Graph Theory in Memory of G.A. Dirac (Annals of Discrete Mathematics)

GRAPH THEORY IN MEMORY OF G.A. DIRAC

ANNALS OF DISCRETE MATHEMATICS

General Editor: Peter L. HAMMER Rutgers University,New Brunswick, NJ, U.S.A.

Advisory Editors: C. BERGE, Universitede Paris, France M. A. HARRISON, Universityof California,Berkeley, CA, U.S.A. V. KLEE, Universityof Washington,Seattle, WA, U.S.A. J.-H. VAN LINT CaliforniaInstitute of Technology,Pasadena, CA, U.S.A. G.-C.ROTA, MassachusettsInstitute of Technology, Cambridge, MA, U.S.A.

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NORTH-HOLLAND -AMSTERDAM

NEW YORK

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GRAPH THEORY IN MEMORY OF GmAm DIRAC Edited by

Lars DOVLING ANDERSEN Department of Mathematics and Computer Science Institute of Electronic Systems Aalborg University, Denmark

IvanTAFTEBERG JAKOBSEN SkanderborgAmtsg ymnasium, Denmark

CarstenTHOMASSEN Mathematical Institute The Technical University of Denmark

BjarneTOFT Mathematical Institute Odense University, Denmark

Preben Dahl VESTERGAARD Department of Mathematicsand Computer Science Institute of Electronic Systems Aalborg University, Denmark

1989

NORTH-HOLLAND-AMSTERDAM

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NEW YORK

OXFORD TOKYO

Elsevier Science Publishers R,V., 1989 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmittedin any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publishers, Elsevier Science Publishers R. V. (Physical Sciences and Engineering Division/, F! 0. Box 103, 1000 AC Amsterdam, The Netherlands.

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Preface

From 2 to 7 June 1985, a meeting was held at Sandbjerg, Denmark, in memory of the graph theorist Gabriel Andrew Dirac who died the year before. Attendance was by invitation only, and 55 mathematicians from 14 countries participated in lectures and discussions on graph theory related to the work of Dirac. This volume contains contributions in honour of the memory of Dirac from participants and others, and should not be seen only as proceedings from the meeting. All the papers have been refereed, and we wish to thank all the referees who have devoted their time to the project. We also thank the people who have helped typing the manuscripts: Susanne Albzk, Karin B. Andersen, Martin Hare Hansen, Allan L. Jensen, Frank Jensen, Susanne Kzseler and Astrid Pedersen. Professor W. T. Trotter Jr. is thanked for his careful editing of the contributions to the problem session. The meeting at Sandbjerg was made possible through donations and contributions from 0 0

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The Danish Natural Science Research Council The Carlsberg Foundation The Ministry of Education The Tuborg Foundation The Otto Mmsted Foundation The Thomas B. Thrige Foundation The University of Aarhus The University of Odense Aalborg University The Technical University of Denmark

The editors express their gratitude to all these sources for their interest and support.

V

vi Finally, we thank the Institute of Electronic Systems, Department of Mathematics and Computer Science, Aalborg University, for financial and technical assistance in preparing the volume.

Denmark, July 1988 Lars Dmling Andersen Ivan Tafteberg Jakobsen Carsten Thomassen Bjarne Toft Preben Dahl Vestergaard

Contents

Preface

Gabriel Andrew Dirac THEEDITORS Hamilton Cycles in Metacirculant Graphs with Prime Power Cardinal Blocks

7

B. ALSPACH The Edge-Distinguishing Chromatic Number of Paths and Cycles K. AL-WAHABI, R. BARI,F. HARARYand D. ULLMAN

17

On the 2-Linkage Problem for Semicomplete Digraphs J . BANG-JENSEN

23

The Fascination of Minimal Graphs R. BODENDIEK and K. WAGNER

39

Optimal Paths and Cycles in Weighted Graphs

53

J. A. BONDYand G. FAN A Note on Hamiltonian Graphs H. J. BROERSMA

71

Uniqueness of the Biggs-Smith Graph

75

A. E. BROUWER Some Complete Bipartite Graph - Tree Ramsey Numbers

S. B U R RP. , ERDOS,R. J. FAUDREE, C. C. ROUSSEAU and R. H . SCHELP vii

79

viii The Edge-Chromatic Class of Graphs with Maximum Degree at Least IVI - 3 A. G. CHETWYND and A. J. W. HILTON

91

On some Aspects of my Work with Gabriel Dirac P. ERDOS

111

Bandwidth versus Bandsize P. ERDOS,P. HELLand P. WINKLER

117

Circumference and Hamiltonism in K 1 ,s-Free Graphs E. F L A N D R II.NFOURNIER , and A. GERMA

131

The Prism of a 2-Connected, Planar, Cubic Graph is Hamiltonian ( a Proof Independent of the Four Colour Theorem) H . FLEISCHNER

141

A Note Concerning some Conjectures on Cyclically 4-Edge Connected 3-Regular Graphs H . FLEISCHNER and B. JACKSON

171

On Connectivity Properties of Eulerian Digraphs A . FRANK

179

Some Problems and Results on Infinite Graphs

195

R. HALIN On a Problem Concerning Longest Circuits in Polyhedral Graphs J . HARANTand H. WALTHER Interpolation Theorems for the Independence and Domination Numbers of Spanning Trees F. HARARYand S. SCHUSTER

211

22 1

Ein zum Vierfarbensatz Aquivalenter Satz der Panisochromie H. HEESCH

229

The Existence Problem for Graph Homomorphisms P. HELLand J . N E S E T ~ ~ I L

255

ix On Edge-Colorings of Cubic Graphs and a Formula of Roger Penrose 267 F. JAEGER Longest &Paths in Regular Graphs H. A. JUNG On Independent Circuits in Finite Graphs and a Conjecture of Erdos and P6sa. P. JUSTESEN

281

299

Contractions to Complete Graphs L. K. JORGENSEN

307

Triangulated Graphs with Marked Vertices H.-G. LEIMER

311

On a Problem upon Configurations Contained in Graphs with Given Chromatic Number F. LESCURE and H . MEYNIEL On Disjoint Paths in Graphs W. MADER Conjecture de Hadwiger: Un Graphe Ic-Chromatique Contraction-Critique n’est pas Ic-RQgulier J. MAYER

325

333

341

A Theorem on Matchings in the Plane M. D. PLUMMER

347

Removing Monotone Cycles from Orientations 0. R. L. PRETZEL

355

Disentangling Pairings in Trees G. SABIDUSSI

363

Colour-Critical Graphs with Vertices of Low Valency H. SACHSand M. STIEBITZ

371

About the Chromatic Number of 1-Embeddable Graphs H . SCHUMACHER

397

A

Problems and Conjectures in the Combinatorid Theory of Ordered Sets W. T. TROTTER A Proof of Kuratowski’s Theorem H. TVERBERG

40 1

417

Finite and Infinite Graphs whose Spanning Trees are Pairwise Isomorphic P. D. VESTERGAARD

42 1

Bridges of Longest Circuits Applied to the Circumference of Regular Graphs H.-J. VOSS

437

On a Standard Method Concerning Embeddings of Graphs K. WAGNER and R. BODENDIEK

453

Construction of Critical Graphs by Replacing Edges W. WESSEL

473

A Brief History of Hamiltonian Graphs R. J. WILSON

487

Erinnerungen an Gabriel Dirac in Hamburg E. WITT

497

Hamilton Paths in Multipartite Oriented Graphs

499

Problems Edited by W. T. TROTTER

515

C.-Q.ZHANG

This Page Intentionally Left Blank

Drawing by J. NeLtEl

Annals of Discrete Mathematics 41 (1989) 1-6 0 Elsevier Science Publishers B.V. (North-Holland)

Gabriel Andrew Dirac The editors

With the passing of Gabriel Andrew Dirac on July 20, 1984, at the age of 59, the international community of graph theorists lost one of its main figures, and Danish university life lost a most stimulating and generous colleague and teacher. Denmark had an early tradition in graph theory through the work of Julius Petersen, but after his death in 1910 graph theory was soon forgotten in this country (however Petersen’s highschool textbooks were used as late as in the early 1960’ies). With Dirac’s move to Aarhus University graph theory was again taken up at a serious level in Denmark. At Aarhus University, in 1966-67 and from 1970, Dirac attracted a large number of students. His lectures were catching. He presented graph theory as a general and important mathematical theory and did so with rigour and meticulous care. Dirac’s thorough style, influenced by the book of Konig (of which he had a very high opinion), was a delight for his students. Dirac was a fascinating person. He had an unconventional view of many matters, he had a penetrating and sharp mind and did not take very much for granted. This held not only for mathematics, but also in everyday political and social life. Therefore it was very enriching to know him and learn from him, even if one did not always agree fully with him. Dirac was a truly international figure. He was born on March 13, 1925, in Budapest, moved to England in 1937 when his mother married the great physicist and Nobel prize laureate P. A. M. Dirac. After wartime service in the aircraft industry and mathematical studies at the universities of Cambridge and London (obtaining his doctorate in 1951 under Richard Rado), he held university positions in London, Toronto, Vienna, Hamburg, Ilmenau, Dublin, Swansea and finally Aarhus. He was a British subject, but also knew Hungarian, German, French and Danish cultures and languages very well. Besides mathematics and a happy family life with his wife Rosemari Dirac and four children Meike, Barbara, Holger and Annette, Dirac’s great

The editors

2

passion was fine art. He took an interest in numerous other things and was a very knowledgeable and multi-faceted person. Dirac is among the most quoted graph theorists. He developed methods of great originality and made many fundamental discoveries. A survey of his main contributions is contained in the obituary by C. Thomassen [J.Graph Theory 0 (1985), 303-3181. By arranging the meeting at Sandbjerg in South Jutland, Denmark, in the summer of 1985, we wished to honour Dirac as a pioneer of graph theory, as an inventive and deep researcher for 35 years, as the most excellent teacher we have known, and last but not least as a good friend with a never ending interest in his students and friends. Dirac continues to be a rich source of inspiration, in mathematical research and presentation, as well as in the way we think about many other aspects of life. We wish to thank Dirac’s family and all participants a t Sandbjerg for making the meeting a worthy memory of G. A. Dirac and his achievements.

The publications of Gabriel Andrew Dirac [11 “The explicit determination of orifice parameters in shock absorbers,” Aircraft Engineering XIX (1947), 258-262. [2] “The vibration of propeller blades,” Aircraft Engineering XX (1948),

322-329, 343. [3] “Note on the colouring of graphs,” Math. 2. 5 4 (1951), 347-353. [4] “Collinearity properties of sets of points,” Quarterly J. Math. (2) 2 (1951), 221-227. [5] “Note on a problem in additive number theory,” J. London Math. SOC.26 (1951), 312-313.

[6] “A property of 4-chromatic graphs and some remarks on critical graphs,” J. London Math. SOC.27 (1952), 85-92. [7] “Ovals with equichordal points,” J. London Math. SOC.27 (1952)) 429-437. [8] “Connectivity theorems for graphs,” Quarterly J. Math. (2) 3 (1952)) 171-1 74. [9] “Some theorems on abstract graphs,” Proc. London Math. SOC.(3) 2 (1952), 69-81.

Gabriel Andrew Uirac

3

[lo] “Map colour theorems,” Canad. J. Math. 4 (1952), 480-490.

[ll] “The structure of k-chromatic graphs,” Fund. Math. XL (1953), 4255. [12] “The colouring of maps,” J. London Math. SOC.28 (1953), 476-480. [13] “Theorems related to the four colour conjecture,” J. London Math. SOC.29 (1954), 143-149. [14] With S. Schuster, “A theorem of Kuratowski,” Indagationes Math. 10 (1954), 343-348. [15] “Circuits in critical graphs,” Monatsh. f. Math. 59 (1955), 178-187. [16] “Map colour theorems related to the Heawood colour formula,” J. London Math. SOC.31 (1956), 460-471. [17] “Map colour theorems related to Heawood colour formula (II),” J. London Math. SOC.32 (1957), 436-455. [18] “A theorem of R. L. Brooks and a conjecture of H. Hadwiger,” Proc. London Math. SOC.(3) 7 (1957), 161-195. [19] “Short proof of a map colour theorem,” Canad. J. Math. 9 (1957), 225-226. [20] “Paths and circuits in graphs: extreme cases,” Acta Math. Acad. Sci. Hungar. 10 (1959), 357-362. [21] “GQnQralisationsdu thQor6me du Menger,” Comptes Rendus Paris 250 20 (1960), 4252-4253. [22] “Un theoreme de rkduction,” Comptes Rendus Paris 251 1 (1960), 24-25. [23] With M. Stojakovic, Problem cetiri Boje (The Four Colour Problem), Matematicka Biblioteka 10,Belgrade (1960).

[241 “Trennende Knotenpunktmengen und Reduzibili tat abstrakter Graphen mit Anwendung auf das Vierfarbenproblem,” J. reine angewandte Math. 204 (1960), 116-131. [25] “bchrome Graphen und vollstandige 4-Graphen,” Math. Nachr. 22 (1960), 51-60.

4

The editors

[26] “In abstrakten Graphen vorhandene vollstandige 4-Graphen und ihre Unterteilungen,” Math. Nachr. 22 (1960), 61-85. [27] “Connectedness and structure in graphs,” Rendiconti del Circolo Matematico di Palerrno Ser. II 9 (1960), 1-11. [28] “On rigid circuit graphs,” Abh. Math. Sem. Univ. Hamb. 25 (1961), 7 1-76. [29] “A contraction theorem for abstract graphs,” Math. Ann. 144 (1961), 93-96. [30] “Note on the structure of graphs,” Canad. Math. Bulletin 5 (1962), 22 1-227. [31] “On the maximal number of independent triangles in a graph,” Abh. Math. Sem. Univ.Hamb. 26 (1963), 78-82. [32] With P. Erdos, “On the maximal number of independent circuits in a graph,” Acta Math. Acad. Sci. Hungar. 14 (1963), 79-94. [33] “On the four-colour conjecture,” Proc. London Math. SOC. (3) 13 (1963), 193-218. [34] “Extensions of Menger’s theorem,” J. London Math. SOC.38 (1963), 148-161. [35] “Some results concerning the structure of graphs,” Canad. Math. Bull. 6 (1963), 183-210. [36] “On complete graphs and complete stars contained as subgraphs in graphs,” Math. Scand. 12 (1963), 39-46. [37] “Extensions of Turh’s theorem on graphs,” Acta Math. Acad. Sci. Hungar. 14 (1963), 417-422. [38] “Percy John Heawood,” J. London Math. SOC.38 (1963), 263-277. [39] “Homomorphism theorems for graphs,” Math. Ann. 153 (1964), 6980. [40] “Valency-variety and chromatic number of abstract graphs,” Wiss. Zeitschrift Univ. Halle 13 (1964), 59-63. [41] “On the structure of 5- and 6-chromatic abstract graphs,” J. reine angewandte Math. 214-215 (1964), 43-52.

Gabriel Andrew Dirac

5

[42] “Graph union and chromatic number,” J. London Math. SOC.30 (1964), 451-454. [43] “Extensions of the theorems of Tur&n and Zarankiewicz,” Proceedings of the Symposium in Smolenice June 1968, Czechoslovak Academy of Sciences (1965), 127-132. [44] “Generalizations of the five colour Theorem,” Proceedings of the Symposium in Smolenice June 1968, Czechoslovac Academy of Sciences (1965), 21-27. [45] “Chromatic number and topological complete subgraphs,” Canad. Math. Bull. 8 (1965), 711-715. [46] “Short proof of Menger’s graph theorem,” Mathematika 13 (1966), 42-44. [47] “Minimally 2-connected graphs,” J. reine angewandte Math. 228 (1967), 204-216. [48] “On the structure of &chromatic graphs,” Proc. Cambridge Phil. SOC.63 (1967), 683-691. [49] “Extension of Konig’s Lemma,” Math. Nachr. 40 (1969), 43-49. [50] “On Hamilton circuits and Hamilton paths,” Math. Ann. 197 (1972), 57-70. [51] “On arbitrarily traceable graphs,’’ Math. Scand. 31 (1972), 319-378. [52] “On systems of paths and circuits in graphs,” Math. Ann. 201 (1973), 133-144. [53] “Uber minimale KP-anfdlige Graphen,” Matematisk Institut, Aarhus Universitet, Preprint Series 20 (1970/71). [54] With C. Thomassen, “Graphs in which every finite path is contained in a circuit,” Math. Ann. 203 (1973), 65-75. [55] “Note on Hamilton circuits and Hamilton paths,” Math. Ann. 206 (1973), 139-147. [56] “On separating sets and Menger’s theorem,” Indagationes Math. 35 (1973), 49-62.

6

The editors

[57] With B. Aagaard Sprrensen and B. Toft, “An extremal result for graphs with an application to their colourings” J . reine angewandte Math. 268-269 (1974), 216-221. [58] “The number of edges in critical graphs,” J. reine angewandte Math. 268-269 (1974), 150-164. [59] With C. Thomassen, “On the existence of certain subgraphs in infinite graphs,” Indagationes Math. 36 (1974), 406-410. [60] “Terminable and interminable paths and trails in infinite graphs,” Discrete Math. 9 (1974), 19-34. [61] “Note on infinite graphs,” Discrete Math. 11 (1975), 107-118. [62] “Remarks on Ic-chromatic graphs and on terminable and interminable trails and paths,” Atti dei Convegni Lincei 17 (1976), 407-421. [63] “Structural properties and circuits in graphs,” Proceedings of the Fifth British Combinatorial Conference 1975, Congressus Numerantium X V (1976), 135-140. [64] With O.S. Nielsen, “On path-amenable graphs” Math. Nachr. 74 (1976), 263-268. [65] “Cardinal-determining subgraphs of infinite graphs,” Contribu ti del Centro Linceo Interdisciplinare di Sci. Mat. 34 (1977), 1-14. [66] “Hamilton circuits and long circuits,” Annals of Discrete Math. 3 (1978), 75-92. [67] “Interminable paths and trails: extreme cases,” Math. Nachr. 82 (1978), 7-19. [68] “Simplicial decompositions of infinite graphs,” Istituto Nazionale di Alta Matematica Francesco Severi, Symposia Mathernatica XXVIII (1987), 159-196.

Annals of Discrete Mathematics 41 (1989) 7-16 0 Elsevier Science Publishers B.V. (North-Holland)

Hamilton Cycles in Metacirculant Graphs with Prime Power Cardinal Blocks B. Alspach Department of Mathematics and Statistics Simon Fraser University Burnaby, B.C., Canada Dedicated to the memory of G. A . Dirac It is shown that every connected metacirculant graph with prime power cardinal blocks, other than the Petersen graph, has a Hamilton cycle.

1

Introduction

Metacirculant graphs were introduced in [2] as a logical generalization of the Petersen graph for the primary reason of providing a class of vertextransitive graphs in which thergmight be some new non-hamiltonian, connected vertex-transitive graphs. The reader is referred to [2] for their basic properties although their construction is now described. Let m and n be two positive integers and a be a unit in the ring of integers modulo n. Let Bi = {u;,o,ui,l, . , u ~ , ~ -for I } i = 0,1, . . . , m- 1 be m mutually disjoint sets of vertices each of cardinal n and called the blocks of the graph being defined. Choose sets of integers Sj {0,1,. . .,n - 1) for j = 0, 1, ..., 1Y.J with the property that amSj = Sj for all j = 0, 1, . . . , [fl, where by definition amSj = { amb: b E Sj }. The set SOhas the additional property that 0 SOand b E SOimplies n - b E SO.If m is even, then Sm12has the additional property that am12Sm/2= -Sm/2 where the negative is taken modulo n. We place edges in the graph as follows. The vertex uoIois adjacent t o uj,k for k E Sj, j = 0, 1, . . . , The remaining edges are obtained by letting the group generated by the two permutations e = (uo,ouo,1 . . U O , ~ - I ) ( u i l o ~ i l i u 1 , n - i ) . . . ( ~ m - i l O u m - l , l . . .urn-1,n-1) and T , where T ( u i , k ) =

..

LfJ.

.

.-

7

B . Alspach

8 ~ ; + ~ , ~act k ,

on the edges incident with UQO. The resulting graph is a metacirculant and is designated by G(m,n,a,SO,S1,. ..,S1,/2~). The Petersen graph is G(2,5,2,{ l, 4}, (0)). It was shown in [3] that if m is odd and n is a prime, then G ( m ,n , a,SO, S1,. ,s[,/2J) is hamiltonian as long as it is connected. The proof depends heavily on D. Marugit’s result [8] which states that every connected Cayley graph on a group which is a semidirect product of a prime order group by an odd order abelian group is hamiltonian. It was shown in [l]that if m is even and n is a prime, then G(m, n , a , SO,S1,. ..,S,/2) is hamiltonian as long as it is connected and not the Petersen graph. This time the proof depends heavily on E. Durnberger’s generalization [6] of MaruBiE’s result in which the former is able to replace odd order abelian by abelian. Recently, D. Witte and K. Keating [7] have generalized the MaruBiE and Durnberger results to obtain the following theorem which is crucial for the proof of the main theorem below.

..

Theorem 1. If G is a connected Cayley graph on a group whose commutator subgroup is cyclic and of prime power order, then G has a Hamilton cycle.

2

Main result

The purpose of this section is to provide a proof of the following result.

..

Theorem 2, Let G = G(m,n, a,SO,S1,. ,SLm/2]) be a metacirculant graph. If n is a prime power, G is connected, and G is not the Petersen graph, then G possesses a Hamilton cycle.

Proof. Let G be a metacirculant graph as described in the hypotheses. The proof proceeds by a sequence of reductions. One may assume n = p “ , p a prime, and e 2 2 .

(1)

The result has already been proved for n a prime in [l]and [3]. Thus, assumption (1) is valid. One may assume amf 1 (mod p e ) ,

(2)

It is shown in [2] that am 1 1 (mod p e ) implies that G is a Cayley graph on the group H generated by Q and r which were described in the introduction. If a = 1, then H is abelian and G has a Hamilton cycle as shown by several people independently (see, for example, [ 5 ] ) . If H is

Hamilton Cycles in Metacirculant Graphs

9

not abelian, then the group (e) generated by e contains the commutator subgroup of H and G has a Hamilton cycle by Theorem 1. Let d be an integer, 1 5 d 5 e , and define a quotient graph G/ p das follows. It has vertices wi,j for 0 5 i 5 m and 0 5 j 5 pe-d - 1 where one can think of wi,j as corresponding to the set of vertices { ui,t : t = j (mod pew') and ui,t E G}. Then wi,j is adjacent to w,.,~if and only if some vertex ui,t in G is adjacent to some vertex uv,uin G where t E j (mod p"-') and u = s (mod pep').

The quotient graph G/pd is isomorphic to the metacirculant graph G ( m , ~ ~ - ~ 5'6, , a 'Si,. , . ,Sim,2j) where

.

a' = a (mod p e P d ) and 1 5 a'

5 pe-d - 1, and

(3)

S i = { j : O < j < ~ ~ - ~ ja =nkd( m o d ~ ~ - ~k )E ,S ; } . It is easy to verify that (3) holds. The quotient graph G/pe was called G / e in [l, 31. If SO= 8, then G has a Hamilton cycle. (4) Suppose first that am f 1 (mod p ) . The quotient graph G/pe is a circulant graph with symbol S = { j : Sj # 8 or S m - j # 0 for 1 5 j 5 m - 1). It is connected because G is connected and, accordingly, Chen and Quimpo's main theorem in [5] implies that G/pe is edge-hamiltonian for m 2 3. Let Gj denote the subgraph induced on Bi by G for i = 0, 1, . . . , m - 1. Since SO = 8, each Gi = the complement of the complete graph with pe vertices. This implies that some S;, i # 0 , must contain an element b f 0 (mod p ) because G is connected. Thus, the edge wowj is in the quotient graph G / p e (the second subscript is dropped in this case as each block of G/pe is a singleton). Let WOW;. . .wo be a Hamilton cycle of G/pe containing the edge WOW; when m 2 3. The case m = 2 will be dealt with shortly. Now lift this cycle of G / pe to a path in G by starting at UO,O and taking the edge to U i , b . Then from ?hi$ take an edge to a vertex ~ jif wiwj , ~ is the next edge of the cycle in G/pe. Continue in this way until returning to a vertex U O , ~E Bo and call the resulting path P. If a f 0 (mod p ) , then the juxtaposition of the paths P,e a ( P ) ,e 2 " ( P ) , ..., e('+l)"(P) results in a Hamilton cycle. If a = 0 (mod p ) , the preceding juxtapositions will not work. Instead, the path P must be modified to ~ a' $ 0 (mod p ) . produce another path P' that returns to a vertex U O , with Since b f 0 (mod p ) and am $ 1 (mod p ) , then amb f b (mod p ) and

rp.,

B. Alspach

10 amb E Si by the property that amSi

= S;. Thus, start P' with the edge

UO,OUi,amb rather than UO,OUi,b and obtain all other edges of P' by taking an edge parallel t o each corresponding edge of P where parallel means the image under the appropriate power of e. The path P' will then terminate a t u ~ ,where ~ I a' = a - b amb. Now a' f 0 (mod p ) when a i 0 (mod p ) and P' gives rise t o a Hamilton cycle in G. In the case that m = 2, notice that -ab E S1 because as1 = -SI (recall that am/2Sm/2 = -Sm/, in a metacirculant graph when m is even). Thus, the 2-path Uo,OU1,bUo,b+(rb is in G. Now b 4-ab = 0 (mod p ) implies that a -1 (mod p ) which, in turn, implies that a2 E 1 (mod p ) . But this contradicts the present case that an f 1 (mod p ) and leads to the conclusion that b ab f 0 (mod p ) . Therefore, juxtapositions of this 2-path under powers of a! as above produces a Hamilton cycle in G. This completes the proof of (4) when am f 1(mod p ) . From (2) and the proof just completed, it may be assumed there is an integer d < e such that am t 1 (mod p d ) but am f 1 (mod pd+'). The proof now proceeds by induction on the exponent e since the result is true when n = p . Consider the quotient metacirculant G/pe-d whose vertices will be labelled q , j , 0 5 i < m and 0 5 j < p d , and whose twist multiplier is denoted a'. Since G is connected, G/pe-d is connected and, by the induction hypothesis, contains a Hamilton cycle H'. Now H' must contain an edge wi,jwrlSwhere s - j f 0 (mod p ) and by suitably operating on H' with powers of e and T , it may be assumed without loss of generality that WO,OWk,bI is an edge of H' with 6' f 0 (mod p ) . Now b' E SL corresponds to some b E Sk and b $ 0 (mod p ) . Now lift H' t o a path P in G that starts with the edge UO,OUk,b and terminates with a vertex ~ 0 where , ~ a E 0 (mod p d ) because H' returns to w o , ~ . Let WObe the set of vertices of Bo that we collapsed t o w o , ~ , that is, WO= { ~ 0 : c, I~ 0 (mod p d ) } . Now if a f 0 (mod #+I), then juxtapositions of P with images of P under powers of e0 will produce a Hamilton cycle. If, on the other hand, a = 0 (mod @I), then start P with UO,OUk,(rmb instead. Use parallel edges to replace the other edges of P to obtain P'. Then P' terminates at UO,+ where a' = a - b amb. Since am = 1 (mod p d ) , a' E WOas required. But a' f 0 (mod p d + l ) for if a' = 0 (mod pd+') held, then amb- b = 0 (mod p d + l ) would be true which would imply am i 1 (mod #+I). Hence, P' can be used t o produce a Hamilton cycle of G. This completes the proof of (4).

+

+

+

If IS01 = 1, then G has a Hamilton cycle. If 15'01 = 1, then the prime p must be 2 and n = 2e,

e

(5)

> 1. In addition,

Hamilton Cycles in Metacirculant Graphs

11

SO = {2"-l} must hold so that each Gj consists of 2e-1 disjoint edges. Therefore, the quotient metacirculant G/2 has 5'4 = 0 and has a Hamilton cycle H' because of (4). Now lift H' as above to a path P in G starting a t UO,O and terminating at either u0,o or ~ 0 , 2 c - 1 . If P terminates a t ~ 0 , 2 e - 1 , then juxtaposing P with a2e-1P yields a Hamilton cycle. If, on the other hand, P terminates at UO,O, let P, be the P. that P begins with U o , o U j , b . Then P, begins with path Q ~ ~ - ' Suppose ~ 0 ~ 2 eU - 1j , b + 2 e - l , Now remove the two preceding edges that begin P and P,, respectively, and insert the edges U O , O U O , ~ ~and -I Uj,bUj,b+2e-l (a's0 = s o as a must be odd). This amalgamates P and P, into a single cycle which must be a Hamilton cycle since P and P, each have length Ze-l. Hence, ( 5 ) is true. If ]Sol 2 3 , then G has a Hamilton cycle. (6)

Again recall the main result of Chen and Quimpo in [5] which states that a connected Cayley graph of degree a t least three on an abelian group is Hamilton connected if it is not bipartite and Hamilton laceable if it is bipartite. First consider the case that p is an odd prime so that each component of G;, i = 0, 1, . ., rn - 1, is not bipartite. Suppose that the components of Go, and thus each G; as well, have cardinal p d , d > 1. Then the quotient metacirculant G/pd has Sh = 0 and by (4) has a Hamilton cycle H'. Each vertex of H' corresponds to a component of some Gi and each edge of H' corresponds to a set of parallel edges joining the vertices of the two components that correspond to the endvertices of the edge. By the Chen-Quimpo Theorem, the subgraph induced on each component is Hamilton connected because the components are circulant graphs. It is then obvious that H' lifts to a Hamilton cycle of G. Now consider the case that p = 2. If the components of Go are Hamilton connected, then the preceding argument works. Suppose that the components of Go have cardinal 2d, d > 2, and induce bipartite graphs, that is, the graphs induced by the components are Hamilton laceable. Again the quotient metacirculant G / 2 d has a Hamilton cycle H' by (4). Each vertex of H' corresponds to a bipartite graph induced by a component of G. If w is a vertex of H', let A , and B,, respectively, denote the bipartition sets of the component corresponding to w. The cycle H' will be lifted in a different manner than either of the two earlier methods. Let WO,O be the vertex of H' whose corresponding component contains UO,O and let UO,O E Awo,o.Choose a vertex UO,b E Bwo,o. There exists a Hamilton path in the component joining UO,O and U 0 , b by the

.

B . Alspach

12

Chen-Quimpo Theorem. Now let w' be the next vertex of H' following w o , ~ . This means that UO,O is adjacent to a vertex 5 in either A,I or B,I, but whichever is the case, U0,b is then adjacent to a vertex y of the opposite bipartition set because of the action of ee-d+l. There is a Hamilton path joining x and y in the component D corresponding to 20'. Join 5 to q o , join y t o U0,b and remove from the Hamilton path in D an edge with endvertices x' and y' distinct from 5 and y. This is possible because ID1 2 4. Now take edges from x' and y' to vertices in opposite bipartition sets in the component corresponding to the next vertex of H'. Join them with a Hamilton path and remove an appropriate edge for extending the partial cycle in G into the next component. Continue working along H' in this way until reaching the last vertex of H' before w o , ~ .Do not remove an edge from the Hamilton path in this component. The result is a Hamilton cycle in G. The proof of ( 6 ) is complete. This leaves the case that [Sol = 2 . Since amSo = So, am f l (mod p e ) and ( 2 ) precludes am 3 1 (mod p"). It is not surprising that this remaining case requires the most intricate proof because the Petersen graph satisfies [Sol = 2 and a2 -1 (mod 5 ) where m = 2.

=

=

When /Sol = 2, G has a Hamilton cycle.

(7)

Let the components of Go have p d vertices, 1 5 d 5 e , so that they are cycles of length p d . The quotient metacirculant G / P " - has ~ Sh = 0 and must be connected. It follows from (4)that G / p e v d has a Hamilton cycle H'. Notice that each vertex of H' corresponds t o a cycle of length pd in G and an edge of H' corresponds to one or more parallel l-factors in G joining the two pd-cycles that correspond to the endvertices of the edge. Hence, the following observation is true.

The subgraph of G corresponding to an edge and its two endvertices in H' contains a graph isomorphic to a generalized Petersen graph.

(8)

The notation G(n,lc) will be used for the generalized Petersen graph with vertices UO, 211, . . ., ~ ~ - 1vo,, v1, . , vn-l and edge set { uivi,u ; u ~ + ~ , viv;+l : i = 0 , 1 , . , , n - 1 with subscripts reduced modulo n } . Since both cycles in the generalized Petersen graph referred to in (8) have length p d , gcd{n,k} = 1. Recall that K. Bannai [4]has proved that G ( n , k ) , when gcd{n, k} = 1, has a Hamilton cycle if and only if G ( n ,k) is not isomorphic to G ( n , 2 ) with n 5 5 (mod 6). But in the latter case there is a compensating result. Namely, in [l]it was shown that there are Hamilton paths in G ( n , 2 ) , when n = 5 (mod 6), from uo to every v;, i # 0. In fact, it is

.

..

13

Hamilton Cycles in Metacirculant Graphs

not hard to show that there is a Hamilton path joining any pair of distinct non-adjacent vertices, but this is not germane to the present proof. Consider lifting the Hamilton cycle H' in G / P " ~ to G. There are two cases t o consider. Let w i , j be the vertex following WO,O in H' and w r , s the vertex preceding w o , ~ . First, assume that the generalized Petersen graph G* induced by the edge Wr,8Wo,o is isomorphic to G(pd,2),pd = 5 (mod 6). In this case start H with u0,o and take an edge from it to a neighbor u ; , j ~in the cycle C ; , j corresponding to w i , j . Continue by taking one of the two Hamilton paths around C i , j starting a t u ; , j , and terminating a t some ui,h+jl. Then take an edge from u;,h+jl to a neighbor on the next cycle corresponding t o the vertex following W i , j on H'. Take a Hamilton path around this cycle starting at the neighbor and then jump to the next cycle using HI. Continue until the cycle C r , s is entered at vertex U ~ , ~From I . remarks above, there is a Hamilton path in G* joining and UO,O unless they happen to be adjacent in G'. Thus, if they happen to be not adjacent in G', H is a Hamilton cycle are adjacent in G*, then use the other Hamilton in G. If u0,o and u r , s ~ path when first travelling around C ; , j . This means that C j , j is exited at a different vertex so that if all subsequent Hamilton paths are given the same orientation ils before, the cycle Cr,s will be entered at a vertex different than U , . , ~ I . Thus, a Hamilton cycle can be constructed in G. It may be assumed that no edge of H' in the quotient graph induces a subgraph containing a generalized Petersen graph of the form G(pd,2 ) with pd 5 (mod 6) or else the argument above establishes the existence of a Hamilton cycle in G. This assumption leads to the second case. First, consider the subcase that H' has odd length. Each vertex of H' corresponds to a circulant subgraph of G (because ee-d is an automorphism of G ) and each is a pd-cycle. Thus, in G , H' corresponds to a sequence of circulant subgraphs each of which is a pd-cycle and such that successive circulants are joined by a 1-factor that is invariant under ee-d. Relabel their vertices as q , o , w q , .. , q P d - l , i = 0, 1, . ., pe-d - 1, so that the vertices whose first index is i 1 correspond to the vertex of IT' that follows the vertices of H' corresponding to those whose first index is i. Hence, e e - d ( v i , j ) = V i , j + l for all i and j . Without loss of generality, it may be assumed that the symbol of the - ~ that is, DO,; is adjacirculant graph on { D O , O , O ~ , ~ ., .. , ~ ~ , }~ isd {l,-1}, cent to v o , i + l for i = 0, 1, . . . , pd - 1. Construct two paths PI and P2 as follows. The initial vertex of PI is DO,O and the initial vertex of Pz is OOJ. Now DO,O is adjacent to v l , j for some j and thus DO,^ is adjacent to v l , j + l . Let ~ 1 , j - k and v l , j - k + l be the vertices immediately preceding the

=

.

+

.

B. Alspach

14

vertices w 1 , j and v l , j + l , respectively, on the cycle 2)1,02)1,kZ)1,2k . . . v1,o. Extend both PI and P 2 by the respective subpaths ' ~ i , j v l , j + k v l , j + 2 k . . .v l , j - k + i and Wllj+l211,j+k+ll)l,j+2k+l ., . v 1 , j - k . Notice that all the vertices in the set { v 1 , 0 , . . ,v l , p d - l } are used and that the second subscript of the last vertex of Pl immediately follows the second subscript of the last vertex of P 2 . That is, the last vertices of the partially constructed paths PI and P 2 have interchanged the role of who comes first. Continue extending PI and P 2 by working along the cycle H' until reaching . ., ~ ~ ~ e - d - ~ , ~ dSince - ~ } .there are an odd number of circulants, the terminal vertex of PI will have the form vmpe-d-l,j and the terminal vertex of P 2 will have the form Vmpe-d-l,j+l. Now Vmpe-d-l,j is adjacent to some vo,;and 2),pe-d-l,j+l is then adjacent to v o , ; + l . If i # 0, then v0,1 to PI and path 2)mpe-d-l,j+lV0,i+l add the path v,pe-d-l,jv0,j2)0,;-1.. 00,;+2. . .oo,o to P 2 thereby producing a Hamilton cycle in G. On the other hand, if i = 0, then at the first stage of the construction of PI and P 2 use the vertices q , j + k and v l , j + l ~ +instead ~ and go around the cycle in the opposite direction. Leave all other extensions unaltered so that the terminal vertices are shifted by 2k. Now complete the cycle as above. Now consider the subcase that H' has even length. The vertex v0,o is adjacent to some v 1 , j which in turn is adjacent to some v 2 , k and so on. Continue in this way constructing a path P that passes through each subcirculant until returning to some vertex W O , ~in {vo,o, v o , ~ , .. . ,v o , p d - l } . If T is relatively prime to p e , then it is easy to see that a Hamilton cycle can be found in G as done earlier. (It is a simple application of the factor group lemma mentioned in [9].) Thus, it may be assumed that T = 0 (mod p ) . In fact, even more can be assumed as is now shown. Let c; be t - j reduced modulo pd where w ; , j v i + l , t is the edge of P corresponding to the edge in H' from the ith vertex to the (i 1)st vertex. Now if any C i , i = 0, 1, . . , , mpe-d - 1 is relatively prime to p , then there is also an edge from vi,j to v i + ~ , j - ~because , am = -1 (mod p") and amS, = S, for all S,. Make this one change in P and the terminal vertex v0,+ will satisfy r' f 0 (mod p). Thus, G has a Hamilton cycle. Thus, it may be assumed that every c; is congruent to zero modulo p . If T = 0, a Hamilton cycle may be found as follows. Let v 2 i I j v 2 i + l , k be an edge of P . Because of (8), there is a generalized Petersen graph induced by the corresponding edge of H' so that it may be assumed that v 2 i , j v 2 ; + l , k is an edge of the geneia.lized Petersen graph. It contains a Hamilton cycle because of K . Bannai's Theorem [4]mentioned earlier. Because of the action of ee-d, there is a Hamilton cycle in the generalized Petersen graph

.

.

.

+

Hamilton Cycles in Metacirculant Graphs

15

. other words, the generalized Petersen graph using the edge v 2 i , j ~ 2 j + ~ , kIn has a Hamilton path from v 2 i , j t o v z ; + l , k . Find these Hamilton paths in the generalized Petersen graphs corresponding to alternate edges of P starting with vo,ovl,j and use the remaining edges of P to connect them together forming a Hamilton cycle in all of G. If each ci is zero, then r is zero and as was just shown this would imply the existence of a Hamilton cycle in G. Therefore, it may be assumed that some ci is nonzero. Let ps be the largest power of p that divides every c;, i = O , l , ..., mpe-d - 1. Clearly, 1 5 s < d. Let G' denote the subgraph of G obtained by replacing each vertex of H' with its corresponding circulant subgraph and replacing each edge of P with the 1-factor between the corresponding circulants where the second coordinates of the edges change by c;. Now @ - d is an automorphism of G' so that the quotient graph G ' / ( Q " - ~ )may " be formed. The edges going between the blocks of this quotient graph have zero change in the second coordinates and from above this implies that there is a Hamilton cycle in the quotient graph G'/(e"-d)s. Now lift this Hamilton cycle in the quotient graph to G' to see what it does there. Start a t the vertex v0,o and follow the Hamilton cycle until reaching a vertex of the form vO,ap~ for the first time. If a f 0 (mod p ) , then the factor group lemma [9] produces a Hamilton cycle in G' which is a subgraph of G. If a = 0 (mod p ) , then find a ci of the form bp' with b f 0 (mod p ) . Such a c; exists by the definition of s. Since am 3 -1 (mod p " ) , -bps is also in the same S j containing bp". Instead of using the edge with ci = bp", use the edge with c j = -bps instead. Make no other changes and the resulting path in G' reaches Wo,apL2bpU instead. Clearly, a - 2b f 0 (mod p ) and the factor group lemma gives a Hamilton cycle in G'. This completes all possible cases and the proof of (7) is complete. There0 fore, Theorem 2 is proved.

3

Conclusion and acknowledgements

The primary question suggested by the result in this paper is whether or not the Petersen graph is the only connected metacirculant without a Hamilton cycle. Indications are that the answer is affirmative. It is likely that the question will not be resolved until the question of whether or not every connected Cayley graph on a group with cyclic commutator subgroup has a Hamilton cycle is resolved. The author wishes to thank the Natural Sciences and Engineering Research Council of Canada whose support is greatly appreciated, the Department of Mathematics, University of Newcastle, Australia for its hospitality

16

B . Alspach

and assistance in the preparation of this paper, and David Witte for several interesting and productive conversations.

References [l] B. Alspach and T. D. Parsons, “On hamiltonian cycles in metacirculant graphs,” Annals of Discrete Math. 15 (1982), 1-7.

[2] B. Alspach and T. D. Parsons, “A construction for vertex-transitive graphs,” Canad. J. Math. 34 (1982), 307-318. [3] B. Alspach, E. Durnberger and T. D. Parsons, “Hamilton cycles in metacirculant graphs with prime cardinality blocks,” Annals of Discrete Math. 27 (1985), 27-34. [4] K. Bannai, “Hamiltonian cycles in generalized Petersen graphs,” J. Combin. Theory ( B ) 24 (1978), 181-188. [5] C. C. Chen and N. F. Quimpo, “On strongly hamiltonian abelian group graphs,” Combinatorial Mathematics VIII (K. McAvaney, ed.), Lecture Notes in Math. 884 (1980), 23-34. [6] E. Durnberger, “Every connected Cayley graph of a group with prime order commutator group has a Hamilton cycle,” Annals of Discrete Math. 27 (1985), 75-80. [7] K. Keating and D. Witte, “On Hamilton cycles in Cayley graphs in groups with cyclic commutator subgroup,” Annals of Discrete Math. 27 (1985), 89-102.

[8] D. MaruBiE, “Hamiltonian circuits in Cayley graphs,” Discrete Math. 46 (1983), 49-54. [9] D. Witte and J. A. Gallian, “A Survey: Hamiltonian cycles in Cayley graphs,” Discrete Math. 51 (1984), 293-304.

Annals of Discrete Mathematics 4 1 (1989) 17-22

o Elsevier Science Publishers B.V. (North-Holland)

The Edge-Distinguishing Chromatic Number of Paths and Cycles K. Al-Wahabi Department of Mathematics University of Qatar Doha, Qatar

R. Bari Department of Mathematics George Washington University Washington D. C., USA

F. Harary*

D. Ullman Department of Mathematics George Washington University Washington D. C., USA

Department of Mathematics University of Michigan Ann Arbor, Michigan, USA

Dedicated to the memory of Gabriel Dirac The edge-distinguishing chromatic number x l ( G ) of a graph G is defined as the minimum number n of colors { 1,2,... ,n} which can be assigned to the vertices V ( G )in such a way that when each edge e = UZI is assigned as its “color” the set of colors {c(u), c ( v ) } , all the edges of G have different colors. We derive a n exact formula for the edge-distinguishing chromatic number of a path and of a cycle. The formulas of Theorems 3.1 and 4.1 appeared in [l]in different but equivalent form. However, we have rederived them in order to provide not only better formulas but also clear, detailed, and complete proofs in this semi-expository note.

1

Definitions

Let G be a graph with vertex set V ( G )and edge set E ( G ) . An r-coloring of G is a mapping C#I from V ( G ) into the set N , = {1,2,. . .,r}. The color of an edge e = uw induced by the r-coloring 6, is q5E(e) = {q5(u),6,(w)}. Thus the coloring 6, induces a coloring q 5 of ~ the edges of G. *Present address: Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003 USA.

17

18

K. A1-Wahabi, R. Bari, F. Haxary and D. Ullman

The coloring 4 is edge-distinguishing, or more briefly admissible if 4~ is one-to-one. The edge-distinguishing chromatic numberX1 of G is the smallest integer r such that G has an admissible r-coloring. This invariant was introduced by Frank, Harary and Plantholt [l].

2

Homomorphisms from P, to I 3 and MT-1 < n < M,.We wish to prove that xl(Cn) = r . It is clear that xl(Cn) 2 r , since n > Mr-1.To prove that xl(Cn)5 r , we need only to prove that there is an admissible r-coloring of C,. It is sufficient, from Corollary 2.2, to find an eulerian subgraph of Iif with exactly n edges. We consider two cases: 1. Let X v- r 5 n < M,,and let H , be an eulerian subgraph of Kf of maximum size. Then we can find the required eulerian subgraph of Kf by removing M,- n 5 r loops from H,.

2. Let M,-1< n < M,- r . In this case, r is odd, since if r were even, it is easily seen that M,-1 = 2 > = M,- r , so our hypothesis is not satisfied. Also, this case does not arise when r = 3. Since r is odd, H , = K:. Let F be a hamiltonian cycle of K,, and let T be a 2-factor of K , - E ( F ) . Then G*= K ; - E ( T ) is connected, since it has a hamiltonian cycle, and every vertex of G* is even, so G*is eulerian. Also, IE(G*)I = M,- r. We can now construct an eulerian subgraph of G with exactly n edges by removing M, - r - n loops from G*. To do this, we must be certain that M,- r - n 5 r , the number of loops in G*.But this follows from the fact that M,- 2r = < = V,-1, since r is odd. 0

9

Corollary 4.1. If r 2 3 and either

2) r is even, and

r2 - r T2 0, z z E E(G). Moreover, by the choice of a', w ( z x ) = a. Thus,

w(wz) = a

for all w E V ( B ' ) .

It follows that any vertex of B' could have been selected as the vertex J'. This immediately implies that G' = B' and that a' = p'. The proof of (i) is completed by setting p = a'. (ii) Let P = woo1 . . .w, be a path in G of maximum weight starting a t 00 = a, and as long as possible subject t o these conditions. Then w, is adjacent only to vertices of P. Let

S = { i : Vizt,

€

E(G)} and j = minS.

Then for each i E S, wow1 ...wiw,wm-l wo = z. By the choice of P,

Summing this over all i E S, we obtain

Let

. . .wi+l

is another path starting at

J . A. Bondyand G. Fan

56

Then

w(C) 2

C w(v;vi+i) t w(v,vj)

2 d t w(vmvj) > d.

i€ s

Thus, G contains the cycle C of weight more than d.

0

Remark 1. It can be seen from the proof of Theorem l(ii) that, even if the condition that w ( e ) > 0 for every e E E ( G ) is dropped, the graph G contains (i) a z-path of weight at least d and (ii) a cycle of weight at least d. Theorem 2 below discusses the corresponding extremal graphs.

A subgraph B is said to be contracted if B is deleted and replaced by a new vertex b joined to every vertex v E V ( G - B ) for which G contains an edge uv for some u E V ( B ) . A pendant triangle is an endblock which is a triangle. Theorem 2. Let G be a connected weighted graph on at least two vertices, z a specified vertex of GI and d a real number. Suppose that w(v) 2 d > 0 f o r every v E V ( G )\ { z } . (i) Then G contains a z-path of weight at least d. Moreover, if G contains no z-path of weight more than d , then the graph G' obtained from G by recursively contracting blocks of weight zero until no such block remains has the following structure: each block B of G' ( a ) includes z , and (b) is weighted so that w(vz) = a for all v E V ( B )\ { z } with vz E E(G) and w(uv) = p for all u , v E V ( B )\ { z } with uv E E ( G ) , where a+@(IV(B)I-2) = d; moreover, (c) i f cr > 0 , then z is adjacent to every vertex of B - z , and i f /3 > 0 , then B - z is complete. (ii) I f d ( v ) 2 2 for every v E V(G)\{z}, then G contains a cycle of weight at least d. Moreover, i f G contains no cycle of weight more than d , then G has a pendant triangle with edge weights zero, zero and d .

Remark 2. In Theorem 2 ( i ) , the description of the graph G' in effect amounts to a complete characterization of the extremal graphs. On the other hand, Theorem 2(ii) provides no such characterization. Indeed, as the following example shows, the problem of characterizing the extremal graphs in Theorem 2(ii) is at least as hard as the problem of characterizing the graphs with Hamilton cycles. Let H be a 2-connected graph on n vertices. Assign a weight of to each edge of H ,where d > 0. Now append t o each vertex v of H an edge vv' of weight d , and attach a pendant triangle with edge weights zero, zero, and d to each vertex v', where the two edges incident with v' are the edges of weight zero. Call the resulting graph G.

5

Optimal Paths and Cycles in Weighted Graphs

57

Then G is a weighted graph on 4n vertices in which w(v) 2 d and d(o) 2 2 for every o E V ( G ) .Moreover, G contains a cycle of weight more than d if and only if H contains a Hamilton cycle.

Proof of Theorem 2. (i) The first assertion is Remark l(i). Thus we only need to prove the second assertion, and it suffices to prove this for graphs that contain no zero-weight block. Let G be such a graph. We use induction on lE(G)I. If (E(G)I = IV(G)l - 1, then G is a tree. Since G contains no zero-weight block, each edge of G is of positive weight. The result follows from Theorem l(i). Suppose now that IE(G)I 2 IV(G)l. If G contains an endblock B which does not include z, let b be the cut vertex of G contained in V ( B ) . From the first assertion, B contains a b-path PI of weight a t least d. Moreover, since G contains no zero-weight block and b # z, there is a (I,b)-path P2 of positive weight. Then PIU P2 is a z-path of weight more than d, a contradiction. This shows that every endblock of G includes z , or equivalently, every block of G includes z. Furthermore, we may suppose that G consists of a single block, for otherwise applying the induction hypothesis t o each block would complete the proof. If every edge of G is of positive weight, then the result follows from Theorem l(i). Suppose that zy E E(G)with w(zy) = 0. Let H = G - zy. We distinguish two cases: CASE1. H contains no z.ero-weight block. By the induction hypothesis, each block of H has the required structure. Case la. H is a block. If z # x, y, then H - z is not complete, and so by (c), /?= 0. Thus, w(xy) = 0 = p. If .z = z,then p(lV(B)I - 2) 2 w(y) 2 d. This implies, from (b), that w(zy) = 0 = a;the same conclusion holds if z = y. In either case, G = H xy has the required structure. Case l b . H has more than one block (in fact, by (a), exactly two blocks). In this case, we must have that w(e) = 0 for every e E E ( H - z), for otherwise an optimal z-path in G would be of weight more than d. This shows that G has the required structure (with LY = d and /3 = 0).

+

CASE2. H contains a zero-weight block. Since G consists of a single block, H is the union of blocks B1, B2, . . . , B,, m 2 2, where z E B1, y E B,, IV(B;)n V(Bi+1)1= 1, 1 5 i 5 m, and IV(B;)n V(Bj)l = 0 for j # i f 1. Without loss of generality, suppose that z f! V(B1 - b), where b is the cut vertex of H contained in B1. If B1 contains a b-path P1 of weight more than d, then extending PI to z yields a z-path in G of weight more than d. Thus, B1 contains no b-path of weight more than d. By the induction hypothesis, B1 has the properties (a), (b), and (c), which implies that B1

J . A. Bondy md G.Fan

58

has a (b,z)-path P of weight d. If there is an edge e of positive weight in some B;, 2 5 i’s m, then e and P lie on a cycle in G of weight more than d, and thus there is a r-path in G of weight more than d. Therefore, w(e) = 0 for all e E U Z a E ( B i ) . This implies that m = 2, y = z , and G = B1 U {zy,by}, where w(zy) = w(by) = 0. The above argument, with the edge zy replaced by the edge by, shows that B1 is a uniformly-weighted complete graph. Thus, G has the required structure (with a = 0 and

P = p7p-I. d (ii) The first assertion is Remark l(ii). We prove the second assertion by induction on IE(G)I. Since d ( v ) 2 2 for all v E V ( G )\ { z } , we have IE(G)I 2 IV(G)I. If IE(G)I = IV(G)I,then each component of G is a cycle and has weight a t least d , with equality only if it contains z. If G contains no cycle of weight more than d, it follows that G must consist of a single triangle with edge weights zero, zero, and d, where w ( z ) = 0. Thus the result holds for IE(G)I = (V(G)I.Suppose now that IE(G)I > IV(G)l. If G is not 2-connected, we apply the induction hypothesis t o a component of G or to an endblock B of G such that z is not an internal vertex of B . This will complete the proof. We may therefore suppose that G is 2-connected. Since w(v) 2 d > 0 for all v E V ( G )\ { z } and IE(G)I > IV(G)I, there are at least two edges of positive weight. If there is an edge e with w(e) 2 d, let e’ be any other edge with w(e’) > 0. By the 2-connectedness of G , there is a cycle containing both e and e’, which is thus of weight more than d. Therefore we may suppose that w(e)

0 for all e E E(G),then, by Theorem l(ii), G contains a cycle of weight more than d. Thus consider zy E E ( G ) with w(zy) = 0. If z # z , then d ( z ) 2 3, by (2) and the hypothesis that w(z) 2 d. The same conclusion holds for y. Hence, d ( v ) 2 2 for all v E V ( G- sy)\ { z } . By the induction hypothesis, G - zy contains a cycle of weight a t least d. However, G - zy does not contain a pendant triangle with weights zero, zero, and d, by (2). Thus, G - z y contains a cycle of weight more than d, and so does G. This completes the proof. 0

3

Weighted generalization of Theorem B

In this, and the next, section we need the following notation: Let C be a cycle in G with a fixed orientation. For any two vertices a and b on C, denote by C [ a ,b] the segment of C from a to b determined by this orientation.

Optimal Paths and Cycles in Weighted Graphs

59

Theorem 3. Let G be a 2-connected weighted graph and d a real number. Let x and y be distinct vertices of G. If w(v) 2 d for all w E V ( G )\ {x, y}, then G contains an (x,y)-path of weight at least d. Proof. Let IV(G)I = n. We use induction on n. If n = 3, let u be the third vertex other than x and y. Then the path xuy is of weight w(u) 2 d. Suppose now that n 2 4 and the theorem is true for all graphs on Ic vertices, 3 5 k 5 n - 1. Let H = G - y be the graph obtained by deleting y from G. We distinguish two cases:

CASE1. H is 2-connected. Choose y' E N(y) \ {x} (note that IN(y)l 2 2 by the 2-connectedness of G ) such that 4Y'Y) = m a 4 4VY) : v E "Y)

\ (41.

Then for all D E V ( H )\ {x},

w ( v , H ) = w(v,G)- W(VY)2 d - ~ ( y ' y ) . By the induction hypothesis, there is an (x,y')-path Q in H of weight at least d - ~ ( y ' y ) .Then P = Qy'y is an (5,y)-path in G of weight at least d.

CASE2. H is not 2-connected. Choose an endblock B of H such that x is not an internal vertex of B. Let b be the unique cut vertex of H contained in B and let B' be the subgraph of G induced by V ( B )U {y}. If yb E E(G),then B' is 2-connected and for all w E V(B')\ {y, b}, w(v,B') = w(v,G) 2 d.

By the induction hypothesis, there is a (y, b)-path P' in B' of weight a t least d. Extending P' in H from b to x, we obtain an (x,y)-path in G of weight a t least d. If yb f! E(G), we add yb to B' and set w(yb) = 0. Applying the above argument to the resulting graph, we obtain a (y,b)-path P' of weight at least d. If d > 0, then P' # yb, since w(yb) = 0; if d = 0, then we can choose P' such that P' # yb, since all we need is that w(P') 2 d. This shows that we still have P' C B'. Extending P' as before completes the proof of the theorem. 0

For later use, we need a slight variation, and direct consequence, of Theorem 3. Corollary 3.1. Let G be a nonsepamble weighted graph on at least two vertices and d a real number. Let x be a vertex of G. If w(v) 2 d for all w E V ( G )\ {x}, then G contains a path of weight at least d from x to any other vertex of G.

J. A. Bondy and G.Fan

60

4

Weighted generalization of Theorem C

In this section, we will prove the following theorem.

Theorem 4. Let G be a %connected weighted graph and d a real number. If w(v) 2 d for every vertex v in G, then either G contains a cycle of weight at least 2d or every optimal cycle in G is a Hamilton cycle. Before proving the theorem, we need the following lemma.

Lemma 1, Let C be an optimal cycle in a weighted gmph G. If there exists a path P in G - C , connecting vertices x and y , such that INc(x)l 2 1, INC(Y)l2 1, and "c(4 u NC(Y)I 2 2, then

Instead of proving Lemma 1, we are going to prove the following more general result, which is a weighted generalization of Lemma 2 in [l].

Theorem 5. Let C be an optimal cycle in a weighted graph G. Suppose that there exists a path P in G - C , connecting vertices x and y , such that INc(x)I 2 1, INc(Y)I 2 1, and INc(x) U Nc(Y)I 2 2. Define

X = Nc(z)\ N c ( Y ) , Y = Nc(y) \ N c ( x ) , and 2 = Nc(z)n Nc(y). Then

unless 121= 1 and either X = 0 or Y = 0, in which cases,

..

Proof, Let A = X U Y U Z and suppose that A = {a1,a2,. , a k } , where are in order around C. For each pair of vertices (a;,a;+l), we shall construct from C two new cycles by replacing the segment C[a;, ai+l] with two (ai,ai+l)-paths. These two paths are defined according to four cases:

a;

(i) a;,a;+l E 2. The two paths are a;xPya;+l

and

a;yPxai+l.

Optimal Paths and Cycles in Weigbted Graphs

61

(ii) ai E 2 and a;+l E X or Y.The two paths are

If a;+l E 2 and

a;

E X or Y ,the paths are defined in the same way.

(iii) a; E X and a;+l E Y or a; E Y and a;+l E X. The two paths are two copies of aixPyai+l or a;yPxa;+l. (iv) a;, a;+l E X or a;, a;+l E

Y.The two paths are two copies of

In each case, we have defined two paths t o replace the segment C[a;,a;+l] and hence formed two cycles (in (iii) and (iv) these two cycles are identical). Since there are k pairs of vertices (a;,a;+l) (i = 1, .. ., k), we obtain 2k cycles. In these cycles, every edge of C is traversed exactly 2k - 2 times; every edge from z or y to 2 is traversed twice, every edge from x t o X is traversed four times and, similarly, every edge from y to Y is traversed four times. Now suppose that the path P is traversed 1 times (we determine I later). Then the average weight of these 2k cycles is 1

-@(k 2k

- l)w(C) + 2

+

+

+

4 2)~t 2w(y, 2 ) 4w(z, X) 4w(y, Y ) !w(P)).

Since C is an optimal cycle, its weight is no less than this average weight. Thus ,

We now determine 1. If 121 2 2, then it is not difficult to see that 1 2 2121; if 121 = 1, X # 0, and Y # 0, then I 2 4 (at least twice from Case (ii) and twice from Case (iii)); if 121 = 0, then, noting that 1Nc(z)l 2 1 and INc(y)I 2 1, we see that X # 0 and Y # 0, and so that I 4 (from Case (iii)). Therefore, 1 2max{4,2121~ unless 121= 1 and either X = 0 or Y = 0. In the case 121= 1 and X = 0, we suppose, without loss of generality, that 2 = {al} and Y = ( ~ 2 , .. .,a k } . Since C is optimal,

J . A. Bondy and G. Fan

62

Similarly,

Also, for all i

#

1,k,

0

This completes the proof of Theorem 5.

Remark 3. It is clear that Lemma 1 is an immediate consequence of Theorem 5 . Proof of Theorem 4. Suppose that there exists a n optimal cycle C in G which is not a Hamilton cycle, and let H be a component of G - C. We consider two cases: CASE 1. H is nonseparable. Choose distinct vertices z and y in H (unless IV(H)I = 1, in which case necessarily 2 = y) such that (i) dC(2) 2 1, &(Y)

L 1, and

(ii) w(z,C) 2 w(y,C) 2 w(v,C) for all v E V ( H )\ {z,y}.

If INc(z)U Nc(y)I 1 2, then, by Lemma 1,

w ( C ) 1 2w(Y,c)

+ 2w ( P) ,

where P is an (z,y)-path in H . If IV(H)I = 1, then

w(P)= 0 2 d - ~ ( 9=)d - w(y,C).

63

Optimal Paths and Cycles in Weighted Graphs Otherwise, by the choice of x and y,

w(v, H ) = w(v) - w(v,C) 2 d - w(y,C)

for all v E V ( H )\ {z),

and so, by Corollary 3.1, we can choose P such that

In either case, ( 6 ) yields that

w(C) 2 2d. Thus we may suppose that N c ( z )U Nc(y) = { u } . Since G is 2-connected, there exists a vertex b E V ( C )\ { a } which is adjacent with some vertex J E V ( H )\ {z,y}. By the choice of x and y, for all v E V ( H ) .

w(w,H) = w(v) - w(v,C) 2 d - w(z,C)

By applying Corollary 3.1 to H , we have an (z,z)-path Q in H of weight

Thus ,

+

w(C)= w(C[a,b]) w(C[b,a])2 2(w(ax)

+ w(Q) + ~ ( b z )L) 2d.

This completes Case 1.

CASE2. H is separable. Let B1 and BZ be two distinct endblocks of H , and let b; be the unique cut vertex of H contained in Bi (i = 1, 2). For i = 1, 2, we choose xi E V ( B ; )\ { b i } such that (i) d c ( s ; ) 2 1, and (ii) w(x;,C) 2 w(v,C) for all v E V ( B ; )\ { b j } . It follows that

w(v, B;) = W ( V ) - w(v,C) 2 d - w(Zi,C)

for all

E V ( B ; )\ { b ; } .

For i = 1, 2, we apply Corollary 3.1 to B; and obtain a n (x;,b;)-path P; in B; of weight w(P;)2 d - w(z;,C). (7)

J . A. B o d y and G.Fan

64

If INc(zl)UNc(z2)1 2 2, then let P be an (zl,zz)-path in H of maximum weight. By (7),

w(P)1 w(P1)

+ w(P2) 1 d -

min(w(z1,C), w(z2,C)).

By Lemma 1,

w(C) 12min{w(zl,C), w(z2,C))

+ 2w(P) 2 2d.

If Nc(z1) = Nc(z2) = {ul}, then there exists a vertex u2 E V(C)\{ul} adjacent to some vertex b € V ( H ) . As (V(B1) \ { b ~ }n) (V(B2) \ {bz}) = 8, b cannot belong t o both V(B1) \ { b l } and V(B2) \ {bz}. We suppose, without loss of generality, that b @ V(B2) \ {b2}. Extending the path P 2 in H from b2 t o b, we obtain an (zz,b)-path P‘ in H of weight

w(P’)L w(P2).

(8)

Since C is optimal,

+ w(C[u2,u11) 1 2 ( w ( w 2 ) + w(P’>+ 4 u 2 b ) ) 1 2(w(u1z2) + w(P‘>).

w(C)= w(C[u1, 2121)

Now, using (8) and then ( 7 ) with i = 2, we have

+

+

w(C) 1 2 ( w ( ~ i z 2 ) w(P2)) 1 2d 2 ( ~ ( ~ 1 -~ 422 )2 , C ) ) = 2d. This completes the proof of Theorem 4.

5

0

Open questions

There are many other theorems on paths and cycles which have weighted analogues. We mention just two. One is Pbsa’s generalization [4] of Dirac’s theorem.

+

Theorem D. Let G be a 2-connected graph such that d ( u ) d ( v ) 2 k for every pair of nonadjacent vertices u and v . Then G contaim either a cycle of length at least k or a Hamilton cycle. This theorem suggests the following problem.

Problem. Let G be a 2-connected weighted graph with w ( u ) + w ( v ) 1k for every pair of nonadjacent vertices u and v . Is it true that either G contains a cycle of weight a t least k or every optimal cycle in G is a Hamilton cycle? Another candidate for generalization to weighted graphs is the following extremal theorem due to E r d k and Gallai [3].

65

Optimal Paths and Cycles in Weighted Graphs

Theorem E. Let G be a 2-edge-connected graph on n vertices and e edges. Then G contains a cycle of length at least

3.

We propose the following conjecture.

s.

Conjecture 1. Let G be a 2-edge-connected weighted graph on n vertices. Then G contains a cycle of weight at least 2w G Remark 4. Call a spanning tree T of a graph G a tritree if every fundamental cycle of T in G is a triangle, and a graph with a t least one tritree a trigraph. It is not hard to show that any cycle of a trigraph uses either zero or two edges of each tritree. Let G be a 2-edge-connected trigraph on n vertices, TI,Tz, .. , T,,,the tritrees of G, and w1,w2,, , w, nonnegative real numbers. To each edge e E E ( G ) ,assign the weight

.

..

w(e) =

c{

Then

wi : e E E ( T i ) } . m

i=l

Let C be a cycle of G. Then C uses a t most two edges of each Ti,and so

Thus, if Conjecture 1 is true, the graph G , with this weighting, is an extremal graph. The following is a weaker conjecture than Conjecture 1.

+.

Conjecture 2. Let G be a weighted graph on n vertices. Then G contains a path of weight at least 2w G Remark 5. To see that Conjecture 2 is implied by Conjecture 1, let G be a weighted graph on n vertices, where n 2 2 (the case n = 1 is trivial), and suppose that G is connected (otherwise, we can consider the nontrivial components of G). Let G’ be the weighted graph obtained from G by adding a new vertex y and joining y to every vertex of G by an edge of weight M, where M > w(G). Then G’ is 2-edge-connected. Let C be an optimal cycle of G’. Since M > w ( G ) and C is optimal, C must contain the vertex y. If Conjecture 1 is true, then

24G’) - 2 M G ) + n M ) - 2 4 G ) W(C)2 IV(G’)I - 1 n n Thus, C - y is a path in G of weight w(C) - 2 M 2

2M.

+. 2w G

66

J . A. Bondy and G. Fan

6

Proof of the conjectures for weighted complete graphs

Towards proofs of Conjectures 1 and 2, we settle the case in which G is a weighted complete graph. Before giving this result, we need some more definitions. A graph is called a vertex-weighted graph if each vertex of the graph is assigned a nonnegative number, called the weight of the vertex. If G is a graph on n vertices v1, v2, . , Vn, then G(wl,w2,. . . ,ton) denotes the vertex-weighted graph obtained by assigning to vertex D; the weight wi, 1 5 i 5 n. If w1 = 202 = - wn = t, then the notation is abbreviated to G(t). Suppose that G(w1, w2,. ,wn) is a vertex-weighted graph. Then the induced (edge-) weighted graph WG(w1, w2,. .,w,) is the weighted graph obtained by setting w(Vjvj) = wi w j for every edge v;vj. So WKn(t)is the uniformly-weighted complete graph on n vertices in which each edge is of weight 2t.

..

...

..

-+

.

Theorem 6. Let G be a weighted complete graph on n vertices where n 2 3. Then the maximum weight of a cycle of G is at least %.& n-l , with equality if and only if G = WK,(wl, w2,. . . ,Wn) for some w1, w2, . . . , w, such that C:,l W j =

s.

Remark 6. A complete graph K n is a trigraph whose tritrees are stars (one centered at each vertex), and the weighting of WKn( w1, w2,. . ,w,) is precisely the weighting described in Remark 4.

.

Proof of Theorem 6. If G = WKn(wl,w2,. . .,w,) and then, for any cycle C of G,

c:=lw; = A 9 n-1

7

s.

and equality holds when C is a Hamilton cycle of G. This shows that the maximum weight of a cycle of G is exactly 2w G Conversely, let G be any weighted complete graph on n vertices. Let C be the set of Hamilton cycles of G. Since G is complete, ICI = and each edge of G is contained in exactly ( n - 2 ) ! Hamilton cycles. Thus the average weight of a Hamilton cycle of G is

67

Optimal Paths and Cycles in Weighted Graphs It follows that the maximum weight of a cycle of G is a t least equality only if

m,with n-l

for all c E C.

w(C)= 2w(G) n-1

(9)

a.

Suppose, now, that the maximum weight of a cycle of G is exactly 2w G We first note that the edge weights of G satisfy the triangle inequality: for any x,y,z E V (G).

w(xy) i- w(xz) 2 w(yz)

(10)

Let P be a Hamilton path in G - x connecting y and z. Then

C = xyPzx

and

C'= yPzy

are cycles of G. Since C is a Hamilton cycle of G, we have, by (9), w(zy)

+ w(xz) + w(P) = w(C) =

n-1

2 w(C') = w(yz) + w ( P ) .

Thus (10) holds. Next, we observe that for any four vertices u, x, y, and z, w(xu)

+ w(yz) = w(xy) + w(uz).

(11)

Let P be a Hamilton path in G - {u,y} connecting x and z, and consider the two Hamilton cycles C = xuyzPx

and

C' = xyuzPx.

BY (9),

w(zu)

+ w(uy)t w(yz)+ w(P) = w(C) = w(C') = w(xy) t w(yu)

+ w(uz) + w(P),

and so (11) holds. We now assign, to each vertex x E V ( G ) ,the weight

w, = +(+Y)

t w(x.4 - W(YZ)),

where y and z are any two vertices other than z. By (lo), w, 2 0. To verify that these vertex weights are well-defined (that is, that w2 does not depend on the choice of y and z ) , it suffices to show that, for any u E V ( G )\ {z,y,z}, the assignment determined by u and z is the same as that determined by y and z , that is,

+

+

$(w(xu) w(xz) - w(uz))= i(w(xy) w(xz) - w(yz)).

J . A. Bondy and G . Fan

68

But this follows directly from (11). It remains to check that for any two distinct vertices x and y, Ws

+ Wy =

(12)

W(Z:y).

Let z be a third vertex. Then, by the definition, 'u)z

+

= +(ZY)

W(5k)

- w(y.z))

and

+ W ( Y 4 - 44), and so (12) holds. Therefore, G = W K n ( W 1 , . .,W n ) . Clearly, we have Wy

=

$(.)(YZ)

w2,.

CZ1Wi = 9, since, if C is a Hamilton cycle of G,

0

This completes the proof of Theorem 6.

e, .sj.

Corollary 6.1. Let G be a weighted complete graph on n vertices. Then the maximum weight of a path of G is at least 2w G with equality if and only if G = WKn(t),where t = Proof. If G = W K n ( t ) and t = ,=I,

then, for any path

P,

and equality holds when P is a Hamilton path of G. This shows that the maximum weight of a path of G is exactly 2w G Conversely, let G be a weighted graph on n vertices. As in Remark 5 , let G' be the weighted graph obtained from G by adding a new vertex y and joining y to every vertex of G by an edge of weight M , where M > w(G). By Theorem 6 and the argument used in Remark 5 , G contains a path of with equality only if G' = WKn+l(wo,w1,. . . ,wn), weight a t least where w is the weight of y. Thus, if G contains no path of weight more than then G = W K , ( t ) , where t = M - WO. Moreover, since a Hamilton path of G is of weight we have t = n-l as required. 0

+.

a,

A,

q,

e,,

Added in proof. The authors have recently proved Conjecture 1, and hence Conjecture 2. Moreover, they have shown that every extremal graph for Conjecture 1 is a weighted union of tritrees, as described in Remark 4,

Optimal Paths and Cycles in Weighted Graphs

69

and that the only extremal graphs for Conjecture 2 are the uniformlyweighted complete graphs and the graphs of weight zero. Conjecture 2 has also been verified (earlier, and by different methods) by A. M. Frieze, C. D. McDiarmid, and B. A. Reed.

Acknowledgement We gratefully acknowledge the support of this research by the Natural Sciences and Engineering Research Council of Canada.

References [l] J. A. Bondy, I. Ben-Arroyo Hartman and S. C. Locke, “A new proof of a theorem of Dirac,” Congressus Numerantiurn 32 (1981), 131-136. [2] G. A. Dirac, “Some theorems on abstract graphs,” Proc. London Math. SOC.(7) 2 (1952), 69-81. [3] P. Erdos and T. Gallai, “On maximal paths and circuits of graphs,” Acta Math. Acad. Sci. Hungar. 10 (1959), 337-356. [4] L. P h a , “On the circuits of finite graphs,” Magyar Tud. Akad. Mat. Kutat6 Int. Kozl. 8 (1963), 355-361.

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Annals of Discrete Mathematics 41 (1989) 7 1 -74 0 Elsevier Science Publishers B.V. (North-Holland)

A Note on Hamiltonian Graphs H. J. Broersma Depart men t of Applied Mat hematics Twente University of Technology Enschede, The Netherlands Dedicated to the memory of G. A . Dirac The closures of the graphs satisfying a sufficient condition by Dirac for the existence of a Hamilton cycle are determined. Infinitely many closures of odd order turn out to be noncomplete.

We use [2] for basic terminology and notations and consider only simple graphs. For a graph G we denote by SG( 5 i ) the number of vertices in G of degree at most i; SG(> i ) is defined analogously. P, will be the path on n vertices, C, the cycle on n vertices and kG the graph consisting of k disjoint copies of G. We first state some known results.

Proposition A. Let G be a graph. If V ( G ) contains a nonempty proper subset S such that w(G - S ) > IS[, then G is nonhamiltonian. Theorem B (Bondy and Chvdtal [l]). Let G be a graph with u 2 3 . If c ( G ) is complete, then G is hamittonian. Theorem C (Chvdtal [3]). Let G be a graph with u 2 3 such that, for each integer i with 1 5 i 5 f(u - l), either S G ( i) ~ 5 i - 1 or SG(> u - i) 2 i 1. Then G is hamittonian.

+

Let E denote the class of graphs G with v 2 5 such that either

G

Kj V

(Kv-2j

+j K , )

with

25j

k u - 1,

or u is odd and 71

H.J . Broersma

72

or

G c Ki(,-q v (21 3, that one of these edges is an edge between two of wo, w1, and wz, where there are .r(G) edges altogether between these vertices if x'(G) = T(G). Then, if p 6 {3,4}, we may argue as above when I E ( H ) )> 2, but with F' instead of F , and show that x'(G) = max(A(G),.r(G)), as required. If p = 3, the same argument works, except that there might be two vertices w,, instead of just one. Finally, suppose that p = 4 and that either IE(H)( = 2 and G has two independent non-simple edges, or that IE(H)I 2 3. Suppose also that x'(G) 2 A(G) 4-1. We must show that x'(G) = max(A(G) 4-~ , T ( G ) ) If. IE(H)I 2 3 then

+

x'(G) - 1 L x'(G \ F ) 5 max( A(G \ F ) 1,.r(G \ F ) ) 5 max(A(G), T(G) - 1)

+

(by induction)

5 X'(G) - 1, so x'(G) = max(A(G),.r(G) - 1). If IE(H)I = 2 then the same argument works, but with F* instead of F . Lemma 6 now follows by induction. 0

3

Proof of the theorem

We can now prove the theorem. The sufficiency is trivial. We shall prove the necessity. Let IV(G)I = 2n 1; if n = 1 then the necessity is trivially true, so we may assume that n 2 2. First, we add edges to G, forming a graph G* with as many edges as possible, subject t o G* being a simple graph, A(G*) = A(G) and G* also containing no overfull subgraph H * with A(H*) = A(G*).

+

The Edge-Chromatic Class of Graphs with Large Maximum Degree

101

Suppose first that G* contains a subgraph HI with A(H') = A(G*) which is maximally non-overfull, that is,

Then, since A(G*) = 2n - 2, either (i) IE(G*)I = A(G*) LiIV(G*)l] = (2n - 2)" (then H' = G*), or (ii) H' = Iizn and G* has an isolated vertex, or (iii) there are two vertices

01

and wz so that

(then H' = G* \ {w~,wz}).

If (i) holds then, by Lemma 1, G* is Class 1, and so G is Class 1. If (ii) holds then clearly G*, and therefore G, is Class 1. So suppose that (iii) holds. The graph G* \ { T J ~ , w ~is} a non-overfull graph of order 2n - 1 and maximum degree 2n - 2, and so, by Lemma 2, G* \ { q , WZ} is Class 1. Edge-colour G* \ {01,02} with 2n - 2 colours. Since IE(G* \ (211, WZ}) = ( n - 1)(2n - 2), it follows that each colour appears on n - 1 edges, and so is missing from exactly one vertex. Furthermore,

I

= (2n - 1)(2n - 2) - 21E(G* \

{2)1,02})1

= (2n - 1)(2n - 2) - (2n - 2)(2n - 2)

= 2n - 2,

+

so d ~ ( 0 1 ) d ~ ( ~ 52 )2n - 2. Therefore, the edges of G* between the sets {vl, v2} and V(G*) \ {v1,02} can be coloured, each edge receiving a different colour. If there are 2n - 2 edges between the sets { w ~ , w z }and V ( C )\ { W I , WZ} then 01 and 02 are non-adjacent, for otherwise

IE(G*)I = 1 + (2n - 2)

+ ( n - 1)(2n - 2) = n(2n - 2) + 1,

A. G . Chetwynd and A. J. W. Hilton

102

so G* would be overfull, a contradiction. If there are fewer than 2n-2 edges between the sets ( ~ 1 , 2 1 2 ) and V ( G * )\ { V I , ~ 2 then ) w102 E E(G*),and 211212 can be coloured with a colour not used so far on w1 and ~ 2 . Therefore, we may suppose that

I E(G*1I < NG)

14IW) IJ

(3)

',

and that

IEV* \ { V l , ' u 2 H I < W)[hlV(G)l - 11

(4)

for each pair ~ 1 , 1 ) 2E V ( G * ) . We may also assume that each pair of vertices of degree less than A(G) = 2 n - 2 is joined by an edge of G*. [Note, this is an important point in our proof here; it seems to be the point which prevents us proving Lemma 1 and the theorem together.] Let the vertices of G* of degree less than 2 n - 2 be ~ 1 . ,. . , I+,,and let the remaining vertices be vP+1,. . . , 02,+1. I f 0 5 p 5 1 then

IE(G*)I 2 $ ( 2 n

+ 1 - p ) ( 2 n - 2 ) 2 2n2 - 2n.

But, by (3), IE(G*)I < ( 2 n - 2 ) n , a contradiction. Similarly, if p = 2 then, by (3),

E(G*)= ${dG*(ZI1)

+ d G * ( V 2 ) + ( 2 n - 1 ) ( 2 n - 2 ) ) < ( 2 n - 2).

d ~ * ( v l 4) dp(V2)

< 2 n - 2. But on the other hand, by (4),

so that

${( 2 n -2)(2n- 1 )- dG* (.I) - d ~ (*V 2 ) ) 5 IE(G*\ { v i ,

V2))

I < (272- 2)(12-

1)

+

so that ~ G * ( v I ) dG'(W2) > 2 n - 2, a contradiction. Thus, p 2 3. Each vertex of {'uP+l,.. . ,02,+1} is joined to at least p - 2 of ( ~ 1 , .. . ,vP}, and so the total number of edges from {vP+l,. ,112,+1} to { q , . ,wP} is a t least (272 1 - p ) ( p - 2). Thus, the average degree of wl, . . . , w p is at least (P - 1) ( p - 2 ) ( 2 n 1 - p ) / p . Then

+

..

+

c

+

v€V(G*)

since

..

( ( 2 n - 2)

- d G * ( V ) ) 5 2 ( 2 n + 1 - 2p),

(5)

The Edge-Chromatic Class of Graphs with Large Maximum Degree

103

Therefore, IE(G*)I 2 3{(2n

+ 1)(2n - 2 ) - 2(2n + 1 - 2 p ) ) = 2n2 - 3n + 2 p - 2.

But, by (3), IE(G*)I < 2n2-2n. Therefore, 2n2-2n-1 2 2n2-3n+2p-2, so that p 5 i ( n 1 ) . Since p 2 3, we also have n 2 5 . Next, we describe how we adjoin to G* a further vertex wo and further edges on the vertices W O , .. ., wp in such a way that a regular Class 1 loopless multigraph Gt of degree 2n - 2 on vertices DO, . . , w ~ ~ +is lformed. After we have described how to construct G t , we shall show that it is Class 1. For 1 5 i 5 2n 1, let 6j = 2n - 2 - d~*(wi)and let 60 = 2n - 2. We may suppose that 60 2 61 2 ... 2 SP+1 = ... = 6zn+1 = 0 (otherwise, we relabel w1, ... , vp). We first note that

+

.

+

i=l

since P

+ 1)(2n - 2 ) - 21E(G*)I 2 (an + 1)(2n - 2 ) - 2{(2n - 2)"

6i = (2n i=l

- 1)

= 2n

> 60.

#[(c%,

To form Gt from G* we first add S i ) - 601 extra edges onto the vertices 81, . .. , wp one by one, at each stage selecting two vertices whose degree is least at that stage t o place the new edge on and keeping the degree of w1 least. After placing these extra edges on w1, . , , wp, if they all have a common vertex then that vertex will be w 1 , and all the vertices joined to w1 by an extra edge will have degrees differing from each other by at most one. , wpl for some pl E (2,. . ,p } . If We may suppose these vertices are w2, the extra edges do not all have a vertex in common then the degrees of all the vertices incident with an extra edge will differ from each other by at most one. In this case, we may suppose that these vertices are v 1 , . . . , w m for some pl E ( 2 , . . . , p } . Then the edges joining 00 t o ( ~ 1 , .. . ,wP} are inserted so as to make the regular loopless multigraph Gt of degree 2n - 2. By the construction we may assume

.

...

and

.

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A. G. Chetwynd and A. J . W. Hilton

[It follows that r n c t ( ' u O , 'up) 2 1 since p 5 !j(nt 1) 5 2n - 2 , ;ts 5 5 3n.l From ( 5 ) , it follows that the number of edges placed on {q, . . ,vp} [ignoring the edges on W O ] is at most

.

+ 1 - 2p) - ( 2 n - 2 ) ) = n + 2 - 2p. From Gt, delete a vertex x E . ., ~ 2 ~ + 1 } Then . x was joined to 4{2(2n

{'up+l,.

at least p - 2 of 01, .. ., v,, had no multiple edges on it, and Gt \ {x} has two vertices of maximum degree 2 n - 2 , apart from WO. If possible, select 2 so that, in addition, in Gt \ {x}, at least one of these two vertices (f 'uo) of maximum degree is in {'up+l,. ,'uzn+l}. If this is not possible, then, again if possible, in addition select x so that there exists a vertex y E { w p + l , . ,w2,+1} such that y is adjacent to one of these two vertices (# vo) of maximum degree in Gt \ {z} and is non-adjacent to the other. The possibility that we cannot choose x to satisfy in addition either of these requirements is considered at the end. In the case when one of these two additional possible requirements for the selection of z can be satisfied, we show that Gt \ {x} can be edgecoloured with 2 n - 2 colours. When G is a multigraph and V' c V(G), let G(V') denote the submultigraph of G induced by V'. Let db(v) denote the degree of 'u in the multigraph Gt(w0,. . ,w,). First, we colour E(Gt(v0,. . . ,w,)) with 2 n - 2 colours. The method by which we do this is not straightforward, and we need to consider several cases.

..

..

.

CASE

I.

&u;)

t mct(wo,v;) 5 2 n - 2 for 1 5 i 5 p .

.

First, we colour the edges of G(v1,. . ,v,) with p colours. We then colour the further edges of Gt('u1,. ,' u p ) which are not in G(v1,. . ,wP) with some further colours. The number of such 'extra' edges is at most n 2 - 2p, so the total number of colours employed thus far is at most ( n 4- 2 - 2p) t p = n 2 - p 5 n - 1 5 2 n - 2. Finally, with wo as a pivot, we colour the 2 n - 2 edges on 'uo by Vizing's original argument in the case when there are multiple edges. This works provided that the numerical condition of this case is satisfied.

..

+

.

+

2. &v;) t rnGt(v0,w;) 2 2 n - 1 for gome i, 1 5 i 5 p . Since the number of 'extra' edges [i.e., edges of E(Gt(w1, ...,' u p ) ) E ( G * ( q , . , ' u p ) ) ] is a t most n + 2 - 2p, it follows that CASE

..

\

The Edge-Chromatic Class of Graphs with Large Maximum Degree

105

Therefore, for some i, 2n - 1 5 d!(v;) t mGt(vO,vi) 5 n

-+

- p t 1t 2mGt(vO, vi)

so that n 1 5 n 4- p - 2 5 2mGt(vo, w;). In other words, vo is joined to v; by at least t 1) edges. Since the degree of vo is 2n - 2, there are at most three such vertices v;. From our definition of pl, it follows that if there is more than one such vertex then pl 5 3 and mGt(vo,v;) 2 i ( n 1) for 1 5 i 5 pl. Further, if for 1 5 i < j 5 3 we have Gt({vl,.. .,v2,+l} \ {v;,vj)) = G*({vi,. . . ,vzn+i} \ (vi, vj}) then

i(n

+

Therefore, 4n plies (6).

- 6 2 2{mGt('UO,2);) t mGt(vO,Ifj) + mGt(vi,vj)}, which im-

CASE2(i). Imct(vo,v;) - mGt(vO,2)j)(5 1 for 15 i < j 5 pl. From the above remarks, pl 5 3, so G t ( { s , . ..,~ 2 ~ + 1\}{vi, vj}) = G*({Q, ...,~ 2 ~ + 1\ }{viyvj}) for 1 5 i < j 5 3, and so (6) is true. Let

Go(vo, ... ,vP) be the multigraph obtained from Gt( wo, . . . ,vP) by replacing each multiple edge joining one of {vo, v1, v2, v3} to one of ( ~ 4 , .. .,vP} by a single edge. Let A0 = A(GO(v0,. . , u p ) ) and TO = ~(G'(v0,. ,vp)). By Lemma 5 , except in some cases (specified in Lemma 5 ) , Go(vo, ...,vP) can be edge-coloured with max(A0, TO) colours, c1, ..., c ~ + , , ~ ) . If 70 > A0

.

..

A. G. Chetwynd and A. J . W. Hilton

106

..

then, since the number of edges of Gt(v1,. .., u p ) \ G*(v1,. ,vp) is at most n - p 1, vo is one of the three vertices between which there are TO edges; we may suppose that vl and v2 are the other two. By Lemma 5 , we can make the edge-colouring of Go(vo,. , u p ) have the property that colours CAo+l, . , cm appear just on the edges joining 01 and v2. In view of ( 6 ) , we can extend this edge-colouring to an edge-colouring of Gt(w0,. ,vP) with the 2n-2 colours c1, .. ,~ ~ - as 2 required. , If A0 2 TO, this is clearly also true. Now suppose that Go(vo,. . . ,vP) is one of the exceptional cases of Lemma 5 . Since (2n - 2) - p 2 (2n - 2) - +(n I) = +(3n - 5 ) 2 5 , in cases (ii)(a)-(ii)(d) of Lemma 5 , (2n - 2) - A0 2 2. In each of these cases, colour an edge joining 211 and 0 2 with CA,+1. From Lemma 5 , it follows that the remaining edges of Go(vo,. ,up) can be edge-coloured with c1, . ,C A ~ We . can then extend this to an edge-colouring of Gt(v0,. . ., u p ) with c1, , ~ ~ - as 2 , required. In case (ii)(e) of Lemma 5, we have p = 4. The only edges of Gt(v0,. , u p ) which are not in GO(v0,. ,vP) are all but one of the edges from vo to v4. Let G i = GO(v1,. . . ,vp), and let H,” = G;(vo,vl, w2,v3). Suppose there are y 1 edges from vg to 214. Then d p ( v 4 ) = 2n - 2 - y. Since IE(G;)I 2 2A(H,”) 3, it follows that for Go = Go(vl, . . . , v p ) , l d ~ o ( ~-; )dGo(vj)I 5 1 (0 5 i < j 5 3), SO d G i ( v i ) 2 2n - 2 - (Z 1) = 2n - 3 - y. Thus, there are at most y 1 edges from each of w l , v2, and v3 to ( ~ 5 , . .. ,v2,+1}, and at most one edge from v4 to ( ~ 5 , .. ,v2,+1}. There are no edges from 00 to ( ~ 5 , . . ,~ 2 ~ + 1 } Thus, .

+

..

..

..

.

+

..

..

. ..

..

..

+

+

+

.

+

.

4y+42 3~+42(2n+l-p)(p-2)=2(2~~-3)= 4n-6.

Therefore, y 2 n - 2 1 3. In this case, we colour one edge between the vertices v1, 02, and v3 with CAo+l and another with CA0+2, and then complete the edge-colouring of Gt(w0,. ,vp) in the same way as in cases (ii)(a)-(d) of Lemma 5, as described above.

..

CASE2(ii). lmGt(v0,vi) - mG+(vo,vj)l > 1 for some i , j E {I,. .. ,n}. All multiple edges are incident with vo or v1, so for 1 5 i < j 5 3 we haveGt({vi,...,vzn+1) \{v;,vj}) = G * ( { ~ I..., , ~ , + l } \ { v ; , q } ) , and so (4) holds. Since (4) holds, r(Gt) < A(G), and so, by Lemma 6, either

.. , u p ) ) = A(Gt(v0,. .. ,vp)), or

0

x‘(Gt(v0,.

0

Gt(vo,.

0

mGt(v0,vl) = 2, p = 4, and

.., u p ) = I 1 f o r s o m e i , j E {I,...,m}. = w4 then mGtt(v0,2(1) 2 2. If 01 # w4 then, by the construction

CASE2'(ii).

ImGt(vO,vi)-mGt(vO,vj)l

If v1 of F , mGt(v0,2)1) = mGtt(v0,vl). In either case, m G t t ( v O , q ) 2 2, and Lemma 6 is applicable. All multiple edges are incident with 00 or v1, so Gt({vl,.*.,vPn+l} \ { v i , v j } ) = G*({vl,...,v2n+l} \ {v;,vj}), and therefore (6') holds. The argument now proceeds without much change from Case 2(ii); one case becomes slightly more complicated: if IE(H)[ = 2, p = 4, x'(Gtt(vo,.. .,v,)) = A(Gtt(v0 ,...,v,)) 1, and there is no one vertex with which each multiple edge is incident, then mGt(?&q) 5 3 and 2n - 2 = dGt(vo) 5 10, so n 5 6. This yields the contradiction that p 5 li(n 1)1 = 3.

+

+

Thus imitating the earlier argument, it follows that (Gt \ F ) \ {vp+l}is edge-colourable with 2n - 3 colours. Clearly, Gt(wg+2,. . .,w2,+1) \ F is edge-colourable with 2n - 3 colours. Finally, we colour the remaining edges of (Gt \ F ) \ {20,+1}; this can be done by applying Vizing's original argument rather carefully. There are now four vertices of maximum degree, namely 200, 201, w2, and 20,+2. The pivot vertices of the fans are always t o be in the set {wp+2, ,w2,+1}. The edge 20~+,.~+2w1 is coloured last but two, the edge wp+T1+~w2 is coloured last but one, and an edge on wp+2 is coloured last of all; Vizing's argument works because at each stage the pivot vertex is joined t o no vertices at which 2n - 2 colours have been used. In this edge-colouring of (Gt \ F ) \ {w,+l}, each colour is missing from exactly one vertex, and each vertex of degree 2n - 4 has exactly one colour missing. Therefore, the edge-colouring can be extended to an

...

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A. G. Chetwynd and A. J . W. Hilton

edge-colouring of Gt \ F with 2n - 3 colours, and so Gt and therefore G’ and thus G can be edge-coloured with 2n - 2 colours. 0 This completes the proof of the theorem.

Acknowledgement The authors started to write this paper whilst they attended a workshop on latin squares at Simon Fraser University, organized by K. Heinrich. They would like to thank her for inviting them and also the Natural Sciences and Engineering Research of Canada and Simon Fraser University who sponsored the event, and the Open University, England, who provided additional support for the authors.

References [l] A. G. Chetwynd and A. J. W. Hilton, “Partial edge-colourings of complete graphs or of graphs which are nearly complete,” Gmph Theory and Combinatorics, Vol. in honour of P. Erdos’ 70-th birthday, Academic Press (1984), 81-98. [2] A. G. Chetwynd and A. J. W. Hilton, “The chromatic index of graphs of even order with many edges,” J . Graph Theory 8 (1984), 463-470. [3] A. G. Chetwynd and A. J. W. Hilton, “Critical star multigraphs,” Graphs and Combinatorics 2 (1986), 209-221.

[4]A. G . Chetwynd and A. J. W. Hilton, “Star multigraphs with three vertices of maximum degree,” Math. Proc. Cambridge Phil. SOC.100 (1986), 303-317. [5] M. Plantholt, “The chromatic index of graphs with a spanning star,” J . Graph Theory 6 (1981), 5-13. [6] M. Plantholt, “On the chromatic index of graphs with large maximum degree,” Discrete Math. 47 (1983), 91-96. [7] V. G. Vizing, “On an estimate of the chromatic class of a pgraph,” Diskret. Analiz 3 (1964), 25-30. [In russian.]

Annals of Discrete Mathematics 4 1 (1989) 111-1 16 0 Elsevier Science Publishers B.V. (North-Holland)

On some Aspects of my Work with Gabriel Dirac P. Erdos Mat hemat ical Inst it U t e Hungarian Academy of Sciences Budapest, Hungary

Dedicated to the memory of my friend and coworker G. A . Dirac Several results and problems from areas of discrete mathematics of joint interest to G. A. Dirac and the author are discussed.

I must have known Dirac when he was a child in the 1930’s, but I really became aware of his existence when I visited England for the first time after the war in February and March 1949. We met in London and he told me of his work on chromatic graphs. Dirac defined a k-chromatic graph t o be vertex critical if the omission of any vertex decreases the chromatic number and edge critical if the removal of any edge decreases the chromatic number. I immediately liked these concepts very much and in fact felt somewhat foolish that I did not think of these natural and obviously fruitful concepts before. At that time I was already very interested in extremal problems and asked Gabriel to prove that for every k an edge critical k-chromatic graph must have o(n2) edges. More precisely define ff’(n) to be the largest integer for which there is a G ( n ; f f ) ( n ) )(i.e. a graph on n vertices and f f ’ ( n ) edges) which is k-chromatic and edge critical. Estimate or determine ff’(n) as accurately as possible. Trivially f g ’ ( 2 n + 1) = fg’(2n 2 ) = 2n 1 and I expected that for k > 3, ff’(n) = o(n2). To my great surprise very soon Dirac showed [3]:

+

f t ’ ( 4 n t 2) 2 (2n t 112

+

+ 4n + 2 .

(1)

After I recovered from my surprise I immediately asked: is (1) best possible? This question is still open. Toft [16] proved that 111

P. Erdos

112

where fi")(n)is the largest integer for which there is a graph G ( n ;f$")(n)) which is k-chromatic and vertex critical. It seems quite likely that for every k24

but as far as I know (3) is still open, and it has not even been proved that the limits exist. I also asked what about f f ' ( n ) and ft)(n). I still hoped that perhaps fi"'(n) = o(n2). In 1970 Toft [14]proved that n

Toft's proof is based on the idea of Dirac for the case k 2 6. Simonovits and I proved that f i e ) ( n )5 $+n which was later improved t o f i " ' ( n ) 5 $. It would be very desirable to improve (4)or my result with Simonovits and t o determine limn+m f i e ' ( n ) / n 2 . By the way, the graph of Toft has many vertices of bounded degree. This lead me to ask: Is there an edge critical four chromatic graph of n vertices each vertex of which has degree > cln? I conjecture with some trepidation that such a graph does not exist. The strongest known result is due to Simonovits [lo] and to Toft [15]. They proved that there is a G ( n ) which is four chromatic and edge critical, each vertex of which has degree > cn1I3. Dirac's six chromatic edge critical graph is regular of degree 5 -t2. It is not impossible that there is a four critical regular graph of n vertices and degree cln, but this seems unlikely to me. As far as I know there is no example of a regular edge critical four chromatic graph of degree L for L 2 6, but I expect that such graphs exist for every 1. By the way all of the early examples of these critical graphs contained abnormally large bipartite graphs and also small odd circuits. At first I thought that this was not an accident and must be so, but I learned that Rijdl found counterexamples some time ago. He showed that there is a constant Cr so that there is a four chromatic edge critical graph on n vertices and c,n2 edges which contains no odd circuit of size 5 2r -t 1. Exact results are not known, of course, though no doubt it can be proved that cr tends t o 0 as T tends t o infinity, but probably showing cr < cr-1 will not be easy.

On some Aspects of my Work with Gabriel Dirac

113

I recently heard from Toft the following conjecture of Dirac: Is it true that for every k 2 4 there is a k-chromatic vertex critical graph which remains k-chromatic if anyone of its edges is omitted. If the answer as expected is yes then one could ask whether it is true that for every k 2 4 and T there is a vertex critical k-chromatic graph which remains k-chromatic if any T of its edges are omitted. Perhaps there is an f(n) so that for every k 2 4 there is a k-chromatic vertex critical graph on n vertices which remains k-chromatic if any f ( n ) of its edges are omitted. If so one could try to determine the largest such f(n). Recently Toft asked: Is there a four chromatic edge critical graph with n vertices and c1n2 edges which can be made bipartite only by the omission of c2n2 edges? The original example of Toft could be made bipartite by the omission of cn edges. Rod1 and Stiebitz [ll]constructed such a graph but it is quite possible that the largest such c1 for which this is possible will be less than &, and it might be of some interest to determine the dependence of c2 from c1. Dirac [5] and I independently noticed the following strengthening of Turin's classical theorem. Denote by f(n;k ( r ) ) the smallest integer for which every G(n;f(n;k ( r ) ) ) contains a complete graph k ( r ) on T vertices. We noticed that then our G(n; f(n;k ( r ) ) )already contains a k ( ~ 1 ) from which a t most one edge is missing. In fact we showed that there is a k ( r - 1) and c,n further vertices y1,. . .yt, t = crn, for which each of the y's is joined - The exact value of c, is known only for T = 3. to every vertex of our k ( ~ 1). Bollobbs and I conjectured and Edwards proved that every G(n;[n2/4] 1 ) contains an edge (21,22) and y's, each of them joined to both 2 1 and 22, and is best possible. It would be very nice if one could prove some analogue to our result for hypergraphs. The simplest problem would be: denote by f(n;rC(3)(4))the smallest integer for which every 3-uniform hypergraph G(3)(n;f( n;! ~ ( ~ ) ( 4 ) ) ) on n vertices and f(n;k(3)(4)) triples contains all four triples of some set of 4 elements. The determination of f ( n ; / ~ ( ~ ) is( na) classic ) unsolved problem of Turin. Is it true that our G(3)(n; f(n;k(3)(n)))contains 5 points 2 1 , z2, 23, y1, y2 and all the triples of the quadruples { q , z 2 , ~ 3 , y 1 }and { ~ 1 , ~ 2 , ~ 3 , ~ Unfortunately 2 } ? nothing is known about this. Our result led me to the following theorem. By the theorem of Koviri, V.T. S6s and /') a complete bipartite graph ~ ( T , T ) . Turin [9]every G(n;c ~ n ~ - ~ contains I showed that every G(n; c2n2-l/') contains a k ( ~ 1 , ~1 ) with at most one edge missing [8]. Simonovits and I have the following problem: Let f(n;H )be the smallest integer for which every G(n;f(n;H ) ) contains a subgraph isomorphic

+

+

+

+

P. Erdos

114

t o the graph H . Is it then true that every G ( n ; f ( n ; H )contains ) at least two subgraphs isomorphic to R? We could not decide this question for C4’s. H = C4. In fact we expect that every G(n;f(n;‘24)) contains Nearly thirty years ago I conjectured that every G ( n ; 3 n- 5 ) contains a topological complete pentagon, i.e. 5 vertices every two of which are joined by paths no two of which have any interior point in common. It is easy to see that if true this conjecture is best possible. I soon found out that Dirac [6] anticipated me; he made the same conjecture before me. The conjecture is still open. As far as I know the best result is due t o Thomassen [13] who proved that every G(n;4n - 10) contains a topological complete pentagon. Despite our many contacts Dirac and I only had one joint paper [7]. This paper was perhaps undeservedly neglected; one reason was that we have few easily quotable theorems there, and do not state any unsolved problems. We prove there, among other results, that if G ( n ; n 3) is planar then it contains two edge disjoint circuits. Finally, let me remind the reader of a nice conjecture of Dirac [2]. This was conjectured also independently and simultaneously by Motzkin. Let there be given n points 21,. ,z, in the plane not all on a line. Then for at least one zi there are - c distinct lines among the (z;,zj), In a weaker form this conjecture has recently been proved by J. Beck [l]and independently by Szemerbdi and Trotter [12]. Dirac gave several beautiful consequences of his theory of critical graphs. For example, he proved in [4] that any graph G on the orientable surface S ,, y 2 1, for which equality holds in Heawood’s inequality

+

..

must contain a complete graph on H ( 7 ) vertices. He also obtained the similar result for non-orientable surfaces. This was one of the most significant contributions to map-colour-theory since Heawood’s pioneering paper in 1890. (The case of the torus was first obtained by P. Ungbr). Dirac made many deep and significant contributions t o several other parts of graph theory than those mentioned above, among them paths and circuits, Menger’s theorem and connectivity, extremal results for contractions and subdivisions, and infinite graph theory. He seemed much influenced by the work of Konig. His own influence is now present everywhere in graph theory. Finally, I want to thank B. Toft for his help in writing this paper and for supplying some references.

On some Aspects of my Work with Gabriel Dirac

115

References [l] J. Beck, “On the lattice property of the plane and some problems of Dirac, Motzkin and Erdos,” Combinatorica 3 (1983), 281-297. [2] G. A. Dirac, “Collinearity properties of sets of points,” Quarterly J. Math. 2 (1951), 221-227. [3] G. A. Dirac, “A property of 4-chromatic graphs and some remarks on critical graphs,” J. London Math. SOC.27 (1952), 429-437. [4] G. A. Dirac, “Map colour theorems,” Canad. J. Math. 4 (1952), 480490. [5] G. A. Dirac, “Extensions of Turin’s theorem on graphs,” Acta Math. Acad. Sci. Hungar. 14 (1963), 417-422. [6] G. A. Dirac, ‘‘Homomorphism theorems for graphs,” Math. Ann. 163 (1964), 69-80. [7] G. A. Dirac and P. Erdijs, “On the maximal number of independent circuits of a graph,” Acta Math. Acad. Sci. Hungar. 14 (1963), 79-94. [8] P. Erdos, “On some extremal problems in graph theory,” Israel J . Math. 4 (19651, 113-116. [9] T. Koviri, V. T. S6s and P. Turbn, “On a problem of Zarankiewicz,” Coll. Math. 3 (1954), 50-57.

[lo] M. Simonovits, “On colour-critical graphs,’’ Studia Sci. Math. Hungar. 7 (1972), 67-81.

[ll] M. Stiebitz, Beitriige mr Theorie der fiirbungskritischen Graphen, Dissertation, Technische Hochschule Ilmenau (1985). [12] E. Szemerbdi and W. T. Trotter, ‘‘ Extremal problems in discrete geometry,” Combinatorica 3 (1983), 381-392. [13] C. Thomassen, “Some homomorphism properties of graphs,” Math. Nachr. 64 (1974), 119-133. [14] B. Toft, “On the maximal number of edges of critical k-chromatic graphs,” Studia Sci. Math. Hungar. 5 (1970), 461-470.

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P. Erdos

[15] B. Toft, “Two theorems on critical 4-chromatic graphs,” Studia Sci. Math. Hungar. 7 (1972), 83-89. [16] B. Toft, “An investigation of colour-critical graphs with complements of low connectivity,” Annals of Discrete Math. 3 (1978), 279-287.

Annals of Discrete Mathematics 4 1 (1989) 117-130 0 Elsevier Science Publishers B.V. (North-Holland)

Bandwidth versus Bandsize P. Erdos Mat hemat ical Inst it U t e Hungarian Academy of Sciences Budapest, Hungary

P. Hell School of Computing Science Simon Fraser University Burnaby, B.C., Canada

P. Winkler Department of Mathematics and Computer Science Emory University Atlanta, Georgia, USA

Dedicated to the m e m o r y of G. A . Dirac The bandwidth (bandsize) of a graph G is the minimum, over all bijections p : V ( G ) + { 1,2,.. . , IV(G)l},of the greatest difference (respectively the number of distinct differences) Ip(v)-p(w)I for vw E E(G)* Weshow that a raph on n vertices with bandsize k has bandwidth e; between k and cnl-,, and that this is best possible. In the process we obtain best possible asymptotic bounds on the bandwidth of circulant graphs. The bandwidth and bandsize of random graphs are also compared, the former turning out to be n - c1 logn and the latter a t least n cz(logn)2.

1

Introduction

The problem of bandwidth minimization was motivated by the needs of matrix manipulations in structural engineering [l, 10, 191: there it is desirable to store the matrices in such a way that their non-zero entries are all as close to the main diagonal as possible. Simultaneous row/column permutations are applied to transform a given (large, sparse, symmetric) matrix to a form in which all non-zeros are in a narrow band of sub- and super-diagonals surrounding the main diagonal. Given a symmetric matrix M = (m,,j) (i, j = 1,.. . ,n ) , one may consider the graph G on n vertices in

117

118

P. Erdos, P. Hell and P. WinJder

which i is adjacent to j just if mi,j # 0. We now give the precise statement of the bandwidth minimization problem in terms of graphs; its correspondence to the above problem for symmetric matrices is self-evident (via the translation just given). Let G be a graph with n vertices. A numbering of G is a bijection p : V(G) -+ (1,. . . ,n}; the numbers Ip(u)- p(w)I for uw E E(G) are called the edge-diferences of the numbering p . The width of a numbering p is its largest edge-difference. The bandwidth of a graph G, bw(G), is the smallest width of any numbering of G, A matrix in which all non-zeros are in a narrow band is convenient for storage and computation. For some applications it may be enough to have all the non-zero entries concentrated in a small number of sub- and superdiagonals. (Such matrices also seem to arise in certain applications, e.g., in queueing network analysis of job line production models, cf. [8]Figures 1 and 2.) The corresponding graph-theoretic analogue is the following: The size of a numbering p is the number of distinct edge-differences of p; the bandsize of a graph G, bs(G), is the smallest size of any numbering of G. Of course, it follows from the definitions that bs(G) 5 bw(G). The actual motivation for the study of bandsize (as opposed to the possible application described above) originated from an investigation of spanning subtrees of the Ic-dimensional hypercube Q k . J. Malkevitch studied spanning subtrees of Q k , and derived a number of their properties; M. Rosenfeld observed that such trees must admit a numbering with k edgedifferences (in fact, with edge-differences 1,2,4,.. . ,2“-’),and was led t o ask if there was a bound to the number of edge-differences required by numberings of trees of maximum degree k. The notion of bandsize is formally introduced in [12],where it is shown that the bandsize of the complete 2. Since the maximum binary tree of height n, Tn,is between and degree in T, is three, this answers Rosenfeld’s question in the negative. Throughout the paper, we reserve the symbol “lg” for logarithms base 2, and “ln” for logarithms base e. The tree Tn has w = 2”+l- 1 vertices; thus its bandsize is roughly c lg w (for f < c < $). On the other hand, it can be shown that the bandwidth of Tnis as high as (cf. [4]for a tree similar to Tn).When the bandsize is so much smaller than the bandwidth, storing the matrix in the form we suggest, with few non-zero diagonals, would seem particularly attractive. In this paper we take up the comparison between bandwidth and bandsize. The largest bandwidth among all graphs of fixed bandsize is asymptot-

i

6,

9+

Bandwidth versus Bandsize

119

ically determined in the next section: a qraph with n vertices and bandsize k can have bandwidth as large as O(nl-r), but no more. Our method also yields best asymptotic bounds for the bandwidth of circulant graphs. In the last section we compare the bandwidth and bandsize of random graphs: it turns out that their values are quite close, n - c1 logn for bandwidth, and a t least n - c z ( 1 0 g n ) ~for bandsize. The bandwidth problem, i.e., the problem of deciding for a given graph G and integer k, whether there exists a numbering of G of width at most k, is well known to be NP-complete, even in the case of trees, [16, 91. On the other hand, if k is fixed, the problem of deciding if bw(G) 5 k can be solved in polynomial time, [17, 111. In contrast to this, the problem of deciding if bs(G) 5 k is NP-complete for every fixed k 2 2, [18]. A variety of other numbering problems have been studied recently, [6]. For instance the minsum (or optimal linear arrangement) problem [6] may be stated as follows: The sum of a numbering p is the sum of all its edgedifferences; the minsum of a graph G is the smallest sum of any numbering of G. The reader may find it amusing to note that the largest size of a numbering of G bears an obvious relation to graceful graphs; this notion, in some sense dual t o the notion of bandsize of a graph, may be called the gracesize of G, gs(G). Because of the famous graceful graph conjecture, it could be interesting to prove non-trivial lower bounds on the gracesize of trees. (In this terminology, the graceful graph conjecture asserts that the gracesize of any tree is equal to its number of edges.)

2 2.1

The extreiiial case General remarks

In this section, and the next, we prove the following theorem:

Theorem 1. Let k be fixed. A graph G with n vertices and bandsize k has bandwidth only O(nl-k). Moreover, this bound is best possible. Let I = { i 1 , i 2 , . . . ,il} be a set of integers, 0 < il < i 2 < . . . < il < n. The linear graph L n ( l ) (or L,(il,iz,.. .,il)) has the vertex set 2, = {0,1,. . . ,n - 1) and the edge-set { u v : Iu - E I } . The circulant graph C,(I) (or Cn(i1,i2,., . ,ir)) also has the vertex-set 2, and its edge-set is { u v : Iu - 01 = i (mod n ) for some i E I } . A graph is linear if it is isomorphic to some linear graph, and is a circulant if it is isomorphic to some circulant graph. Note that each L n ( l ) is a spanning subgraph of

VI

Cn(1).

P. Erdos, P. Hell and P. Winkler

120

Let G be a graph with n vertices. It follows from the definitions that the bandwidth of G is the smallest integer I such that G is isomorphic to a spanning subgraph of L,( 1,2,. . . , I ) (the isomophism taking a vertex numbered x to the vertex z - 1 of Ln( 1,2,. , . ,1 ) ) , and the bandsize of G is the smallest integer I such that G is isomophic to a spanning subgraph of some Ln( il ,i2, . . . ,il). Our proof of Theorem 1 (or rather of Theorem 2, below) depends heavily on the use of the natural “circular” extension of these notions: The circular bandwidth of the graph G (still with n vertices), cbw(G), is the smallest I such that G is isomorphic t o a spanning subgraph of C,(l, 2,. . . ,I); the circular bandsize of G, cbs(G), is the smallest I such that G is isomorphic to a spanning subgraph of some C,(il, i2, . . ,il). (Circular bandwidth shall play a central role in our proof; circular bandsize is introduced only for symmetry.) We shall also need the notion of “circular length”; formally the (circular) n-norm of a positive integer i, Ililln, is the unique lbl, - 5 6 _< F, such that i = a n 6. (Thus for i in Z,, the n-norm of i is the distance from 0 t o i in the graph Cn(l).) If we call the norm of a numbering the largest norm of its edge-differences, then the circular bandwidth of a graph is the smallest norm of its numberings. (If we call the normsize of a numbering the number of distinct norms of its edge-differences, then the circular bandsize of a graph.is the smallest normsize of its numberings.) The usefulness of circular bandwidth is due to the fact that

.

9

+

cbw(G) 5 bw(G) 5 2cbw(G). The first inequality follows directly from the definitions. To prove the second inequality it is enough to show that bw(C,( 1,2,. . . , l ) ) 5 21 :

+

Number all vertices x = O , l , . . . , - 1 by 2z 1, and all vertices y = [:I,, . . , n - 1 by 2(n - y). It is straightforward to verify that this is a numbering of Cn(1,2,. . . ,I) and that the edge-difference of any edge uv with w = u 1 (mod n ) is either 1 or 2; hence any edge-difference is at most 21. In the next subsection we shall prove the following result:

+

.

Theorem 2. bw(Cn(il,i2,. . ,ik)) I 4nl-k. Proof of Theorem 1 (from Theorem 2). A graph G with n vertices and bandsize k is (up to isomorphism) a spanning subgraph of L,(I) for

Bandwidth versus Bandsize

121

By Theorem som I with k elements; hence a spanning subgraph of Cn(I). 2, its bandwidth is only O ( n l - i ) . This bound is best possible, as for n = mk, the graph G = Ln(1,m,m2,...,mk-1 ) has bandwidth at least

(for k fixed and m + m). To see this, note that any two vertices of G can be joined by a path . m edges (at most m steps of type mk-', at most steps of at most of each of the types mk-2,.. . ,m, 1). Thus, whatever the numbering of G, the shortest path joining the vertices numbered 1 and n must have some edge-difference a t least

9

-n2- -1. v

m

2 k+l

mk-l

- 1. 0

Note that

e 4 is

maximized for k = Inn; hence

Corollary 1. The ratio of1bandwidth to bandsize for a graph on n vertices cannot ezceed (&) .4n1-

2.2

G.

The bandwidth of circulants

Here we prove Theorem 2; we restrict our attention to connected circulants (as the bandwidth of a graph is the maximum bandwidth of its components). In fact we shall show that (for any fixed k) .max bw(Cn(il,i2,. $1r...rlk

.., i k ) )

= @(nl-~).

. . ,i k ) )

= n(nl-i),

1

(2)

The lower bound, ma? bw(C,(il, i 2 , .

il

p...j$k

follows the same way it did for linear graphs. The remainder of this subsection contains the proof of cbw(Cn(i1,i2,. . .,ik)) 2 2n1-i,

(3)

which implies (2) and Theorem 2 because of (1). In other words, we seek an isomorphism of any Cn(i1,i 2 , . .. ,ik) onto a subgraph of cn(172,. . . , I ) with 1 5 2nl-k.

P. Erdos, P. Hell and P. Winkler

122

Lemma 1. Let n, k, and il,i2,,. . , i k be given positive integers. Then there exists an integer m with 0 < m < n such that Ilm ijll,, 5 n'-k for all j = 1 , 2 , . . ,k .

.

Proof. Note that the n-norm satisfies the triangle inequality. Let S ( n ,k) denote the torus [O,n)kwith entries taken modulo n. Let

-

A = { ( m i1,m. i2,.

..

,

m

a

ik)

E S(n,k) :0 5 m < n},

and for each m let 1 2

1

B, = { ( y l , y 2 , . . . , y k ) E S ( n , k ) :l l y j - m - i j l l n 5 - - . n l - r forall j } . The (k-dimensional Euclidean) volume of each B, is

Since the combined volume of the B,'s is nk (the volume of S ( n , k ) ) , and . . ,,zk) . in some since they are all closed sets, there is a point 2 = ( q , ~ BI, n B,u, m' # m". We then take m = Im'- m"l so that 0 < m < n and we have

for all i = ij, j = 1 , 2 , . . . ,k, as required. We are grateful to Mikl6s Simonovits for pointing out that Lemma 1 may also be derived from Dirichlet's theorem on simultaneous diophantine approximation, [2, p.1591. Because of its relation to [14], this may allow us to find the "multiplier" m figuring in Lemma 1 efficiently (cf. [14. p.5245251). Also note that Lemma 1 is sufficient to imply that

when n is prime. In fact, a long as the m from Lemma 1 is relatively prime to n, the mapping taking 2 to rn x mod n is a bijection 2, -+ 2, and hence an isomorpism of C,,(i1, i2, . . ,ik) onto a spanning subgraph of C n ( m i l , m i 2 , . . . ,m ik); and therefore onto a spanning subgraph of C,(l, 2 , . . . ,1) with I = Lnl-kJ.

-

.

-

Bandwidth versus Bandsize

123

If m and n are not relatively prime, then the above mapping is not a bijection. Nevertheless there always exists a bijection accomplishing our aims. This is not hard to see by arguing that the above mapping x + rn - x takes exactly g points of 2, onto each of the points g,2g,. .,n g = 0; thus it can be made bijective by local perturbations. To define such a bijection explicitely we can use the following facts:

.

Lemma 2. Let g = gcd(m,n) and d =

P.

(a) Each x E 2, can be uniquely written as and 0 5 r < d.

I

= dq

(b) Each x E 2, can be uniquely written as x = mu and 0 5 v < g.

+ r with 0 5 q < g

+ v with 0 5 u < d

+

Proof. Each x can be written as 2 = dq T with 0 5 T < d; since I E 2, and dg = 0 in Z,, q may be assumed to satisfy 0 5 q < g. The uniqueness in (a) follows from the fact that gd = n = lZ,l. Each I can also be written as x = mu v with 0 5 v < m; evidently u < d because md = * n 2 n. Since a m E g (mod n) for some a , v may be assumed to satisfy 0 5 v < g. 0 The uniqueness in (b) follows by the same argument as in (a).

-

+

Let F : 2, -+ 2, be defined as follows : if x = dq t r with 0 5 q < g and 0 5 T < d, then F ( I ) = m r q. According to Lemma 2, F is well. defined and a bijection. It remains to verify that F takes C,(il, i2,. . . ,ik) onto a spanning subgraph of Cn(l, 2 , . . . ,I) with 1 5 2n’-f;.

+

Lemma 3. If 111 - ~‘11,

= i, then [IF(.)

- F(x’)1In 5 Ilm - ill, t Ilgll,.

+

Proof. Let z = dq r , 0 5 q < g, 0 5 r < d , and x‘ = dq t T d(q f ) T’ where 0 5 T ’ < d. Case 1: 0 5 q t f < g. Then F ( x ) = m r q and F(x’) = mr‘ t q t f. Hence

+ +

+i =

+

I~F(I)

-

- F(z’)~], = l)m ( T - T ’ ) - f 1, = Ilm * ( d f - i) - f 1, = Ilm ‘ i f 1, L Ilm ’ ill, t Ilslln.

+

(Note that f 5 g because dg E 0 (mod n).) Case 2: 0 5 q t f - g < g. Then x’ = d(q f - g) T’ and F(d)= m . r ’ t q t f

+-

l l ~ ( x -) F(x’)lln =

=

+ - g. Hence Ilm - - r’) - f + glIn = Ilm. ( d - f - i) - f + 41, Ilm - i + f - glln L Ilm - ill, t 11g1In. (T

0

P. Erdos, P. Hell and P. Winkler

124

Lemma 4. Ifgcd(n,il,iz, ...,ik) = 1, and i f m < n has each llm.ij[l,,5 1 1 n1-F ( j = 1,2,. . . ,k ) , then g = gcd(m,n) 5 nl-b. Proof. If all Ilm. ijll,, = 0 then each m ‘ij is a common multiple of m and n, hence also a multiple of Therefore is a divisor of all ij as well as of n; according to our assumption = 1, contrary to g 5 m 5 It. O n the other hand, note that g divides m i j - a n for each j = 1 , 2 , , . . ,k, and every integer a. Assume that Ilm. ijll,, # 0; since llm. ijll,, is either r n . i j - a . n o r a . n - r n . i j for someintegera,gdivides JJrn.ij\l,,and hence 1 0 g 5 Ilm i j l l , , 5 nl-r as claimed.

a

y.

-

.

-

Proof of (3) (and thus of (2) and Theorems 2 and 1). Lemmas 1, 3 and 4 imply that the mapping F given above is an isomorphism of C,,(il, i2,. . ,ik) onto a spanning subgraph of C,,(l, 2,. . .,I), where 2 5 2n1-t, provided gcd(n,il, i 2 , . . .,ik) = 1. Since this condition is always satisfied for connected circulants C,( il, i 2 , . ,ik), (3) has been proved. 0

.

..

Remark. Let da(G) denote the number of vertices of G of distance s from a fixed vertex. We have studied, jointly with Martin Farber, the behaviour of d3(C,,(i1, i 2 , . . ,ik)) and d3(L,,(i1,iz,. . . ,ik)) for fixed k and believe that there exists a constant c depending only on k such that

.

d a ( ~ , ( i l , i 2 , . . . , i k )) 5 cn’-r

1

( 4)

for all n, s, and i1, i 2 , . . . ,ik. This would then offer another proof of Theorem 1: indeed any breadth first numbering (or so-called “level algorithm” in the terminology of [20])of L,,(il, i2,. . ,ik) would have width at most 2 c . n l - i . When k = 1 (4) is obvious (the constant c is 1 or 2 depending on the choice of the starting vertex); we were also able to prove (4) for k = 2 and k = 3. It turns out that to complete the proof of Theorem 1 this way, it would suffice to show that, for fixed k,

.

3

The random case

Our object in this section is t o compare the bandwidth and bandsize of random graphs. For 0 < p < 1 and n a positive integer, let G,,, be a graph with vertex set V,, = {1,2,. . . ,n} and edges defined randomly as follows: for each pair { i , j } of vertices with i # j the edge i j is included

Bandwidth versus Bandsize

125

(3

with probability p and excluded with probability q = 1 - p ; the choices are made independently. This construction has been studied extensively, especially by Erdos and Rdnyi [7]; see also BollobAs [3]. We wish t o compute bounds on bw(Gn,p) and bs(G,,,) for almost all G,,, when n is large. For simplicity we concentrate below on the case p = ?j; the results are easily generalized. Note that when p = the graph G,,,,which we denote below simply by G,, has the special property that every graph on the labelled set V, is equally likely to occur. The following theorem is proved in [13].

4

Theorem 3. With probability approaching 1 as n n - (2 t f i t o(1)). Ign

+ 00,

< bw(Gn) < n - (2 + &- o(1)) - Ign.

As we found, a number of recent papers studied the bandwidth of random graphs, [5, 13, 15, 20, 211. Theorem 3 is stronger than similar results in [20, 211, and weaker than the most general version of [13]. We have stated it in this way for simplicity, and also because this was the form of the result we had before discovering [13]; we had the constant 2 t 6in the lower bound, but not in the upper bound. (In [15] the authors study the average bandwidth of trees, which turns out to be between c l f i and czfilog n.)

Theorem 4. With probability approaching 1 as n bs(G,) 2 n Proof. Fix c

>

(A+

o(1))

00,

.

& and t = [c(lnn)21; we show that

Pr(G, has a numbering of size less than n

- t ) -P

0.

It suffices to show that Pr(G, has a numbering omitting lengths n - d l , . .. , n - d t ) O 0 contradicts the claim. There is no i-critical wg-set C, for otherwise C n M is i-critical, contradicting the minimality of M . Similarly, there is no j-critical y2i-set C, since then M - C would be i-critical, contradicting the minimality of M , and there is no j-critical vr-set, since d(M,-e) > 0.

CASE2. There is no critical wr-set, but there is a critical xZi-set M . Let M be maximal, and suppose that M is i-critical for some i = 1, 2, . . ., k. Then there exists an edge yw such that y $ M . We claim that yw and wz can be split off. Indeed, by assumption there is no critical wT-set. Similarly, there cannot be a critical @-set C, since C is i-critical; then C U M is also i-critical, contradicting the maximality of M , and if C is 0 j-critical ( j # i), then d(M,iT) > 0, contradicting the claim. The following theorem is immediately implied by Theorem 4.1.

Theorem 4.2. In an Eulerian digraph G = (V,E), the maximum number t of puirwise edge-disjoint A-paths is k i=l

.

where V,,V,, . ., Vk are disjoint subsets of V and V;. n A = {w;} (i = 1, ., , k). Furthermom, t = C d;, and if M ; denotes the minimal set for which Mi n A = {w;} and 6(M;)= d;, then the sets M; (i = 1, 2, . .. , k) are pairwise disjoint, and they form an optimal solution to (3).

2,

.

Remark. Lov&z’s theorem on A-paths of an undirected Eulerian graph [7] easily follows if we apply Theorem 4.2 to any Eulerian orientation of G. On the other hand, the converse derivation is not difficult either. Let G = (V,E ) be an Eulerian digraph, and let denote the underlying undirected Eulerian graph. We show that there are t directed edge-disjoint A-paths in G, provided that there are t edge-disjoint A-paths in E. For a directed A-path or circuit P , we say that an internal node w of P is in-bad (on P ) if both edges of P incident to w enter u; w is said to be out-bad if the two incident edges on P leave w. Let b(P) denote the number of bad nodes of P . Let P be a decomposition of E into undirected A-paths 5,Pz, , Pt and some circuits such that

z

...

x ( b ( P ) :P E

P)

On Connectivity Properties of Eulerian Digraphs

191

is minimal. We claim that every path P is a directed path in G (although not necessarily a simple path). Suppose, indirectly, that a path P E P contains an in-bad node v (say). Since G is Eulerian, v must be an out-bad node of another member Q of P. At least one of the two possible switches of P and Q at v gives rise to another partition P' of E into t (possibly not simple) A-paths and some circuits for which C ( b ( P ) : P E P') is smaller, a contradiction. See Figure 4.

Figure 4. Switch of P and Q at v.

Remark. From an algorithmic point of view, the edge-disjoint A-paths problem and its capacitated version (when to every edge a non-negative capacity is assigned such that ec(v) = &(v) for every v E V) can be solved quite analogously to the two-commodity flow problem analysed in Section 3. We leave out the details. Theorem 4.3. Let G = ( V , E ) be an Eulerian digraph and vx E E . There exists an edge yv E E such that splitting 08yv and vx does not reduce dc(s,t ) for any s , t E V - v. Proof. Call a set C critical with respect to a pair of nodes s,t (# v) if C is a tT-set and d c ( s , t ) = e(C). If C is critical for s,t , then C is critical for t , s . The set C is said to be critical with respect to s , t in if I { s , t } n CI = 1 and c ( s , t ) = d(C). Obviously, C is critical in G if and only if C is critical in 5. The next lemma concerns 5.

c

Lemma. I f X and Y are critical, then either (i) X n Y and X U Y are critical and d ( X , Y ) = 0 , or (ii) X

- Y and Y - x are critical and d(x,T)= 0.

A . Frank

192

Proof of Lemma. Let X be critical for q , ~ and , let Y be critical for y l , y2. We have three possibilities t o consider. (a) XnY separates oneof the pairs { q , z 2 } , {yl, yz}, and XUY separates the other. (A set A is said to separate 2 and y if ( A n (2,y}( = 1.) Then

d(X)

+ d(Y)= d ( X n Y )+ d ( X u Y )+ 2d(X,Y)

+

2 d ( X ) t d(Y) 2d(X,Y) from which (i) follows. (b) X - Y separates one of the pairs { q , x 2 } , {yl,yz}, and Y - X separates the other. Applying (a) to the sets X and p,we obtain (ii). (c) X separates { y l , y ~ } ,and Y separates { t 1 , ~ 2 } ,and one of the pairs {XI, ZZ}, {y1,y2} is separated by both X n Y and X U Y . Now

Suppose that the pair (y1, y2} is separated by both X Then we have 4x1 = d(Y)= C(Y1, Y2) and

+

n Y and XU Y .

+ + 2 C(Yl,Y2) + C(Y1, Y2) + 24x9 Y )

d ( X ) d(Y)= d(X n Y ) d ( X u Y ) 2d(X, Y )

+

+

= d ( X ) d(Y) 2d(X, Y ) from which (i) follows. 0

This completes the proof of the lemma.

The lemma implies the corresponding assertion for G. Therefore, if X and Y are critical in G, and the edge w t enters both, then X U Y is also critical. Hence, there is a unique maximal critical tv-set M (if there is one at all). If no such M exists, any edge yw can be split off. If we have such an M , there is an edge yw with y 4 M . Indeed, if M is critical for s, and t and the required edge yw did not exist, then

a contradiction. By the construction of M , the edges yw and split off without reducing dc(s,t) for any s , t E V - w.

212

can be 0

On Connectivity Properties of Eulerian Digraphs

193

Remark. W. Mader proved [9] that, given a not necessarily Eulerian digraph G = (V, E) and a node v such that dc(s,t ) 2 k for every s, t E V - v and e(v) = 6(v), a pair of edges yv,vz can be split off such that the connectivity from any s € V - v to any t € V - v continues t o be at least k. Remark. B. Jackaon [5] also proved Theorems 4.1 and 4.3.

References [l] L. R. Ford and D. R. Fulkerson, Flows in Networks, Princeton Univ. Press, Princeton, N. J. (1962).

[2] S. Fortune, J. Hopcroft and J. Wyllie, “The directed subgraph homeomorphism problem,” Theoretical Computer Science 10 (1980), 111121. [3] A. Frank, “Edge-disjoint paths in planar graphs,” J. Cornbin. Theory (B) 39 (1985), 164-178. [4] T. C. Hu, “Multicommodity network flows,” Operations Res. 11 (1963), 344-360. [5] B. Jackson, “Some remarks on arc-connectivity, vertex splitting and

orientation in digraphs.” Preprint (1985). [6] R. M. Karp, “On the computational complexity of combinatorial problems,” Networks 5 (1975), 45-68. [7] L. LovBsz, “On some connectivity properties of Eulerian graphs,” Acta Math. Acad. Sci. Hungar. 28 (1976), 129-138. [8] W. Mader, “A reduction method for edge connectivity in graphs,” Annals of Discrete Math. 3 (1978), 145-164. [9] W. Mader, “Konstruktion aller n-fach kantenzusammenhangenden Digraphen,” Europ. J . Combinatorics 3 (1982), 63-67.

[lo] H. Okamura, “Multicommodity flows in graphs,” Discrete Applied Math. 6 (1983), 55-62. (Proc. London Math. SOC.(3) 42 (1981), 178192).

[ll] H. Okamura and P. D. Seymour, “Multicommodity flows in planar graphs,” J . Combin. Theory ( B ) 31 (1981), 75-81.

194

A. Frank

[12] B. A. Papernov, “Feasibility of multicommodity flows,” Studies in Discrete Optimization (A. A. Friedman, ed.), Isdat. “Nauka,” Moskow (1976), 230-261. (In Russian). [13] N. Robertson and P. D. Seymour, “Graph minors - a survey,” Surveys in Comb~naton’cs(I. Anderson, ed.), London Math. SOC.Lecture Notes 103 (1985), 154-171. [14] B. Rothschild and A. Whinston, “Feasibility of two-commodity network flows,” Operations Res. 14 (1966), 1121-1129.

[ 151 A. Schrijver, “Min-max results in combinatorial optimization,” Mathematical Progmmming - The State of the Art (A. Bachem, M. Grotschel and B. Korte, eds.), Springer Verlag (1982), 439-500. [16] P. D. Seymour, “Disjoint paths in graphs,” Discrete Math. 29 (1980), 293-309. [17] P. D. Seymour, “Four-terminus flows,” Networks 10 (1980), 79-86. [18] P. D. Seymour, “On odd cuts and plane multicommodity flows,” Proc. London Math. SOC.(3) 42 (1981), 178-192.

Annals of Discrete Mathematics 41 (1989) 195-210 0 Elsevier Science PublishersB.V. (North-Holland)

Some Problems and Results on Infinite Graphs R. Halin Mathematical Seminar University of Hamburg Hamburg, FRG

Dedicated to the memory of G. A . Dirac The structure of infinite graphs is investigated. Especially locally finite and so-called bounded graphs are examined, as well as infinite maximal planar graphs.

1

Introduction

G.A. Dirac was among that minority of graph theorists who have a genuine interest in infinite graphs. I had several discussions with him on this topic, especially when I enjoyed an invitation by him t o h h u s in 1977. Some of the material presented in this paper was suggested by these conversations. With gratitude and respect I dedicate this report on some of my current research to his memory. In Section 3 the concept of maximal planar graph is discussed. There are several equivalent statements in the finite case which however split up highly if carried over to infinite graphs. It becomes evident that only the concept based on Kuratowski’s theorem and its extension to countable graphs (see Dirac and Schuster [2]; it is attributed to P. Erdos by these authors) leads to a sensible definition of maximal planar graphs in the infinite case. Further in Section 3 some results of [5] are reported and extended. In Section 4 the structure of countable maximal planar graphs is investigated especially with respect to infinite paths. Every 3-connected planar graph has a locally finite spanning subtree. Hence every infinite maximal planar graph contains a 1,oo-path. We characterize those maximal planar . it is shown that an infigraphs which do not contain a 2 , ~ - p a t h Further nite maximal planar graph has a VAP-free representation in the plane iff it 195

196

R. Halin

has only one end (see below for definitions); then it has a t most 2 vertices of infinite degree. In Section 5 we discuss a possible classification of infinite graphs by reducing “modulo finiteness”; the concept seems especially useful for locally finite graphs because they can be represented by trees. In Section 6 we introduce “bounded graphs”, a class of countable graphs generalizing the locally finite graphs. A countable graph is called bounded if to every function f : V(G)-+ 1zv there is a sequence 71,72,73,. of natural numbers which, for each 1,oo-path U C G, majorizes the sequence of f-values of the vertices in U apart from a finite number of exceptions. The problem of characterizing the bounded graphs by forbidden subgraphs is solved for the class of trees.

..

2

Terminology and notation

We follow the standard terminology and notation in graph theory; some additional remarks are necessary. All graphs in this paper are undirected and do not contain loops or multiple edges; they are allowed to be infinite. A graph is looally finite if all its vertices have finite degree. A one-way infinite path is briefly denoted as a 1,w-path, or simply as a PI,^. Similarly, a 2, w-path, or a is a two-wuy infinite path. 1,oo-paths U , U’ in a graph G are separated by a finite T C G if there are infinite subpaths of U and U‘ which belong t o different components of G - T . 1,ca-paths U,U’ in G are called equivalent if they cannot be separated by a finite T ; we then write U -G U’. N G is an equivalence relation; any equivalence class with respect to N G is called an end of G. U NG U’ is equivalent with the existence of a third 1,ca-path which meets both U and U‘ infinitely often [3]. A graph G is muzimal in a class I’ of graphs (or with respect to a graph theoretical property defining I’), if G E I’ and any graph arising by adding a new edge (Lee,which joins two non-adjacent vertices of G) to G no longer belongs to I’. If @ is a set of finite graphs, I?(@) denotes the class of all graphs (finite or infinite) not containing a subdivision of any F E @. The class of maximal graphs in I?(@) is denoted by I?(@). Using Zorn’s lemma one easily proves that each G E I?(@) can be extended, by adding new edges, to an element of f(@)[5, (3.1)) The elements of f(@)have a simplicia1 decomposition with finite or countable prime members (see [5]; [8], Chap. 10; or [13] for more details).

Some Problems and Results on Infinite Graphs

197

By G * H we denote the disjoint union of graphs G and H , in which, in addition, every possible edge is drawn between V ( G )and V ( H ) . In G * n, n a cardinal, n is understood as the edgeless graph with n vertices. A graph G is called planar if i t can be represented (drawn) in the plane. Sometimes we identify an (abstract) planar graph with one of its representations in the plane, If we use the attribute ‘%lane” we refer to a special representation (figure) in the plane. A VAP-free representation of a graph in the plane is a representation such that the vertex set (as point set in the Euclidian plane) does not have an accumulation point.

3

On planar embeddings and maximality of countable graphs

If G is a finite graph with a t least 3 vertices, then the following statements are equivalent:

(a) G has a maximal plane representation $J.This means: It is not possible t o draw an edge (= Jordan arc) between two non-adjacent vertices without meeting +(G) in an inner point of this arc. (b) G has a plane representation $ which triangulates the plane. This means: .RR2 - $(G) has, as its connected components, regions (= open and connected point sets) whose boundaries are triangles of G. (c) G is a maximal planar graph (i.e., maximal in the class of planar graphs). (d) G is maximal in

I’(K5, K 3 , 3 ) .

This equivalence is no longer valid at all in the case that G is countable. Of course each embedding of a maximal planar graph must be a maximal plane representation, hence (c) ==+ (a). But for instance PI,^ has a maximal plane representation, hence (a) does not imply (c). For let PI be an edge [u,b],represented as a vertical line segment in the plane. Assume the finite path P, starting in u to be represented in the plane by vertical and horizontal line segments. Then continue P, to a path P,+l by adding a path in its end vertex # a such that this continuation is represented by horizontal or vertical line segments and each point of P, has on both sides (left and right) points of this continuation in distance < 2-,. (This can be done by tracing P, “very closely” on both sides). The

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union of the P, gives a representation of in which no additional edge can be drawn. are e.g. such that all points of the Other “foolish” embeddings of plane are accumulation points of the vertex set. (Let D be a countable dense subset of R2 such that each horizontal and vertical line contains only finitely many points of D ;then by traversing D using horizontal and vertical line segments we find a straight line representation of PI,^ such that V ( P l , , ) 2 D.)Also, we can draw PI,, in such a way that a given finite or countable number of connected components of R2- PI,^ arises. (c) does not imply (b), for the “iterated cube” CJ is a maximal planar graph without a triangle. (Q is defined as follows: Let Q1 be a rectangle in the plane. Assume Q n to be constructed as a finite plane graph in which each region is bounded by a rectangle. Then in each of these rectangular regions a further rectangle is drawn, the vertices of which are connected to the vertices of the boundary by 4 additional edges such that a cube graph arises. The union of the Q n is Q.) Of course every triangular embedding gives a maximal plane representation, so (b) implies (a). But (b) does not imply (c). For instance let a straight line representation of K 4 be given with vertices a , 6 , c , d , vertex a inside the (bounded) triangle b, c, d. Let a sequence of points pi # a , b on the segment be given which converges to a , and draw all edges [pi,c], bi,d ] . Then the resulting configuration is a plane graph which triangulates the plane, but this graph is not maximally planar, because an edge [a,6]can be added. By the Erdos-Dirac-Schuster extension [2] of Kuratowski’s theorem to countable graphs we see that (c) and (d) are equivalent, and along this line only it makes sense to investigate the maximality of countable planar graphs. We call the elements of I ’ ( K s , K 3 , 3 ) formally planar. Among the (maximal) formally planar graphs those of cardinality 5 No are the (maximal) planar graphs. In [ 5 ] , Satz 4 the structure of maximal formally planar graphs is described. We call an edge (or a K z ) in a graph G of type 1 if it is contained in a non-separating triangle of G; otherwise it is of type 2. Then we have:

Proposition 3.1. Every maximal formally planar graph G has a decomposition of the form

G = UGx X -1, b,d

Dies bedeutet: In samtlichen Vierfarbungen von F,, tragen die beiden Randgegenecken A und C des Randkreises C, von F,, verschiedene Farben. In dieser F,, gehe jetzt der Randkreis C4 = ( A , B , C, 0)durch Identifikation A 3 C in das Kantenpaar ( B , A 1 C , D ) iiber; F,, ist hierbei in die Triangulation Tv-l ubergegangen. Laut Induktionsvoraussetzung ist Tv-l vierfarbbar. Es liege daher jetzt T,,-1 mit einer seiner nT, Vierfarbungen gefarbt vor. Nun werde die Identifikation A = C wieder aufgehoben, wobei alle Ecken ihre Farbe behalten und die jetzt wieder verschiedenen Ecken A und C beide die Farbe behalten, die sie bei der Identifikation hatten. Jetzt

Ein zum Vierfarbensatz Aquivalenter Satz der Panisochromie

237

liegt eine Vierfarbung der ursprunglichen F,, vor, fur die wenigstens eine der beiden Ungleichungen n u > -1 bab

oder

n u 21 bad

gilt. Das steht im Widerspruch zu dem Gleichungspaar in (15). Hierdurch ist bewiesen, dafl es keine Figur F,, gibt, die die Bedingungen (15) erfullt. Aus Symmetriegrunden ist hiermit auch der zweite der 11 Falle (14) erledigt, dessen Definition sich aus (15) durch Platztausch von n a mit bad

n

bcb

ergibt.

Da beim Aufbau des Widerspruchs nur die zwei Gleichungen in (15), aber keine der zwei Ungleichungen benutzt wurden, ist zugleich bewiesen, dafl es auch fiir den Fall 11 von (14) mit seiner Bedingung r(F,,) = 0 keine Figur gibt, die diesen Fall 11 verwirklicht. Hieraus folgt, wie beilaufig bemerkt sei, 5 5 w 5 v - 1, r(F,,,) 2 3 gilt, sind auch noch alle Figuren F,, vierfarbbar. Satz 4.1. Unter der Vomussetzung, dajl fur alle Figuren F,,

5

Die Falle Nr. 3, 4 sowie 7 bis 10 aus den 11 Fallen (14)

Hier wird mit derjenigen Verwirklichung von r ( F ) = 2 begonnen, bei der n u >1, bad

n u > -1; b,b

dies sind die Definitionsbedingungen fur den Fall 3 aus den 11 Fallen von (14). Die Einsicht, dai3 keine Figur F,, existiert, die (17) genugt, wird durch einen Blick auf die einfachsten Eigenschaften von Kempeketten in F,,Farbungen gewonnen: Kempeketten sind maximal zusammenhangende zweifarbene Teilgraphen in einem mit hochstens 4 Farben gefarbten Graph. Mit den Kempeketten eines Farbpaares werden meist zugleich auch die Kempeketten des komplementaren Farbpaares betrachtet. Ein Paar komplementarer Paare der 4 Farben wird als Farbpaarwahl bezeichnet. Es gibt die 3 Farbpaarwahlen ab, cd; ac, bd; ad, bc.

(18)

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dz

Fur eine Figur F, mit Randkreis Cq = existiert bei jeder Farbung wenigstens eine Farbpaarwahl derart, dafl das eine der 2 Randgegeneckenpaare (2.B. A , C) nur mit Farben des einen Farbpaares - sagen wir: a , c, - das andere ( B , D ) nur mit Farben des komplementaren Farbpaares b,d - gefarbt ist. In (7), z.B., trifft das bei jeder der 4 Farbungen 1, 2, 3, 4 fur die Farbpaarwahl ac, bd zu. Wegen der (in der Definition geforderten) Maximalitat des Zusammenhangs einer jeden Kempekette liegen daher fur dieses Farbpaar immer die beiden Randecken genau eines der beiden Gegeneckenpaare in einer Kempekette. Diese Kempekette trennt zugleich immer die beiden verschiedenen komplementaren Kempeketten voneinander, deren jede genau eine der beiden ubrigen 4 Randecken enthalt. Vertauscht man in einer dieser beiden letzteren Kempeketten deren beide Farben miteinander, so wird dabei (bekanntlich) die Farbungsvorschrift (siehe Abschnitt 2, Anfangssatz) nirgends verletzt. Man erhalt dadurch also eine andere Farbung der F,, bei der auch genau eine der 4 Randecken ihre Farbe in die andere ihres Kempekettenfarbpaares gewechselt hat. Iterierung dieser Operation fuhrt auf die Ausgangsfarbung einschlieDlich Ausgangs-Randfarbung zuruck. Das heifit: Den F,-Farbungen ist eine Puarstruktur eigen, an der auch die 4 Randfarbungen partizipieren. Fur die Randfarbungen lauten die moglichen Paare (Randfarbungsnumerierung siehe (7)): 1 - 3 oder 1 - 4 oder 2 - 3 oder 2 - 4. Weil, wie gesagt, bei der Umfarbungsoperation nur immer genau eine Randecke eines Gegeneckenpaares die Farbe zu wechseln vermag, gibt es kein Paar 1 - 2 und kein Paar 3 - 4, das durch die besprochene Operation hatte entstehen konnen, Das Gesagte ist darum nunmehr folgendermaflen zu formulieren: S a t z 5.1. Bei jeder Figur F,, far bu ng

5

5 v, tritt mit einer Farbung der Rand-

1 wenigstens auch eine Furbung mit Randfarbung 3 oder 77 n Y? 79 2) n 2 3 ?’ n 79 97 97 n 77 3 1 ” 99 Y9 n # n 4 1 ” )>

4 auj 4 ”, 2 ”, 2 ”,

Eine einfache Folgerung aus diesem Satz lautet: Satz 5.2. Es gibt keine Figur

F,, deren samtliche Furbungen dieselbe

Randfiirbung haben. Hierdurch ist fur die 4 F a l e Nr. 7 bis 10 der 11 Falle (14) - dies sind genau die samtlichen 4 Falle mit r(F,) = 1 - bewiesen, dafi es keine Figur F, gibt, durch die einer von ihnen verwirklicht wurde.

Ein zum Vierfarbensatz Aquivalenter Satz der Panisochromie

239

Ferner ist der Fall Nr. 3, mit T(F,) = 2, der in (17) definiert ist, infolge von Satz 5.1 ohne mogliche Verwirklichung. Ebenso ist auch der aus (17) durch Vertauschung der ersten beiden Randfarbungen mit den letzten beiden Randfarbungen hervorgehende Fall Nr. 4, na >1, bab

n a > -1 , bed

laut Satz 5.1 ohne Verwirklichung durch eine Figur F,.

6

Beschreibende Formulierung des letzten der 11 Falle (14)

Wie schon im Schluflsatz von Abschnitt 3 gesagt, bleibt jetzt fur F, bei Ansetzen von (13) allein der Fall Nr. 5 (mit seinem symmetrischen Gegenstuck Nr. 6) von den samtlichen 11 Verwirklichungsmoglichkeiten (14) zu untersuchen. Definiert ist Fall 5 durch die folgenden 4 Bedingungen - 2 Gleichungen und 2 Ungleichungen -

Fall 6 durch na >1, bab

n u >1. beb

Unter der in Abschnitt 3 vor Ungleichung (13) ausgesprochenen Induktionsvoraussetzung, daB fur alle Figuren F ! , 5 5 w 5 v - 1, die Ungleichung (12) gilt, wird jetzt das Vorliegen einer Figur F, angenommen, fur die (13) in der Form der 4 Bedingungen (20) gilt. Diese Bedingungen besagen: Fur F, ist Satz 4.1 in folgender Weise erfullt : Zunachst: Wegen der 2 Gleichungen in (20) tragen die Randkreisecken A und C des Randkreises Cq = .“d) in allen Vierfarbungen von F, dieselbe Farbe. Alsdann gilt wegen der 2 Ungleichungen in (20): Es gibt wenigstens eine Vierfarbung von F,, in der auch B und D gleichgefarbt sind, und es gibt ferner ebenfalls wenigstens eine weitere Vierfarbung von F,, in der B und D verschiedengefarbt sind. Wird jetzt aus F, durch Hinzufugen der Diagonale (B,D) die Triangulation T, E T,- erzeugt, so tragen ebenfalls in siimtlichen - wegen der

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240

letzten Ungleichung in (20) sicher existierenden - Vierfarbungen der T,,die Ecken A und C dieselbe Farbe. Wird indessen, anstelle von ( B , D ) , die andere Diagonale, ( A , C ) , in den Randkreis von F, eingezogen, so liegt TvI vor. Jetzt ist es - wegen der beiden Gleichungen in (20) - nicht moglich, auch fur die Kante ( A ,C) die Farbungsvorschrift (siehe Anfangssatz des Abschnittes 2) zu erfullen, daB auch deren beide Ecken A und C verschiedene Farben tragen. Dies heiat aber: Ti ist nicht vierfarbbar, T,,I ist funfchromatisch. Damit steht man vor einer neuartigen Situation: Erste Erfahrungen mit ihr vermitteln den deutlichen Eindruck, dafl eine Erledigung auch dieses letzten Falles von (14) in einem ahnlichen Rahmen wie fur die 9 in Abschnitt 4 und 5 abgehandelten Falle nicht erkennbar ist. Zwar lassen sich mit geringen Mitteln bemerkenswerte Eigenschaften solcher T, beweisen, z.B. dafl eine funfchromatische T,, keine Ecke des Grades 4 enthalt. Die Hauptfrage indessen ist die Frage nach der Existenz von Figuren F,,, die (20) genugen. Nun gibt es eine Algorithmik, in der diese Hauptfrage direkt und unkompliziert formulierbar ist; sie gehort indessen einem m.W. bisher wenig entwickelten Gebiet an. Wie schon ervahnt, handelt es sich dabei um die Panisochromie; von ihr wird darum in Abschnitt 7 ein Abrifl gegeben. Dieser Abrifi kann kurz sein, weil der (unbewiesene) Satz aus dem Gebiete der Panisochromie, dessen Beweis das in Abschnitt 3 bis 5 Gebrachte zu einem Beweis des Vierfarbensatzes erganzt, in seiner Formulierung einem noch recht einfachen Teilgebiet der Panisochromie zugehort. Diese Formulierung geschieht in Abschnitt 8, womit das Ziel dieser Arbeit erreicht ist.

7 7.1

Kurzer Abrifi der Panisochromie Allgemeine Einfuhrung

In der Graphenfarbungstheorie sind meist die Eclcen die Trager der Farben (fur planare Graphen durch Dualisierung die “Lander”). Die Farbungstheorie hat die Frage zum Gegenstand, wie in verschiedenen Graphen und Graphenklassen die “Farbungsforderung” verwirklicht wird. Diese Farbungsforderung lautet: Die beiden Ecken einer jeden Kante tragen verschiedene Farben. Die Objektklasse, in deren samtlichen Elementen die Farbungsforderung zu verwirklichen ist, ist also die Klasse der adjazenten Eckenpaare eines Graphen. Hat man dabei die planare Problematik - sei es fur endliche Graphen

Ein zum Vierfarbensatz Aquivalenter Satz der Panisochromie

241

(Kugel) oder unendliche (euklidische Ebene) - im Auge, so ist man besonders an den Ergebnissen bei mazimal planaren Graphen, d.i. bei Triangulationen T interessiert. Ist, im Kugelfall, die Eckenzahl v , so gilt die Eulersche Polyedergleichung (1) mit 2e = 3 f ;

(22)

hieraus ergibt sich die Anzahl der [Kanten oder der] adjazenten Eckenpaare einer jeden T, zu (siehe Textzeile 2 auf Seite 1) e = 3v - 6. Diese 3v - 6 Eckenpaare sind also, wie schon gesagt, der genaue Objektbereich, auf dem die Farbungsforderung operiert. Nun gibt es in der T, im ganzen (g) = &(v - 1 ) Eckenpaare; das heifit: es gibt in einer T, 1

(i)- ( 3 -~6) = -(v - 3 ) ( -~ 4) = ( " T ~ ) 2 nichtadjazente Eckenpaare. Direkt sind diese nicht betroffen durch die Farbungsforderung. Ihr gegenuber ist daher als allgemeines Verhalten eines beliebigen dieser ('i3) nichtadjazenten Eckenpaare anzunehmen, daB in den , der Tv die beiden Ecken dieses (im allgemeinen zahlreichen) n ~ Farbungen Paares bei einigen dieselbe Farbe, bei den ubrigen verschiedene Farben tragen. Die Beobachtung zeigt nun, daB es Graphen T, gibt, fur die die gewohnliche Farbungsforderung, in allen Farbungen verschiedene Farben zu tragen, uber die adjazenten Eckenpaare hinaus auch fur weitere Eckenpaare erfiillt ist. Jedes solche Eckenpaar wird panheterochromes Eclcenpaar zur Farbenanzahl x (hier meist x = 4 ) genannt (vgl. das bekannte Beispiel des Ikosaedergraphen, 7.2, Bsp. 3). Ferner zeigt die Erfahrung, dafi fur nichtadjazente Eckenpaare in Graphen auch das (fur adjazente Eckenpaare ausgeschlossene) Kontrare vorkommt, namlich Gleichgefarbtheit der beiden Ecken in allen Farbungen, das ist: deren Panisochromie. Dabei ist dies Vorkommen nicht auf Paare beschrankt, es zeigt sich auch bei Eckentripeln, -quadrupeln, . . ., -nl-tupeln. In jedem solchen Falle sagt man, der Graph enthalte eine nl-elementige Menge panisochromer Ecken mit n1 2 2 . Ja, unter diesen gibt es Graphen, in denen je eine solche Menge zu zwei oder drei der x Farben, bis, hochstens, zu den samtlichen x Farben vorliegt (Beispiel siehe 7.2, Bsp. 6). In einem Graph mogen V,W ein panisochromes Eckenpaar sein, und es gebe in dem Kreis der mit W adjazenten Ecken wenigstens eine Ecke X mit einem Abstand 2 2 von V . Als Folge der Panisochromie von V,W ist das Eckenpaar V , X panheterochrom. Jede in ahnlicher Weise als Folge

H . Heesch

242

vorkommender Panisochromie auftretende Panheterochromie werde dependente Panheterochromie genannt. Hingegen heiBt jede in einem Graph ohne ein panisochromes Eckenpaar vorkommende Panheterochromie independent. Fur independente Panheterochromie ist der unmittelbar vorher erwahnte Ikosaedergraph ein Beispiel. Siehe auch die Weiterfiihrung dieser Unterscheidungen 7.3.4!

7.2

Beispiele

Beispiel 1. Die ungeraden Doppelpyramiden (als Panisochromie-Besp.): Der Aquator werde durch 2n - 1 Ecken in 2n - 1 Kanten geteilt; jede Ecke wird mit dem Nord- und dem Sudpol durch je eine weitere Kante verbunden. Der Aquatorkreis besitzt 82n-1 = (22("-1)- 1)/3 Eckenfarbungen mittels 3 Farben. Durch Hinzufugen einer vierten Farbe fur jede der beiden Polecken hat man die samtlichen ebenso vielen 4-Farbungen der ungeraden Doppelpyramiden; die beiden Polecken bilden die zweielementige panisochrome Eckenmenge.

Beispiel 2. Nachdem die 7-seitige Doppelpyramide Tg,l schon unter Beispiel 1 fie1 und eine T9,2 erst unter Beispiel 4 erwahnt werde, mogen hier die beiden ubrigen der 4 moglichen Tg besprochen werden: T9,3 entsteht aus F7 in Bild 4, Zeile 3, und aus FG in Bild 4, Zeile 5, durch Identifizierung der beiden Randkreise. T9,3 hat 5 Ecken des Grades 4, d.i. n4 = 5; n5 = 126 = 2. Die beiden Ecken A , C des Grades 5 bilden ein panisochromes Paar.

Bild 5. T9,4

T9,4 entsteht (Bild 5) durch Einsetzen je einer Ecke des Grades 4 in jede der 3 rechteckigen Seitenflachen eines dreiseitigen Prismas. Es ist 72.4 = 3 n5 = 6;die Anzahl Farbungen ist nrs,, = 2. Die Ecken des Grades 4 bilden eine dreielementige panisochrome Eckenmenge.

Beispiel 3. Der Ikosaedergraph mit der Anzahl 10 seiner Farbungen hat kein panisochromes Eckenpaar. Uber das bereits in Abschnitt 7.1 uber

Ein zum Vierfarbensatz Aquivafenter Satz der Panisochromie

243

ihn Gesagte hinaus werde weiter vermerkt: In allen 10 Farbungen sind simultan die Ecken eines jeden seiner 6 Eckenpaare des Abstandes 3 verschiedengefarbt. Bei x = 4 Farben gibt es (?$ = 6 verschiedene Paare ungleicher Farben. In jeder Ikosaederfarbung kommt jedes dieser 6 Paare, ah, ac, a d , bc, bd, cd in genau einem der 6 genannten panheterochromen Eckenpaare des Abstandes 3 vor. 6 ist also die Maximalanzahl denkbarer panheterochromer Eckenpaare in irgendwelchen 4-gefarbten Graphen. Fur panheterochrome Eckentripel = 4. Meines Wissens ist uber hierzu ist die Maximalanzahl nur (t) = gehorige Verwirklichungen nichts bekannt. Der Ikosaedergraph, eine der 87 Triangulationen 7'12 (siehe Anzahlliste nach G1. (3)) ist nicht die kleinste (das ist v-minimale) Kugeltriangulation : trianmit independenter Panheterochromie. Diese ist vielmehr T ~ JMan guliere einen Aquatorkreis c 6 einmal (Sudhalbkugel) mittels einer Ecke v 6 und ein zweites Ma1 (Nordhalbkugel) mittels einer VsV5-Kante. Es gibt bei nT,,, = 6 Farbungen kein panisochromes Eckenpaar. Wohl aber ist jedes der 3 diametralen Eckenpaare des Cs panheterochrom und also independent panheterochrom.

(t)

Beispiel 4. Hier wird eine bestimmte Ti5 betrachtet (Bild 6) mit n ~ , ,= 74. Das (einzige) panisochrome Eckenpaar in Ti5 hat Abstand 3. A

A

Bild 6. Bild 7 und Bild 8 zeigen die 36 Farbungen der Ti5 zur Randfarbung 1 = bfb. In Bild 9 und Bild 10 sind die 19 Farbungen der Ti5 zur Randfarbung 4 = b:b gezeichnet. Diese stehen mittels der Symmetrie der Ti5 in ersichtlicher bijektiver Zuordnung zu den ebenfalls 19 Farbungen zur Randfarbung 3 = @, im ganzen gibt es also nT,, = 36+2.19 = 36+38 = 74 Farbungen der 7'15.

244

H. Heesch

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Ein zum Vierfarbensatz Aquivalenter Satz der Panisochromie

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Ein zum Vierfarbensatz Aquivalenter Satz der Panisochromie

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Bild 10. In allen bisherigen Beispielen zur Panisochromie gab es (wenigstens) ein panisochromes Eckenpaar des Abstandes 2. Ein solches gibt es nicht in der T15. Deren einziges panisochromes Eckenpaar ist F, J mit dem Abstand 3.

Beispiel 5. Unter den (laut Liste nach G1. (3)) 25 Triangulationen 2'1 findet sich eine, T11,9,(Bild ll), in der 2 Paare panisochromer Ecken so vorkomrnen, daf3 je eine Ecke des einen Paares mit je einer des anderen Paares adjazent ist. Es liegt also Panisochromie eines Kantenpaares vor. Es ist n ~ = 10. ~ Die ~ beiden , ~ panisochromen Eckenpaare sind F , I und G, J ; das panisochrome Kantenpaar ist ( F ,G), (I,J ) . In jeder Farbung der T11,gtragt das Eckenpaar F , G dasselbe Farbpaar - und zwar ausschliealich in derselben Reihenfolge - wie auch das Eckenpaar I,J. Die Symmetrie der ist eine Vierergruppe: Spiegelung an der Ebene des Funfecks ( A , B , C, D ,E ) und an der d a m orthogonalen Ebene durch A , H , IC und drei Kantenmittelpunkte, Drehung urn K urn die Schnittgeraden der beiden Spiegelebenen. Die erstgenannte Spiegelung permutiert die beiden panisochromen Kanten, die letztgenannte permutiert innerhalb jeder der beiden Kanten die Ecken.

H. Heesch

248 B

E Bild 11. T11,g

Beispiel 6 . Obwohl fur die Fragen in dieser Arbeit die in Unterabschnitt 1.1 fur Triangulationen Tv getroffene Einschrankung sachgerecht ist, gibt es Operationen wie z.B. das Kontrahieren von Kanten, die zum Erkennen von Eigenschaften panisochromer Graphen dienlich sind, bei deren Ausfiihrung indessen jene Einschrankung durchbrochen wird, indem sich als Kontraktat eine “gewohnliche” Triangulation ergibt mit Ecken vom Grad 3 sowie mit beidseits (noch mittels innerer Ecken) triangulierten Dreiecken. Ein in dieser Weise begriindetes Oszillieren zwischen den beiden T,-Begriffen werde (ohne weiter erwahnt zu werden) bei derartigen Betrachtungen akzeptiert, darunter auch bei diesem Beispiel 6, auf das in 7.1 hingewiesen wurde, da es die dortige Bemerkung illustrieren moge: Werden 2 Exemplare des Tetraedergraphs 1 3 vertices and m < edges. We shall proceed to derive a number of structural properties of the graph H, which will eventually imply that it cannot exist, thereby proving the theorem. It follows from our earlier remarks that H is a core.

(3

3

The structure of triangles

Our first goal is to prove that each edge of H belongs to a unique triangle. We do this in a sequence of steps: ( A l ) H contains a triangle.

(A2) H contains no K 4 . (A3) Each vertex of H belongs to a triangle. (A4) Any two vertices of H have a common neighbor.

(A5) There is no homomorphism S + H . (Figure 1.) (A6) H contains no K,' . (Figure 1.) (A7) Each edge of H belongs to a unique triangle.

259

The Existen ce Problem for Graph Homomorphisms

S

Figure 1. This now follows from (A4) and (A6). In particular, the neighborhood of any vertex of H induces a perfect matching. Our next objective is to investigate the interconnections among the triangles of H .

(A8) In H , any triangle abc and edge cc' (c' a subgraph .'2 (Figure 2.)

# a, b,

c) are contained in

3

U Figure 2.

(A9) For any homomorphism U are adjacent in H .

4

H (Figure 2), the images of i and j

(A10) In H , any two triangles, abc and ab'c', are contained in a subgraph P (= K3 x K 3 ) . (Figure 3.)

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U

b

C

P Figure 3.

4

The structure of squares

From now on, we base our considerations on a fixed vertex r , chosen to be a vertex of maximum degree in H. By (A7), the neighborhood of T consists of k (say) disjoint edges ala;, u2uk, .. , ,a&. Let R denote the subgraph of H induced by the remaining vertices V ( H )- { r ,a l , a ; , . ,ak,a i } ; according t o (A4), each vertex z of R is adjacent to some ai. By (A7), each edge uv of R belongs t o a triangle uvw; if w = ai, we label the edge uv by aj. (Of course, the whole triangle uvw could belong to R, in which case none of the edges uv, uw, and vw would be labelled.) If v in R is adjacent to some a; then the edge a;v lies in a triangle ajvw where w is also in R; hence, v is incident with an edge labelled ai. Note that (A6) implies that each edge obtains a t most one label, and that two edges of the same label cannot intersect or have two of their endpoints adjacent. We shall state this briefly as follows.

..

(Bl) Two edges of the same label cannot be incident or adjacent.

A similar proof to that of (Bl) shows that no vertex can belong to both an edge labelled ai and an edge labelled a:. For any i # j, we can apply (A10) to the two triangles TU;U: and raja: t o conclude that there is in R a four-cycle with edges consecutively labelled ai, a j , a:, and a;. (Figure 4.) Such a four-cycle will be called a square; there may, of course, be fourcycles in H (or even in R) which are not “squares.” There are a t least (:) squares, and they may intersect. Their structure is analyzed in this section, and it leads to a proof of Theorem 1. (B2) The squares are edge-disjoint.

The Existence Problem for Graph Homomorphisms

26 1

T

Figure 4.

(B3) H is2k-regular. It follows from the proof of (B3) that each labelled edge belongs to a square (thus to a unique square), and that H contains exactly one square labelled a;aja:ag for each i # j .

(B4) Each vertez of R belongs to a square. (B5) The subgraph RL of R formed by the labelled edges is bipartite. (B6) Each component of RL is complete bipartite. We conclude from ( B l ) and (B6) that each label aj (or a:) occurs at most once in a component of RL. It also follows from (B6) and (A7) that an unlabelled edge of R joins two vertices of diferent components of RL. Note that the unique triangle containing an unlabelled edge of R has both other edges unlabelled (and in R).

(B7) Suppose x y z is a triangle in R with all three edges unlubelled, x is incident with an edge labelled air yv is any edge of R such that v is incident with an edge hbelleda:, and z is incident with an edge labelled a j . Then v is also incident with an edge labelled a j . If uv is a n unlabelled edge of R, and if u is incident with an edge labelled ai while v is incident with an edge labelled a:, we mark the edge uv by the index i. Note that each edge obtains at most one mark.

( B 8 ) Every unlubelled edge of RL is marked by exactly one index i. (B9) Each vertex of R belongs to at least two squares.

P. Hell and J. NeSetNl

262

(B10) H does not exist. The proofs of (Al)-(AlO) and (Bl)-(BlO) involve a number of indicators, sub-indicators, and edge-sub-indicators, some of which are illustrated in Figure 5.

I kl

A, J

J

I

I Figure 5.

Remark. The situation is less clear for directed graphs. Even a conjecture anticipating which H-coloring problems are polynomial and which are NPcomplete does not suggest itself. Only a few results are known [2,17]. There are some simple digraphs H (paths, cycles, transitive tournaments, etc.) for which polynomial H-coloring algorithms exist [2, 171. Typically, they make

The Existence Problem for Graph Homomorphisms

263

use of results of the following type: there is a homomorphism D + H if and only if there is no homomorphism H’ + D (for some fixed digraph H’, depending on H) [2, 7, 13, 191. (These results may be viewed as proving that H-colorability is in NP n co-NP, and are, in some sense, prototype results of this type; this line of study is pursued in [13,19].) There are also a few classes of digraphs H with NP-complete H-colorability problems [17]. We note in passing that many NP-complete H-coloring problems may be produced by using the construction (of G*)from the proof of Lemma 1, with a suitable choice of the indicator I. Specifically, let (I,i, j ) be a digraph indicator such that for graphs G and H, there is a homomorphism G -+ H if and only if there is a homomorphism of digraphs G*+ H*. Such indicators are called ‘strongly rigid’ in the terminology of [8]; they can be constructed to satisfy many additional properties - assuring for example that H* is an acyclic, or even balanced, digraph. (A digraph is acyclic if it has no directed cycles; it is balanced if it has the same number of forward and backward arcs on any cycle.) In any event, if D = H* for such an I and a non-bipartite H then the D-coloring problem is also NP-complete. Thus, there are balanced (and hence also acyclic) digraphs H for which the H-coloring problem is NP-complete. Acyclic digraphs H with NPcomplete H-coloring problems were also constructed by S. Burr, and by W. Gutjahr and E. Welzl (personal communications). Finally, it should be mentioned that any digraph H such that the ‘symmetric part’ H s of H (all pairs uv of vertices for which both uv and vu are arcs of H) is a non-bipartite graph also results in an NP-complete H-coloring problem. This is an easy corollary of Theorem 1 and the observation that a graph G admits a homomorphism G --+ H s if and only if it (viewed as a digraph) admits a homomorphism G -+ H.

Acknowledgements We are grateful to David Kirkpatrick and Emo Welzl’for many inspiring conversations and helpful ideas.

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264

References [l] M. Albertson, P. Catlin and L. Gibbons, “Homomorphisms of 3-chromatic graphs 11,” Congressus Numerantium 47 (1985), 19-28. [2] G. Bloom and S. Burr, “On unavoidable digraphs in orientations of graphs,” J . Graph Theory 11 (1987), 453-462. [3] G. A. Dirac, “Homomorphism theorems for graphs,” Math. Ann. 153 (1964), 69-80. [4] W. D. Fellner, “On minimal graphs,” Theoretical Computer Science 17 (1982), 103-110. [5] M. R. Garey, D. S. Johnson and H. C. So, “An application of graph coloring to printed circuit testing,” IEEE Trans. Circuits and Systems 23 (1976), 591-598. [6] M. R. Garey and D. S. Johnson, Computers and Intractability, Freeman (1979). [7] R. Haggkvist, P. Hell, D. J. Miller and V. Neumann-Lara, “On multiplicative graphs, and the product conjecture,” Combinatorica 8 (1988), 71-81. [8] 2. Hedrlin and A. Pultr, “Symmetric relations (undirected graphs) with given semigroups,” Monatsh. f. Math 69 (1965), 318-322. [9] P. Hell, “Absolute planar retracts and the Four-color conjecture,” J. Combin. Theory (B) 17 (1974), 5-10.

[lo] P. Hell and J. NeSet81, “Cohomomorphisms of graphs and hypergraphs,” Math. Nachr. 87 (1979), 53-61. [ll] P. Hell and J. NeSetiil, “On the complexity of H-coloring,” Simon Fraser University, School of Computing Science Technical Report T R 86-4. To appear in J . Combin. Theory (E). [12] D. S. Johnson, “The NP-completeness column: An ongoing guide,” J. of Algorithms 3 (1982), 89-99.

[131 P. KomArek, “Some new good characterizations of directed graphs,” Cas. pro Pest. Mat. 51 (1984), 348-354.

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[14] L. A. Levin, “Universal sequential search problems,” Problems o f Information Transmission 9 (1973), 265-266. [15] F. T. Leighton, “A graph coloring algorithm for large scheduling problems,” J . Res. Nat. B. Standards 84 (1979), 489-496. [16] H. A. Maurer, A. Salomaa and D. Wood, “Colorings and interpretations: a connection between graphs and grammar forms,” Discrete Applied Math. 3 (1981), 119-135. [17] H. A. Maurer, J. H. Sudborough and E. Welzl, “On the complexity of the general coloring problem,’’ Information and Control 51 (1981), 123-145. [18] J. Neietfil, “Representations of graphs by means of products and their complexity,” MFCS (1982), 94-102. [19] J. Neietfil and A. Pultr, “On classes of relations and graphs determined by sub-objects and factor-objects,” Discrete Math. 2 2 (1978), 287-300. [20] G. Sabidussi, “Graph derivatives,” Math. 2. 76 (1961), 385-401. [21] K. Wagner, “Beweis einer Abschwachung der Hadwiger-Vermutung,” Math. Ann. 153 (1964), 139-141.

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Annals of Discrete Mathematics 41 (1989) 267-280 0 Elsevier Science Publishers B.V. (North-Holland)

On Edge-Colorings of Cubic Graphs and a Formula of Roger Penrose F. Jaeger * LS D IMAG Grenoble, France

Dedicated to the memory of G. A . Dirac We study a formula, due to Roger Penrose, which gives the number T(G) of edge-3-colorings of a cubic connected plane graph G in terms of certain families of cycles of G. We present an equivalent formula which gives T(G)in terms of the set of embeddings of G on orientable surfaces. Then, using the theory of bicycles and leftright paths of Rosenstiehl and Shank, we obtain another equivalent formula, which refers only to algebraic properties of G. Finally, we prove the new formula for all cubic connected graphs, thus generalizing Penrose’s formula to non-planar graphs.

1

Introduction

The physicist Roger Penrose presented in 1969 a paper, entitled “Applications of negative dimensional tensors” [l],where he obtained a number of remarkable formulas for the number of edge-3-colorings of cubic connected plane graphs. As the title of the paper suggests, Penrose’s methods of proof are quite original, and they appear to be very powerful. We shall not discuss these methods here. T h e purpose of the present work is t o study and generalize one of Penrose’s formulas. In Section 2 we present this formula and give a n interesting interpretation in terms of the set of embeddings of a cubic connected plane graph on arbitrary orientable surfaces. We also *Full address: Laboratoire de Structures Discrktes, Institut IMAG, BP 68, 38402 St. Martin d’Hbres Cedex, France.

267

F. Jaeger

268

exhibit a counter-example to a natural extension of the formula to nonplanar graphs. In Section 3 we show how the formula can be expressed in purely algebraic terms by using the theory of bicycles and left-right paths of Rosenstiehl[2] and Shank [3]. Finally, in Section 4 we prove the validity of the algebraic version of the formula for arbitrary cubic connected graphs. The definitions not given here will be found in [4] or [5], and we consider only finite undirected graphs, which may have loops or multiple edges.

Presentation of the formula

2 2.1

Statement of the formula

Let G = (V,E ) be a connected cubic plane graph; T ( G )denotes the number of edge-colorings of G with 3 colors (edge-3-coZorings, for short). As is well known, the statement that T ( G )# 0 whenever G is bridgeless is equivalent t o the Four Color Theorem, and this is a rather powerful motivation for the study of non-trivial expressions for T(G). Let Y be a subset of V . Following [l],we shall associate to Y a set the additional amount of of cycles of G, which we shall denote by C(Y); formalization introduced here will be helpful in the sequel. Consider the algorithm A described below, which defines a cycle ( e l ,v1,. . . ,ek, W k ) of G (ei E E , vi E V , i = 1, ..., Ic): (u)

Select an edge el of G and choose a travel direction on e l .

( 6 ) Traveling on e; with the specified direction leads to the vertex

0;.

( c ) Perform a right turn at if vi belongs to Y , and a left turn otherwise; this leads on the edge ei+l with a specified travel direction.

( d ) If ei+l is equal t o el and the travel direction assigned to ei+l in step ( c ) is the same as the travel direction assigned to el in step ( a ) , stop. The resulting cycle is ( e l , q ,. . , e , , v ; ) .

.

( e ) Otherwise go to step ( 6 ) with i replaced by i

+ 1.

We define C ( Y )as the set of cycles of G which can be obtained from algorithm A , with the convention that two cycles corresponding to the same directed cyclic sequence of edges and vertices are identical. It is then easy to check that for each edge of G with a specified direction, there exists exactly one cycle of C(Y)which takes this edge in this direction.

On Edge-Colorings of Cubic Graphs

269

Example 1. If Y = V then C(Y)is the set of face-boundaries of G taken in the clockwise direction. Example 2. Let G be the graph of Figure 1 and assume that Y = {v}. Then C ( Y )consists of 3 cycles: (f,w,g, w,f,w, e, v), ( 9 ,w)and ( e , w).

Figure 1.

Example 3. Let G be the graph of Figure 2 and assume that Y = {v}. Then C(Y)consists of the unique cycle ( e , w,g, v , f,20, e , w , g , w,f,w).

Figure 2.

We may now state the formula we want to study ([l],p. 240):

For instance, in Example 2, IC(Y)I = 3 for all Y V and this immediately gives T ( G )= 0; in Example 3, IC(Y)I = 1 if IYI = 1, IC(Y)I = 3 otherwise, and both sides of ( Fl ) equal 6.

2.2

Interpretation in terms of embeddings on orientable surfaces

A theorem of Edmonds [6] allows t o represent any 2-cell embedding (ernbedding, for short) of a connected graph G on an orientable surface in terms

F. Jaeger

2 70

of rotations. For this purpose, each edge of G is considered as a pair of oppositely directed edges. For a vertex w of G, a rotation ofv is a cyclic permutation nu on the set of directed edges incident to v and directed away from w. A rotation system o f G is a family ( ru,w E V ) , where rVis a rotation of v. To every rotation system of G corresponds an embedding of G on some orientable surface; the face-boundaries of this embedding can be determined using the algorithm A introduced in the previous section, with the only difference that in step (c) the turn a t w; is performed in such a way that ei+l (directed away from wi) is the image under A, of ei (also directed away from w;). Conversely, every embedding of G on some orientable surface corresponds in this way to a unique rotation system. For a detailed treatment of this classical “permutation technique,” the reader can refer to [7]. Now, if G is a cubic connected plane graph, we may associate to each subset Y of V a rotation system n(Y)= ( A ~ w, E V ) such that A, geometrically corresponds to a counter-clockwise rotation around w if w E Y , and to a clockwise rotation otherwise. Let I ( Y ) be the embedding of G corresponding to n ( Y ) . It is easy to check that the set of face-boundaries of I(Y)is precisely C(Y).Let g(Y)be the genus of the embedding I ( Y ) (that is, the genus of the associated orientable surface). Then, by Euler’s formula:

Then Penrose’s formula ( F l ) becomes

or, equivalently

(- 1)Iy141-s(y).

T ( G )= YGV

Example 4. Let G be the complete graph on 4 vertices embedded in the plane. This graph has precisely two embeddings of genus zero (corresponding to the opposite rotation systems A(V) and n(B)), and each contributes 4 t o the right-hand side of (F2). By Euler’s formula, the other embeddings must be of genus 1, and their total contribution to the right-hand side of (F2) is -2. Thus, both sides of (F2) equal 6.

Remark. Since the correspondence I between subsets of V and orientable embeddings is one-to-one, (F2) can be viewed as a summation over the set of

On Edge-Colorings of Cubic Graphs

271

orientable embeddings of G (where embeddings corresponding t o opposite rotation systems are considered distinct).

2.3

Failure of a geometric generalization

Let G = ( V , E )be an arbitrary cubic connected graph and e = ( e,, o E V ) V , let A(Y,e) = ( A,, v E V ) be the be a rotation system of G. For Y rotation system of G such that A, = (e,)-' if o E Y , x , = e, otherwise. Let I(Y,e) be the embedding of G corresponding to n(Y,e), C(Y,e) be the set of face-boundaries of this embedding, and let g(Y,e) be its genus. Let

Thus, Penrose's formulas (Fl) and (F2) assert that T ( G ) = Q ( G , e ) whenever e defines a planar embedding of G. Note that the set of rotation systems of G is { n(Y,e) I Y G V }, with in particular e = n(8,e). Furthermore, it is easy to see that for all Y G V , V , A ( Y , A ( Z , ~=) A(Y ) Z , e ) (where denotes the symmetric Z difference of sets). Hence, we also have C(Y,n(Z,e)) = C(Y 2 , ~and ) g(y,~ ( 2 el), = g(Y 2,el. Consider now the rotation system e' = ~ ( 2 e)., Then

-+

+

"+"

+

Q(G,e')= YCV

YCV

and thus, setting Y' = Y

+ 2: (-1)Iy'+Zl4'-g(Y',Q)

Q(G,e') =

= (-l)lZ1&(G,e).

Y'GV

Hence, for any two rotation systems e and e', IQ(G,e)I = IQ(G,e')I, and we shall denote this number by Q(G). Now we may adapt an argument used by Penrose in [l]for the study of another formula to show that Q(G)is not in general equal to T ( G ) .For this purpose, consider the two drawings in Figure 3 of the complete bipartite graph K 3 , 3 in the plane.

F. Jaeger

272

X

X

t

t

Figure 3. Note that the two drawings define the same graph G on the vertexset {x,y, z, t, u , v}. Let e (respectively @’) be the rotation system of G which geometrically corresponds to the clockwise rotation around each vertex in the plane drawing of Figure 3(a) (respectively 3 ( b ) ) . Then obviously Q(G,e) = Q(G,e’) because the two drawings are essentially identical (this idea could be made more precise with the notion of isomorphism of maps on orientable surfaces). On the other hand, e and e’ coincide on exactly 3 vertices x,z , u, that is, e’ = n({y, t , v}, e ) ,and hence Q(G,e ) = -Q(G, e’). It follows that Q(G,e) = Q ( G ,e‘) = 0 and hence Q ( G )= 0 # T( G) . In the next section we present an algebraic expression of Penrose’s formula (Fl).

3 3.1

An equivalent algebraic form Spaces and graphs

Let X be a finite set; P ( X ) denotes the set of subsets of X . For A , B in P ( X ) , we denote by A t B the symmetric difference of A and B . For A in P ( X ) and a in GF(2) we define a A by: a A = 0 if Q = 0, a A = A if a = 1. Then P ( X ) together with the two operations defined above is a vector space over GF(2). We shall call any subspace of P ( X ) a space on X. For instance, a family ( Ai, i E I ) of subsets of X generates a space on X which we denote by ( Ai, i E I ) . In particular, for A C X , ( { a } , a E A )

On Edge-Colorings of Cubic Graphs

273

is a space on X which we identify with P ( A ) . We define the scalar product A B E GF(2) of two subsets A , B of X as equal to 0 if [ An BI is even and equal to 1 otherwise. If F is a space on X , then the set { A E P ( X ) I VB E F,A . B = 0 ) is alsoa spaceon X which is denoted by F’. For instance (X)’ is the space of subsets of even cardinality of X . It is known that (F’)’ = F,and 3 and F1 are said to be orthogonal spucas on X . Let G = ( V , E ) be a graph. The boundary of an edge e with ends v, v’ is d ( e ) = {v} (0’) (so that d ( e ) = 8 if e is a loop). For F E the boundary of F is d ( F ) = C e E F a ( e ) . Thus, 0 is a linear mapping from P ( E ) to P ( V ) . The kernel of 8, denoted by C(G),is the cycle space of G. For S C V , the set of edges of G with exactly one end in S is called the cocycle o f S and is denoted by w(S). Since w ( S ) = ~,,,o({v}), w is a linear mapping from P ( V ) to P(E). The image of o,denoted by K(G), is the cocycle space of G. It is easy to show that C(G) and K(G) are orthogonal spaces on E . The space C(G)n K(G) is called the bicycle space of G and is denoted by B(G). Its elements are called bicycles of G (see [2], [3]).

+

3.2

s

Bicycles and left-right paths

Let G = ( V , E ) be a connected plane graph. Consider algorithm A’ below which defines a cycle ( e l ,v1, . . . ,ek, vk) of G (ej E E , vi E V , i = 1, . .. ,k):

( a ’ ) Select an edge el of G, choose a travel direction on el and a turning behavior for el which can be R (right) or L (left). (6’) Traveling on ej with the specified direction leads to the vertex w;.

v; (that is, enter the rightmost edge) if current behavior is R, and a left turn otherwise; this leads on edge ei+l with a specified travel direction; change the turning havior (that is, the behavior for ej+l is chosen as different from behavior for e i ) .

( c ’ ) Perform a right turn at

the the bethe

(d’) If ei+l is equal to e l , and the travel direction and turning behavior assigned to ei+l in step (c‘) are the same as the corresponding parameters assigned to el in step (a’), stop. The resulting cycle is

( e l 3 v1 (el)

* * * 3

ei, vi).

Otherwise go to step (b’) with i replaced by i

+ 1.

F. Jaeger

274

The cycles of G produced by algorithm A’ are called the left-right paths of G. They were first studied by Rosenstiehl[2] and Shank [3]. Let us adopt the convention that two left-right paths corresponding to the same directed cyclic sequence of edges and vertices, or to two opposite such sequences, are identical. Let then L R ( G ) = {PI,. , ,P p } be the set of left-right paths of G. It is easy to check that each edge of G appears twice in the elements of L R ( G ) ; to be more precise, it appears either twice in one of the Pi’s and not in the others, or once in two different Pi’s and not in the others. In fact, it can be shown that the Pi’s are the face-boundaries of an embedding of G on a (not necessarily orientable) surface. Let us call the set of edges which appear exactly once in P; the core of Pi. It is clear that the core of P; belongs to C(G). By considering the geometrical dual G* of G, it is easy to see that, similarly, the core of Pi belongs to K(G) (this is because LR(G*) can be identified with L R ( G ) while C(G*)can be identified with K(G)). Hence, the core of Pi is a bicycle of G. The following stronger result is proved in [2] and [3]: The cores of any T - 1 of the Pi’s form a basis of the bicycle space B(G). We shall retain the following corollary:

.

Proposition 1. Let G be a connected plane graph with exactly r left-right paths. Then dim B(G) = T - 1. We now use this result to derive a new expression for Penrose’s formula (Fl).

3.3

An algebraic formula equivalent to ( F l )

Let G = ( V , E ) be a connected plane cubic graph. For F 2 E , we denote by G : F the graph obtained from G by subdividing each edge of F , that is by replacing each edge of F by a path of length 2.

Proposition 2. For all subsets Y of V , IC(Y)I = ILR(G:E - w(Y))I. Proof. Let Y be a subset of V . It is clear that the graph G : E - w(Y) is bipartite. More precisely, we may color the vertices of this graph with two colors R and L in such a way that each edge has one end of each color, with the vertices of degree three corresponding in G to the vertices of Y (respectively V - Y) colored R (respectively L ) . Consider now a left-right p a t h P = ( e l , q , ..., e k , v k ) ofG:E-w(Y). Wenote that in theexecution of algorithm A‘ of Section 3.2 which produces P , either for all i = 1, . . . , k, the turning behavior a t vi is equal to its color, or for all i = 1, . .., k,

275

On Edge-Colorings of Cubic Graphs

the turning behavior a t wi is distinct from its color. Moreover, if the sequence ( e l ,~ 1 , .. . ,e k , w k ) is in one of these situations, the opposite directed cyclic sequence ( e l ,vk,. . .,e2, w1) is in the other situation. We recall our convention that two opposite sequences correspond to the same left-right path. Hence, we may identify the set L R ( G : E - w ( Y ) ) with the set of face-boundaries of the embedding associated to the rotation system which geometrically corresponds t o a counter-clockwise plane rotation a t vertices colored R and to a clockwise rotation at vertices colored L . Finally, it is immediate, using the straightforward correspondence between this embedding of G : E - w ( Y ) and the embedding I ( Y ) of G defined in Section 2.2, that LR(G : E - w ( Y ) ) is in one-to-one correspondence with the set C ( Y ) 0 of face-boundaries of I ( Y ) . This completes the proof. Now, using Propositions 1 and 2, the formula ( F l ) becomes (-1)

T ( G )= (3)i'"'

IY 121+dim &3( G:E-w( Y))

YGV

Since G is cubic, and the graph obtained from G by identifying the vertices of V - Y into a single vertex has an even number of vertices of odd degree, IYI = Iw(Y)I (mod 2) for all Y V . Thus, the formula (Fl) becomes 1 31VI-1

T ( G )= ( 5 )

C(-1)1

w(Y)l2dimB(G:E- w ( Y ) )

YCV

Finally, we observe that for K E K(G) there exists exactly two subsets of V whose cocycle equals K. Hence, the formula ( F l ) is equivalent to the following 1 ;Ivl-2

T ( G )= ( 2 )

c

(-1)

IKl2dim B(G:E-K)

(F3)

KEK(G)

In the next section we prove that this formula holds for all cubic connected graphs G.

4

The main result

Theorem. Let G z (V,E ) be a cubic connected graph, and let T ( G ) be the number of edge-3-colorings of G. Then

W )= ( 12 )W l - 2

c

(-1)

K€IC(G)

IK12dim &3(G:E-A')

F. Jaeger

276

For a fixed K in K ( G ) ,let us first evaluate dimB(G : E - K) in terms of spaces on E . Denote by E' the edge-set of G : E - K, and let q5 be the unique linear mapping from P ( E ) to P(E')such that: for e in K , 4({e)) consists of the edge of G : E - K corresponding to e; and for e in E - K , 4({e)) consists of the pair of edges in series in G : E - K corresponding to e. It is clear that the restriction of 4 to C(G) is an isomorphism from C(G) t o C ( G : E - K ) . Let BK(G)= + - l ( B ( G : E - I i ) ) . Thus,BK(G) is aspace on E isomorphic to B(G : E - K) and we may write

Now, let us present a convenient description of BK(G). We may write successively for a subset X of E

,XE f?jh'(G)iff $(X)E C(G : E - Ii)n K(G : E - I Ok. So 2 5 T I 2k - 1 remains to be considered. We shall treat the alternatives: X is contained or not contained in a triangle. Assume first that XI is not in any triangle in G. The graph G' obtained through contracting XI with one of its neighbours satisfies IG'l = n - 1 and e(G') = e(G) - 1. Hence by ( 2 ) and the induction hypothesis in n, G' _> O k (G' has more than f ( n - 1,k) edges), and hence also G 1 O k ,as it is easy to see that to every O kin G' there corresponds an O k in G. Assume next that X1 is contained in a triangle C(X1, X2, X3). Then 3

c v ( X i ) 5 2 k - 1+ 2 ( n - 1) = 2n + 2 k - 3. i=l

P. Justesen

304 Define G" = G - XI- X2 - X 3 . w e get

e(G") t 2n + 2k - 6 2 e(G) 2 f(n, k) and hence by (3), Furthermore,

e(G") 2 f ( n - 3, k - 1). e(G") 2 g(n - 3 , k - 1) t 1.

Proof of (6): Assume on the contrary that

g(n - 3, k - 1) t 1 > e(G"). Then, by (5) and (l), n