , but not to z , hence XS is a coclique of size n - r. That entails n - r S a ( M ) ,the independence number of M. But a ( M )< r since M # K,.So n > 2r - 2 implies that there is a vertex in G with at least r neighbors (by Corollary 3). Thus a ( M ) S n - r k r - 1. The case a ( M ) = r is trivial. If a ( M )= r - 1, M is a star v + since there are no isolated vertices (assumption justified by Proposition 2), and by Proposition 5 we have n s r + 1, a contradiction. 0

-

I suspect the restriction n > 2r - 2 in the lemma is removable, with the exception of M = K , U K2, where G = CP, has degree 4 = r and order n = 6 = 2r - 2.

8. Wide uniform distribution

Suppose each induced subgraph of G of order q contains the same number of copies of M, where 4 is a fixed number 5 r. Let us call this wide uniform distribution of M . The smallest possible 4 is the width of the distribution. (This idea was proposed to me by Erdos.) We can apply all the previous results with model M U Kq-, and obtain the properties of uniform distribution for a model of order q. But by Proposition 2, if n 5 4 + r then M is uniformly distributed. So there is no need to consider wide distribution separately except for small n. As an application we generalize a result of Sir66 141 (for simple G) and Boshk [1,2] - the case r = 2 of the following consequence of Proposition 6. Corollary 9. Suppose every q vertices of G support the same positive number of copies of K . Then G = p K n unless G has order n < q + r.

For r = 2 the nontrivial G are the regular graphs of order q + 1. An example

T. Zaslavsky

664

+

of order n = q 1 valid for any r between 3 and q is any vertex-transitive multigraph (except p K , ) that contains an r-clique. I do not know any nontrivial examples of higher order with width q > r 3 3.

References [I] J. Boslk, Induced subgraphs, in: Proc. 6th Hungarian Coll. on Combinatorics, Eger (1981) submitted. [2] J. Bosak, Induced subgraphs with the same order and size, Math. Slovaca, submitted. [3] C.D. Godsil, Hermite polynomials and a duality relation for the matchings polynomial, Combinatorica 1 (1981) 257-262. [4]J. Sirlh, On graphs containing many subgraphs with the same number of edges, Math. Slovaca 30 (1980) 267-268. [S] T. Zaslavsky, Generalized matchings and generalized Hermite polynomials, in: Proc. 6th Hungarian Coll. on Combinatorics, Eger (1981), submitted.


LIST OF PROBLEMS SUBMITTED DURING THE CONFERENCES Problem 1 (L. Batten) A geodesic subgraph of a (finite, simple, connected) graph G is a subgraph A with the property that any geodesic of two points of A is in A. It is easy to see that G provided with its geodesic subgraphs is a closure space. Consider only those graphs G for which this closure space satisfies the Jordan-Dedekind chain condition: All maximal chains of closed sets have the same length (examples: trees, complete graphs). Calculate jd(u, b ) , the number of such graphs having u points and b edges. Note. For a tree, jd(u, b ) = rv, the number of trees on u points.

Problem 2 (A. Benhocine and A.P. Wojda) Let n and p be integers such that 2 s p S n. We call D (n,p ) a digraph formed by two dipaths (x = a l , .. . ,ap = y ) and (x, b l , .. . , b,-,, y ) such that a,# b,. We proved that if D is a digraph of order n 3 3 and size e ( D ) a ( n - l ) ( n - 2) + 2 then D contains D ( n ,p ) , 2 zs p S n except if D is isomorphic to M ( n ) (the digraph with n vertices and (n - l)(n - 2) + 2 edges containing K , L and one pendent vertex, such that all edges of M ( n ) are symmetric) or n = 5 and D is isomorphic to K : from which we reject a symmetric cycle of the length 3. Problem 2.1. Prove the similar theorem for antisymmetric digraphs. Problem 2.2. We conjecture that for every sequence (a,, . . . , a n ) such that a, = 1 or a, = - 1 in every digraph of order n and size e ( D )3 ( n - l)(n - 2) + 2 there exists a cycle xl,. .., x n , x , such that ( x , , x , + , ) E E ( D if) a, = I, ( x # + x,) ~ , E E ( D )if a, = - 1, except if al = * * = a, ;or D isomorphic to M ( n )or n = 5 and D isomorphic to N.

-

Problem 3 (V. Chvatal) An undirected graph is called adorable if it admits an acyclic orientation such 665

List ofproblems

666

that no induced path with four vertices is directed in the way indicated below:

0

\

(either)

/

.-

,

\

0

Adorable graphs are strongly perfect and have another nice property related to perfection; the class of adorable graphs includes all the comparability graphs, chordal graphs and complements of chordal graphs. Is there a good algorithm for recognizing adorable graphs?

Problem 4 (W. Deuber)

n persons 1,. . .,n play the following game on a directed graph G with special point x o E V(G): Starting in xo the players in turn move along arcs. If player i reaches an endpoint, the game is finished and player i obtains rank n, player i - 1 obtains rank n - 1, player i - 2 obtains rank n - 2. Every player is supposed to maximize his rank (thus making the game coalition free). For n = 2 , ‘kernels’ and ‘Grundy functions’ may be applied in order to determine the game. In particular, the solution is well known for the game with piles of matches. For n > 2 the case of games with piles of matches is understood, by generalizing the Grundy approach. Nothing seems to be known for arbitrary graphs G.

Problem 5 (M. Deza)

- -

A subgraph G of S, has type L (where L = {lo,I,, . . . ,L,}, lo < 1, < < l,-,, is a set of integers) if any a E G fix f ( u )E L points. Kyota proved my conjecture that I G 1 divides (n - C). We say that g is L-shurp if I G 1 is equal to (n - li). For L = {0,1,. ..,r - 1) it is exactly sharply #-transitive groups. Problem 5.1. Show that G has exactly lo + 1 orbits if lo # 0. Problem 5.2. In general, classify #-sharp groups (the known cases are only L ={s,s + 1,. , ., s + #}, L ={s,s + 2 } and L = {s,s +3}).

nIiEL

List ofproblems

667

Problem 6 (S. Fiorini and J. Lauri) Definitions 6.1. (i) Sk is a surface of genus k (orientable or nonorientable). (ii) Graph G triangulates s k if G embeds in S, in such a way that each face is bounded by a 3-circuit. (iii) G has a p-representation in S, ( p 3 4) if G embeds in s k in such a way that each face except one is bounded by a 3-circuit; the exceptional face is bounded by a p-circuit. (iv) p ( u ) = valency (degree) of u E V ( G ) . Conjecture 6.2. G triangulates

s k

if G - u has a p(u)-representation in Sk,

Vv E V(G).

The conjecture is true for So.

Problem 7 (J.L. Fouquet) Conjecture 7.1. Soit G = ( X , U ) fortement connexe.

k

= max { d + ( x ) , d - ( x ) } . XEX

1

Soit h ( G )= min{p G Phamiltonien}. Alors h ( G ) S k. Problem 8 (G. Hahn)

Let n and k be given (positive) integers. It is not difficult to see that there is a least m = sr(P,,, k ) such that in any K,,, whose edges are coloured using no colour more than k times, there is a path P,, on n points all of whose edges are of different colours. Conjecture 8.1. If n > 2k then sr(P,,, k ) = n.

This is true for k = 2 (proof by counting P,,’s).

Problem 9 (A. Hilton)

Let x,(G) be the path-chromatic number of a graph, i.e., the minimum of colours needed so that each colour class is the vertex disjoint union of paths. Is it true that

668

List of problems

(This problem is due to J. Akyama.)

Problem 10 (A. Hilton) Let n be even, and let n - 2 edges of K , be coloured properly. Show that this partial colouring can be extended to a proper edge colouring of K,, with n - 1 colours.

Problem 11 (A. Hilton) If n elements in an n X n matrix are specified, then it is known exactly when this can be completed to form a latin square. What if more than n elements are fixed?

Problem 12 (F. Jaeger) Soit V la classe des graphes on orie tts cubiques sans isthmes. Pour G E %' on note G *le graphe orient6 symCtrique obtenu en remplaqant chaque artte de G par deux arcs opposCs, qui seront dits former un doublet; trois arcs issus d'un mtme sommet ou trois arcs arrivant en un mtme sommet forment un triplet. Pour G I ,G2dans %' on Ccrit G ,3 G2s'il existe une application de I'ensemble d'arcs de G: sur celui de G: qui preserve les doublets et les triplets. It s'agit d'une relation d'ordre dont je connais trois ClCments minimaux: le graphe H3 a deux sommets et trois arttes parallttles, K4 et le graphe de Petersen P. Conjecture 12.1. I1 n ' y a pas d 'autres e'le'ments minimaux. Cette conjecture implique celles du 5-flot de Tutte, de la double couverture par des couplages de Fulkerson et de la double couverture par des cycles. I1 parait intiressant d'Ctudier une conjecture plus faible, a savoir: Conjecture 12.2. (%, 2 ) a un nombre fini d 'e'lkments minimaux.

List of problems

Problem 13 (J.M. Laborde)

Let a (0,)be the stability number of the n-cube. It can be shown (M. Mollard) that a(Q,)=k

n2" x where k , E [1,2].

Is it true that limn-.z k, = l? Recall that B ( G ) is the minimum cardinality of B C V(G) such that every vertex of G is in B or joined by an edge to some vertex in B. Problem 14 (H. Meyniel)

Soit G un graphe, 9'la famille des stables de taille maximale de G, T un transversal de Y supposC minimal. Soit h la taille de la plus grande clique en laquelle G est contractible. Quelle est la meilleure fonction de h f ( h ) telle que I TI g ( h ) ? Problem 15 (E. Pin)

Let E be a finite set such that cardE = n. If f :E rank f = card f ( E ) .

+E

is a mapping define

Conjecture 15 (temp-Pin). Let 1 s k =s n and let A be a subset of E E , the set of all mappings from E to E. Assume rhat the semigroup (under composition) generated by A contains at least one element of rank S ( n - k ) . Then there exists an element of r a n k s ( n - k ) of the form a 1 0 a 2 0 ~ ~ - owhere a p , a l ,..., ap € A and p =sk'. Problem 16 (H. Sachs)

I have posed the following problems many times, but, as far as I know, there has been no progress. I shall therefore pose them again, using the most intuitive formulation. A ball configuration B is a finite set of unit balls of a metric space such that the balls are allowed to touch but not to intersect (see Fig. 1).

&@

Fig. 1.

670

List ofproblems

A coloration of (the balls of) B is called feasible iff any two balls of B that touch have different colours. Let x(B)denote the minimum number of colours required for a feasible coloration of B and let x,,be the maximum of x ( B ) extended over all ball configurations B of the euclidean n-dimensional space. It is easy to show that x. exists. All facts known about x. are the following. (i) x,= 2 (trivial); (ii) x 2 = 4 (this can easily be proved without using the Four Colour Theorem); (iii) 5 s ,y3S 10. Problems. Determine x3,and determine good bounds for xn for n > 3. Obviously, the problems can be generalized in many ways.

Problem 17 (H. Grotsch) Draw a finite set S of topological circuits in the plane such that (i) the circuits are allowed to intersect but not to touch; (ii) any point of the plane is contained in at most two circuits of S ; (iii) the number of intersections is finite. Clearly, the plane graph G generated by S is finite and regular of degree four (see Fig. 2).

Fig. 2.

Let, as usual, x denote the chromatic number with respect to vertex colouring. Conjecture 17.1. x ( G ) s 3. The problem is unsettled even in the case of geometrical circuits.

ANNALS OF DISCRETE MATHEMATICS VOl. 1:

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