Graph Colouring and Variations

GRAPH COLOURING AND VARIATIONS ANNALS OF DISCRETE MATHEMATICS General Editor: Peter L. HAMMER Rutgers University, Ne...

Author: D. de Werra | A. Hertz

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GRAPH COLOURING AND VARIATIONS

ANNALS OF DISCRETE MATHEMATICS

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

Advisory Editors: C. BERGE, Universite de Paris, France M.A. HARRISON, University of California, Berkeley, CA, U.S.A. V KLEE, University of Washington, Seattle, WA, U.S.A. J. H. VAN LINT California Institute of Technology,Pasadena, CA, U.S.A. G. C. ROTA, Massachusetts Institute of Technology;Cambridge, MA, U.S.A.

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD *TOKYO

39

GRAPH COLOURING AND VARIATIONS

D. de WERRA A. HERTZ Departemen t de Ma thema tiques Ecole Polytechnique Federale de Lausanne Lausanne, Switzerland

1989

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD .TOKYO

Elsevier Science Publishers B.V., 1989

IC)

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise; without the prior permission of the copyright owner. No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein.

ISBN: 0 444 705 333

Publishers: ELSEVIER SCIENCE PUBLISHERS B.V. P.O. Box 103 1000 AC Amsterdam The Netherlands

Sole distributors for the U. S.A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas New York, N.Y. 10010 U.S.A.

Reprinted from the Journal Discrete Mathematics, Volume 74, Nos. 1-2, 1989

Library of Congress Cataloging-in-Publication Data Graph colouring and variations1 [edited by] D. de Werra. A. Hertz. cm. -- (Annals of discrete mathematics ; 39) p. Bibliography: p. ISBN 0-444-70533-3 1. Map-coloring problem. I. Werra, D.de. II. Hertz, A. Ill. Series. QA612.18.G73 1989 514'.3--dc 19

PRINTED IN NORTHERN IRELAND

88-28842 CIP

SPECIAL DOUBLE ISSUE GRAPH COLOURING AND VARIATIONS Guest Editors: D. de WERRA and A. HERTZ CONTENTS A. HERTZ and D. de WERRA, Foreword C. BERGE, Minimax relations for the partial q-colorings of a graph K. CAMERON, A min-max relation for the partial q-colourings of a graph. Part II: Box perfection P. DUCHET, On locally-perfect colorings M.O. ALBERTSON, R.E. JAMISON, S.T. HEDETNIEMI and S.C. LOCKE, The subchromatic number of a graph A. HERTZ and D. de WERRA, Connected sequential colorings A.J.W. HILTON, Two conjectures on edge-colouring B. BOLLOBAS and H.R. HIND, A new upper bound for the list chromatic number C.T. HOANG and N.V.R. MAHADEV, A note on perfect orders F. JAEGER, On the Penrose number of cubic diagrams R.E. JAMISON, On the edge achromatic numbers of complete graphs H.A. KIERSTEAD, Applications of edge coloring of multigraphs to vertex coloring of graphs M. KUBALE, Interval vertex-coloring of a graph with forbidden colors J. MAYER, Hadwiger's conjecture ( k = 6): Neighbour configurations of 6-vertices in contraction-critical graphs H. MEYNIEL, About colorings, stability and paths in directed graphs J. MITCHEM, On the harmonious chromatic number of a graph S. OLARIU, Weak bipolarizable graphs C.T. HOANG and B.A. REED, &comparability graphs H. SACHS and M. STIEBITZ, On constructive methods in the theory of colour-critical graphs E. SAMPATHKUMAR and C.V. VENKATACHALAM, Chromatic partitions of a graph M.M. SYStO, Sequential coloring versus Welsh-Powell bound S.K. TlPNlS and L.E. TROUER, Jr., A generalization of Robacker's theorem A.D. PETFORD and D.J.A. WELSH, A randomised 3-colouring algorithm

1 3

15 29

33 51 61 65 77 85 99 117 125 137 149 151 159 173 201 227 241 245 253

Discrete Mathematics 74 (1989) 1-2 North-Holland

1

FOREWORD Graph coloring has been a field of attraction for many years; a wide collection of papers has been dedicated to the study of chromatic properties of graphs. Initially such problems were just a kind of game for pure mathematicians; it was in particular the case of the famous four color problem. However, as people were getting used to applying the tools of graph theory for solving real-life organizational problems, chromatic models appeared as a quite natural way of tackling many situations. Among these are timetabling problems, or more generally scheduling with disjunctive constraints (pairwise incompatibility between jobs), clustering in statistics, automatic classification, group technology in production (partitioning a collection of parts into families of parts which are as similar as possible in their production process), VLSI design, etc. The theory of perfect graphs and particularly the perfect graph conjecture of Claude Berge provided a strong impetus for the development of the theory of coloring. Several papers in this volume are dealing with special classes of perfect graphs which are characterized by chromatic properties. A natural extension of coloring problems - motivated by a polyhedral formulation of optimization in perfect graphs - consists in expressing an integral vector in a polyhedron as a sum of integral vectors contained in a smaller polyhedron. This extension is considered in some contributions of the present volume. Besides this, colorcritical graphs have been a focusing point in many research works; such graphs, having some inherent structure, can hopefully be characterized by more and more properties. Many other variations and extensions of the basic node (or edge) coloring problem have been proposed: for instance a node or an edge may receive a set of “consecutive” colors, or a color chosen in a given set of admissible colors. A color class may be extended from a stable set of nodes to a union of disconnected cliques. Such types of variations are presented in the volume. Related optimization problems are also discussed; among those is the maximum q-coloring problem: find the largest number of nodes which can be colored with q colors in a given graph G (with chromatic number larger than q). Edge coloring problems form a special case of node coloring; deciding if there exists an edge coloring using A(G) colors in a simple graph G with maximum degree A(G) is an NP-complete problem. However, many results giving bounds of the edge chromatic number based on the degrees of the nodes and on some properties of the graphs can be obtained. This volume contains a few papers in this direction. Many algorithmic approaches have been developed and have provided large 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

2

D.de Werra, A. Hertz

classes of graphs for which coloring problems can be solved in polynomial time. Such results will certainly be extremely useful for applications. The papers presented in this text do not provide an exhaustive survey of the various fields of chromatic optimization; in particular they do not describe the numerous fields of applications of colorings. We hope however that they will bring to the reader a view of some facets of this active research area. We will have reached our aim if the contributions collected here will stimulate new investigations in the chromatic properties of graphs and hypergraphs. In such a wide field, it is difficult to unify notations. We have nevertheless encouraged the authors to use the definitions of Claude Berge, Graphs (North-Holland, 1985). Most of them followed these lines and the terms used in different ways are usually defined in the papers or given in appropriate references. Finally we would like to thank the authors who have been the active contributors to this volume. Our gratitude extends also to the many anonymous referees who have devoted much of their time to improve the quality and readability of the papers. We wish to acknowledge the enthusiastic support and the encouragements of Peter L. Hammer, Editor-in-chief of Discrete Mathematics. The help of North-Holland in preparing this volume is also gratefully acknowledged. Many thanks are due to Mrs. A.-L. Choulat for having handled the manuscripts with great care. Lausanne, June 15, 1988.

A. Hertz D. de Werra

3


MINIMAX RELATIONS FOR THE PARTIAL 4COLORINGS OF A GRAPH C . BERGE C.M.S., E.R. 175 Combinatoire, 54 Bd Raspail, F75270, France A parfial q-coloring of a graph is a family of q disjoint stable sets, each one representing a “color”; the largest number of colored vertices in a partial q-coloring is a number aq(C), extension of the stability number a ( G )= a,(G). In this note, we investigate the possibilities, for 1 s 9 =sy ( G ) , to express a q ( C )by a minimax equality.

1. Optimal partial q-colorings Let G be a simple graph with a finite vertex-set X and with chromatic number y(G), and let q be an integer, 1 S q S y ( G ) . A partial q-coloring of G is a family 9,= (Sl, S,, . . . , S,) of q disjoint stable sets. If x E Sj, we shall say that the

vertex x is of color ( i ) . All the vertices need not have a color. The q-coloring 9, is optimal if the number of “colored vertices” lUirqSiI is as large as possible. The number of colored vertices in an optimal q-coloring will be denoted by a,(G), so that cul(G) = a ( G ) , the stability number of G. Consider a family V = (C, l j E J ) of cliques (complete subgraphs). The qcoloring Yqand the clique family ‘G: are associate if we have simultaneously: (A,) S j f l S j =0,

(A3) Sifl Cj # 0

CJl C j = O for i # j ;

for alli and all j .

Let G = ( X , E ) b e a g r a p h o n X = { x , , x , , . . . , x n } . Put Q = { 1 , 2 , . . . , q}, and let K , be the complete graph on Q. The Cartesian sum G K , is a graph on the Cartesian product X X Q, where ( x , i ) and ( y , j ) are joined if x = y and i # j , or if [ x , y ] E E and i = j . Every stable set So of G + K , defines a partial q-coloring as follows: color the vertex x of C with color i if and only if ( x , i ) E So. Conversely every partial q-coloring of G defines a stable set of G + K , . Thus, we have the following basic result:

+

Lemma 1. For every graph G and every integer q, there exists a one-to-one correspondence between the maximum stable sets of G K , and the optimal partial q-colorings of G , and

+

a,(G) = a(G

+ K,).

0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V.(North-Holland)

C. Berge

4

Lemma 2. A partial q-coloring Y, = (Sl, &, . . . , S,) is optimal if and only if for every partial q-coloring 9;= (Si,S;, . . . , S;), there exists an injective mapping u :US] +USi such that (1)

ux

E

{x} u r x

xES;

(2)

ux E S j

i#j

1

*ux = x

Clearly, a partial q-coloring Y, satisfying this condition is optimal, because if we consider for 9’; a partial q-coloring which is optimal, we obtain (since u is injective):

The converse follows immediately from Lemma 1 and the “maximum stable set Lemma” ([l], Chap. 13, 01): A stable set B is maximum if and only if every stable set S disjoint from B can be matched into B.

Lemma 3. Let G be a graph, and let q be an integer, 1s q s y ( G ) . Then every partial q-coloring which has an associate clique-family is optimal. S2, . . . , S,) be a partial q-coloring of G , and let So = Uirq(Six Let 9,= (Sl, {i}) be the stable set of C + K, defined as in Lemma 1. The existence of a clique-family % satisfying (Al), (A2),( A 3 )allows to construct a clique-partition of G + K , as follows: Take the cliques { x } x Q for all x E X - U j C j , and the cliques Cj x {i} for all i and j. Clearly, these cliques are pairwise disjoint and cover X; furthermore, by (A3),each clique of the partition contains one element of So, and only one. This shows that So is a maximum stable set. From Lemma 1, it follows that .4., is optimal. We denote by 8 ( G ) the least number of cliques which partition the vertex-set of G.

Theorem 1. Let G be a graph with chromatic number y ( G ) , and let q be a positive integer, q < y(G). Then the following conditions are equivalent: (1) a ( +~K,) = e(c + K ~ ) ; (2) every optimal q-coloring has an associate clique-family ; (3) some optimal q-coloring has an associate clique-family ; (4) if M denotes a clique-partition of G , then

Proof. (1) implies (2). Assume that a,(C + K , ) = 8 ( G + K , ) . Let Y q = (Sl, S,, . . . , S,) be an optimal q-coloring of G. By Lemma 1, the set So =

Minimax relations for partial q-colorings

5

+

(Si x { i } ) is a maximum stable set of G K q . Let %,, be a minimum clique partition of G K q , so that I %,I, = ISo/.Some cliques are of the type { x } x 0,with x E B (for some B c X) and the others are of the type Cj x { i }. We may replace each 0,by Q, and assume that the Ci satisfy:

+

(I)

cjn Cj. = 0

(2)

U Cj = X - B

for j # j';

i

for all i ;

Ci f l Sj # 0 for all i and all j . (3) From (2) and (3) we see that for the subgraph Gx-Binduced by X - B , each Sj - B is a maximum stable set, and each %'= (Cj 1 j) is a minimum cliquepartition; therefore any minimum clique-partition % = (Ci \ j) of G X p Bsatisfies (A,), (A2)and (A3).This shows that Yqhas an associate clique-partition %.

( 2 ) implies (3). Obvious. ( 3 ) implies (4). I f we consider a q-coloring 9,and a clique-partition M, we have: (i)

I u s/= c SEYq

c

I C ~ U S I ~min{lCI,q}.

CEM

CEM

Let 9,be some optimal q-coloring. Let % be the clique family associate with 9,. Let M be the clique-partition obtained from V by adding the singletons { x } for Then equality holds in every vertex x which is not covered by a member of Yq. (i). The property (4) follows immediately. (4) implies (1). Let pqand M be a q-coloring and a clique-partition of G for which (i) holds with equality. Since the inequality (i) is true for all 9,and all M , the q-coloring pqis optimal. Let % be a clique-partition of C K , obtained from

+

M

- b y replacing each CEM with [ C I > q by C x { l } , c x { 2 } , . . . , c x { q } , and - by replacing each vertex x which belongs to a E M with lcl s q by { x } x Q. Clearly, % is a clique-partition, and

c

Hence, a(G

+ K , ) = a ( C )= 1 % 1 =

e(G + K,).

0

Remark. In Theorem 1 we assume that q < y(G), because for q = y(G), the condition (1) is always true. Indeed, by Lemma 1 every graph G with chromatic number y(G) = y satisfies

a(G + K y ) = e(C + K y ) = n.

C. Berge

6

The complement of a graph C is a graph G with the same vertex-set, two vertices being joint in G if and only if they are not joined in G. We have:

+

Corollary. If a graph G satisfies a(G K 4 ) = B(G + K 4 )f o r some q < y(G), then its complement G satisfies a(G K r ) = B ( G K r ) f o r some r < y ( G ) .

+

+

+

+

Proof. a(G K 4 ) = B(G K 4 ) is equivalent to the existence of a partial q-coloring having an associate clique-partition. Consequently the result follows from the symmetric role played by the cliques and the stable sets in the conditions (AlL (A*),(Ad. 0 Examples. We study first the Haj6s graph H (a triangle inscribed in a hexagon) with two different optimal 2-colorings (Fig. 1 and Fig. 2). This graph is perfect, but we cannot prove the optimality of a 2-coloring by associate clique-families. No such associate family does exist: In both Fig. 1 and Fig. 2, the cliques C , and C2cover all the uncolored vertices, but are not disjoint; therefore (Cl, C,) is not an associate family. On the other hand a 2-coloring of the complement H of the Haj6s graph is represented on Fig. 3, and its optimality follows from the existence of the associate family (Cl). The main problem we shall study here is: for which classes of graphs does an optimal q-coloring have an associate clique-family? H H 0

c2

--

.

r

c - -

-.

\ c1

/

I

\

I

\

I

I

I

\

I

\

/

\

Fig. 2. cu,(H) = 4.

Fig. 1 . cu2(H) = 4.

-

-

_

-

/

H

Fig. 3. cu2(fi) = 5.


7

2. Existence theorems due to the perfectness of G + K q Let G, denote the subgraph of G induced by a set A of vertices. Recall that a graph G is perfect if u(GA)= 6(GA) for every A c V ( G ) .A chain p is odd (resp. even) if the number of edges in p is odd (resp. even). A chain p is chordless if its vertices induce an elementary chain. A graph G is a parity graph if for x , y E V ( G ) , all the chordless chains joining the vertices x and y have the same parity. For an elementary cycle, we say that two chords [ x , y ] and [ z , t ] cross if the vertices x , z , y , t are encountered in this order on the cycle. It is easy to show: A graph is a parity graph if and only if every odd elementary cycle of length 2 5 has two crossing chords (Burlet and Uhry [ 5 ] ) . These parity graphs have been considered first by Sachs [15], who has shown that every parity graph is perfect. This follows also from a more general result of Meyniel [13]. Burlet and Uhry [5] gave a polynomial time recognition algorithm for a parity graph. Many graphs considered in the literature belong to this class.

Example 1. A bipartite graph, defined by two vertex-sets X and Y, is a parity graph, because every chain joining x E X and x ’ E X is even, and every chain joining x E X and y E Y is odd. Example 2. Let G be a graph such that each block (“maximal 2-connected subgraphs”) is a clique. For x , y E V ( G ) , there is only one chordless chain that joins the two vertices x and y , therefore C is a parity graph. If T is a tree, its line-graph L ( T ) has the following property: each vertex of L ( T ) which does not correspond to a pendant edge of T is an articulation vertex, and each block is a clique. So L ( T ) belongs to this class. Note that one can easily show: Each block of G is a clique if and only if G satisfies the two following properties: (i) each cycle of length 2 4 has a chord (ii) G does not contain as an induced subgraph the graph C ; (cycle of 4 vertices plus one chord). Let us mention a few graphs which, are obviously not parity graphs. For k 2 2 , denote by C2k+lthe odd elementary cycle of length 2k 1; denote by C ; k + l any graph obtained from C 2 k + l by adding one chord (Fig. 4). Denote by CIS‘the graph obtained from C5 by adding two non crossing chords (Fig. 5). We see on the figure that the two vertices a and b are joined by an odd chordless chain and by an even chordless chain. Hence a parity graph does not contain a C 2 k + l , nor a c ; k + 1 , nor a C;. In fact, Burlet and Uhry have shown in [ 5 ] : A graph is a parity graph if and only if it does not contain as an induced subgraph any one of the following configurations : C 2 k + l : cycle of length 2k + 13 5 c ; k + ,: cycle of length 2k + 12 5 plus one edge cycle of length 5 plus two non-crossing chords. Cl;:

+

C. Berge

8 a

a

ci C5 b

d

b

C

Fig. 4.

Fig. 5.

In order to show that a q-coloring of a graph G has an associate clique-family, we know from Theorem 1 that it suffices to show that a(G + K q ) = 8(G + K q ) , which can follow from the perfectness of the graph G + K,. Various necessary and sufficient conditions for the perfectness of the Cartesian sum G + H have been found independently by Ravindra and Parthasarathy [14] and by de Werra and Hertz [7]; for H = K q , the result is simpler:

Theorem 2. Let G be a graph, and let 2 G q S y(G). The Cartesian sum G perfect if and only if q = 2, and G is a parity graph ; or q 3, and G is a graph whose blocks are cliques.

+ K , is

Proof. The condition is necessary. Let q = 2 . Let G be a graph which is not a parity graph. Then G contains a C;, or a C2k+lwith k 3 2, or a C&+l (as shown by Burlet and Uhry [ 5 ] ) . In the first case, C;+K 2 contains an odd chordless cycle: a2, b2, b l , c l , d l , d2, e2, a2 represented by the arrows on Fig. 6. Similarly an odd chordless cycle exists in C2k+l+ K 2 and in c;k+,+ K 2 (remove in Fig. 6 the edges [a2, c2] and [ a l , cl], and then the edges [c2, e2] and [ c l , e l ] ) . In all cases, the graph G + K 2 contains an odd chordless cycle, and consequently cannot be perfect. I

a2

e2

I

I

Fig. 6 .

a

I

C’j+K:,


9

C; + Kg

Fig. 7.

Let q 2 3, and let G be a graph where some blocks are not cliques. Then G contains a C; or a c k with k 2 4 (see note after Example 2). If G contains a c k with k odd 3 5 , we see as above that the Cartesian sum Ck K2 contains a C k + 2 . If G contains c k with k even 24, then C, K3 contains a C k + , (see the arrows in Fig. 7). If C contains a C;, we see that Ci + K3 contains a C7 (see Fig. 7). Thus, in every case, the graph C + K, contains an odd chordless cycle, and, consequently is not perfect. The condition is sufficient. Let q 3 3, and let G be a graph whose blocks are cliques. Then G does not contain a C;. The graph G + K, does not contain a C ; , because otherwise, at least two vertices of this induced C; would have the same vertical projection (but not the four of them), and every possibility leads to a contradiction. By the same argument, G + K , does not contain a C 2 k + l with k s 2. Then, the perfectness of G K, follows from a theorem of Ravindra and Parthasarathy which asserts that a graph which contains no C; and no C 2 k + l with k 3 2 is perfect (for a complete proof, see Tucker [16], Theorem 2). For q = 2, the proof is similar.

+

+

+

The line-graph of a tree satisfies all the requirements of Theorem 2, thus any optimal q-coloring of such a graph has an associate clique-family. However, this argument does not help to construct an associate clique-family, and we can prove more directly:

Theorem 3. Let G be the line-graph of a tree, and let q S y(G). Then, for every optimal q-coloring, an associate clique-family can be efficiently constructed. Furthermore, the graph G K , is perfect.

+

Proof. Let p, denote an optimal q-coloring of a line graph G = L ( T ) of a tree T . We have to show that p, has an associate clique family. If q = y(G), pq is a q-coloring of C and any maximum clique forms an associate clique family. So assume q < y(G). For a vertex a of the tree T, denote by w T ( a )the set of all edges incident to a : a set of edges 4 contained in o T ( a )is

C. Berge

10

called a star of center a . We shall construct a partial q-coloring (El, E2, . . . , E q ) for the edges of T , and a family of stars (4 I j EJ), such that: ( A ; ) 4 n Fk = 0 for j # k ; (A;) every edge e $ UEi s is contained in some 4; (A;) 16 f l Eil= 1 for all i, j . Clearly, ( A ; ) , (A;) and (A;) are equivalent to the axioms (A,), ( A2 ) ,(A3 )which characterize an associate clique-family in G = L ( T ) . Put

B = {x Ix

E

V ( T ) ,d T ( x )> 4).

The subgraph TB of T induced by B is a “forest” (union of disjoint trees). Let b , be a pendant vertex of some tree in TB. Label with a “0” the edge of T B incident to b , and d T ( b l )- q - 1 other edges of T which are incident to b , . Let T, be the forest defined by the unlabelled edges of T, we have d T , ( b l= ) q. Put Bi = I.{ x E V ( T ) ,d ~ , ( x ) > q ) . Then consider a pendant vertex b2 of a tree in the forest T B , , and define as before B2; and so on. The procedure terminates with a forest Tk such that d T k ( x s ) q for every vertex x ; since Tk is a bipartite graph, it follows from Konig’s Theorem that there exists a q-coloring of its edges. This defines a partial q-coloring, and every star F; = w T ( b i )contains all the q colors. Clearly, every uncolored edge of T is contained in some F;, and the E’s are disjoint. This shows that the E’s satisfy (A;), (A;), (AS) and achieves the proof. 0

3. Other classes of graphs In this section, we shall show that some known properties of balanced hypergraphs give easily some new classes of graphs for which the optimal q-colorings have an associate clique-family. An hypergaph H = ( E lE,2 , . . . , E m ) on X = {xl, x 2 , . . . , x , } is a collection of non-empty finite sets (called “edges”) whose union is X (the “vertex-set”); a cycle of fengfh k is a sequence ( x , , E l , x 2 , E2,. . . , Ek,xI)where the xi’s are different vertices, the Ej’sdifferent edges, and x i , x i + , E E, for all i . The hypergraph H is balanced if every cycle of odd length has an edge which contains at least three vertices of the cycle. Clearly, a bipartite graph is also a balanced hypergraph. The balanced hypergraphs have been introduced to generalize the main properties of the unimodufar hypergraphs (hypergraph whose incidence matrix edge-versus-vertex is totally unimodular). Let H be a balanced hypergraph with n vertices and m edges. Let A be its m X n incidence matrix, whose columns represent the edges, and whose rows represent the vertices. Let A * be its transpose. We denote by d = ( d l , dZ,. . . , d,) and y = ( y l , y,, . . . , Y , ~ )two m-dimensional vectors whose coordinates are non-negative integers; we denote

11


by c = (cl, c,, . . . , c,) and t = (tl, t,, . . . , t n ) two n-dimensional vectors whose coordinates are non-negative integers.

Property 1 (Corollary of Berge-Las [12]). For every d E N”, max{x diyi/y E N“, Ay

G

Vergnas [4]; generalized by Lovasz

1) = ,in[

2 tilt E N“, A*t 2 d

1.

Property 2 (Theorem of Fulkerson-Hoffman-Oppenheim [S]). For every d E N“, max

Ic

tilt

E N”, A*t

sd

Property 3. A hypergraph H = ( E l , E,, . . . , En)with “rank” r ( H ) = max lEil is balanced if and only if every partial subhypergraph H’ (obtained from H by removing vertices and edges) is colorable with r ( H ’ ) colors so that no two vertices belonging to the same edge of H ’ have the same color. For the proofs of these results, see [3], Chapter 5.

Theorem 4. Let G be a graph with vertex-set X and let q < y ( G ) . Let H ( G ) be the hypergraph on X defined by the maximal cliques of G. If H ( G ) is balanced, then every optimal partial q-coloring of G has an associate clique-family. Proof. Let H ( G ) = ( E l ,E,, . . . , E m )be the hypergraph of the maximal cliques of G , with incidence matrix A , and assume it is balanced. Let fi = ( E l , . . . , Em, F , , . . . , F,) be the hypergraph obtained from H ( G ) by adding an edge 8 = { x i } for i = 1, 2, . . . , n , and let A be the incidence matrix of fi. Clearly, l? is balanced. Let q be a positive integer, and let ij = (q, q, . . . , q, 1, 1, . . . , 1) be an (rn + n)-dimensional vector with m coordinates equal to q and n coordinates equal to 1. Let t = (t,, t,, . . . , tn) be an integral vector satisfying (A)*Zs(5. and maximizing C t i . Clearly, every coordinate ti is equal either to 0 or to 1. Put S = {xj/l S j S n, ti = 1). Thus, S defines a maximum set of vertices of H with the property: the subhypergraph Hs induced by S has a rank r(Hs) = maxEEH,[El at most equal to q. Since Hs is balanced, S is then a maximum set of vertices of G which can be colored with q colors (by Property 3). Hence:

Let jj = (y,, y,,

. . . , y,,

z , , I,,

. . . , I,) be an (m + n)-dimensional integral vector

C. Berge

12

satisfying Ar

3 (1, 1,

. . . , 1 ) and minimizing the scalar product

m

n

(~l,~)=qC~i+Czj* ;=I

j=]

Since y defines a covering of H minimizing a linear function with positive coefficients, all its coordinates are G l . Put Zo = { i / lG i s m, y j = l }

Jn = { j / l ~j s n, zj = I } . Thus, we have:

Then, we can write (using Property 2), IS( G

C IS n E;I+

ielo

IS n 41s q jeJo

IZ,~ + lJnl= (4, r) =

ti = 1st.

Hence each of these inequalities holds with equality, and, consequently, IS n E;l = q

(i E 10);

(2)

ISnFil=l

(jEJn).

(3)

(S

n EJ n (S n E j . )= 0 (i, i' E Z,, i #i n ).

(4)

Let pq be a q-coloring of G,s.For i E 10 the clique E j contains q different colors, by (2). Every uncolored vertex is contained in at least one of E,'s by ( 1 ) and (3). We may assume that the Ej's are pairwise disjoint: otherwise, we may replace each Ej by I?; c Ei without violating ( l ) ,(2), (3), (4). Thus, (El, E2, . . . , Em)constitute an associate clique family. This achieves the proof. 0

Corollary 1. Let G be a graph whose maximal cliques constitute a unimodular hypergraph. Then every optimal partial q-coloring of G is characterized by the existence of an associate clique family. This follows from Theorem 4, since every unimodular hypergraph is balanced. Note that this statement was obtained first by other methods by Cameron [ 6 ] .

Corollary 2. Let G be the line-graph of a bipartite multigraph. Then every optimal partial q-coloring of G has an associate clique family. The result follows from the Corollary 1, since the hypergraph H ( G ) of the maximal cliques has no odd cycles, and therefore is unimodular.


13

Corollary 3. Let G be an interval graph. Then every optimal partial q-coloring of G has an associate clique family. (Same argument).

Corollary 4. Let G be a comparability graph with no induced K5 minus two dkjoint edges. Then every optimal partial q-coloring of G has an associate clique family. It is easy to show that H ( G ) is balanced. Thus the result follows from Theorem 4. Corollary 4 is in fact a special case of a theorem of Greene and Kleitman, reformulated as follows:

Theorem of Greene and Kleitman. Every optimal partial q-coloring of a comparability graph has an associate clique family. In fact, the Greene-Kleitman theorem [lo] states that the directed graph G of a partial order satisfies:

where M denotes a path-partition. Since every path induces a clique (by the transitivity) and every clique is spanned by a path (by Redei’s Theorem), this is equivalent to the condition (4) in Theorem 1, and we have shown that (4) is equivalent to (1). The Greene theorem [9] can be reformulated as follows:

Theorem of Greene. In the complement of a comparability graph, every optimal partial q-coloring has an associate clique family. Note that all the graphs mentioned in these results are perfect. If G + K , is perfect, it is well known that the determination of a maximum stable set of G Kq is a polynomial problem. But if G is perfect, is the determination of an optimal q-coloring also a polynomial problem?

+

References [l] C. Berge, Graphs (North-Holland, New York, 1985). [2] C. Berge, Minimax relations for normal hypergraphs and balanced hypergraphs, Ann. Discr. Math. 21 (1984) 3-19. [3] C. Berge, Hypergraphs (North-Holland, 1988). French version: Hypergraphes, combinatoires des ensembles finis (Gauthier-Villars, Paris, 1987). [4] C . Berge and M.Las Vergnas, Sur un thCor&rnedu type Konig pour Hypergraphes, Ann. N.Y. Acad. Sc. 175 (1970) 32-40.

14

C. Berge

151 M. Burlet and J.P. Uhry, Parity graphs, Ann. Discrete Math. 21 (1984) 253-277. [61 K.B. Cameron, A minimax relation for q-colorings, Discrete Mathematics 73 (1988). [71 D. de Werra and A. Hertz, On perfectness of sums of graphs, Report ORWP 87/13, Swiss Federal Institute of Technology in Lausanne (1987). [8] D.R. Fulkerson, A.J. Hoffman and R. Oppenheim, On balanced matrices, Math. Programming Study l(1974) 120-132. (91 C. Greene, Some partitions associated with a partially ordered set, J. Combinat. Theory A 20 (1977) 669-680. [lo] C. Greene and D. Kleitman, The structure of Sperner k-families, J. Combinat. Theory A 20 (1976) 80-88. [Ill A.J. Hoffman and D.E. Schwartz, On partitions of a partially ordered set, J. Combinat. Theory B 23 (1977) 3-13. [12] L. Lovasz, Normal Hypergraphs, Discrete Math. 2 (1972) 253-267. [13] H. Meyniel, The graphs whose all cycles have at least two chords, Ann. Discrete Math. 21 (1984) 115-120. (141 G . Ravindra and K.R. Parthasarathy, Perfect product of graphs, Discrete Math. 20 (1977) 177- 186. [ 151 H. Sachs, On the Berge conjecture concerning perfect graphs, in: Combinatorial Structures (Gordon and Breach, New York,1970) 377-384. [I61 A. Tucker, Coloring K,- e free graphs, J. Combinat. Theory B 42 (1987) 313-318.


15

A MIN-MAX RELATION FOR THE PARTIAL qCOLOURINGS OF A GRAPH. PART 11: BOX PERFECTION Kathie CAMERON* Department of Management Sciences, University of Waterloo, Waterloo, Ontario, Canada N2L 3G 1

This paper examines extensions of a min-rnax equality (stated in C. Berge, Part I) for the maximum number of nodes in a perfect graph which can be q-coloured. A system L of linear inequalities in the variables x is called TDI if for every linear function _a_ such that _c is all integers, the dual of the linear program: maximize {cx: ;x satisfies L } has an integer-valued optimum solution or no optimum solution. A system L is called box TDI if L together with any inequalities 1 c x 6 LC is TDI. It is a corollary of work of Fulkerson and LoQasz that: where A is a 0-1 matrix with no all-0 column and with the l-columns of any row not a proper subset of the l-columns of any other row, the system L ( G ) = (A& s 1, g 3 0) is TDI if and only if A is the matrix of maximal cliques (rows) versus nodes (columns) of a perfect graph. Here we will describe a class of graphs in a graph-theoretic way, and characterize them as the graphs G for which the system L ( G ) is box TDI. Thus we call these graphs box perfect. We also describe some classes of box perfect graphs.

1. Introduction Consider a graph H for which every induced subgraph G of H satisfies the following min-max equalities for every positive integer q (equivalently, for every positive integer q < w ( G ) , the maximum size of a clique in G).

I=

(1.1) maximum{lSI: S E V ( G ) ;V clique C in G , IS n CI s q } (1.2) minimum{q

1x1+ IV(G) - U XI:Xis a set of cliques in G } .

(1)

( V ( G ) denotes the node-set of G. In this paper, cliques need not be maximal.) Restricting q to be 1 in the above would say H is perfect. Also then, (using the Perfect Graph Theorem 123,241) a set S as in (1.1) is the same as a set which can be partitioned into no more than q stable sets; that is, a partial q-colouring. Thus: For a perfect graph G, (1.1)

[=

(1.3) maximum{lSI: S E V ( G ) ,S is a partial q-colouring in G}.

* Research supported by the Natural Sciences and Engineering Research Council of Canada 0012-365X/89/$3.50 01989, Elsevier Science Publishers B.V. (North-Holland)

K. Cameron

16

Fig. 1.

It is clear that: For any graph G,

l(1.4) minimum Note that the X of (1.2)and the V of (1.4)may be taken to be node-disjoint. Thus, for perfect graphs, the equality (1) is the same as (4)in ([3], this volume) which says (1.3)= (1.4)- Lov6sz [25] called a graph H q-perfect if for each induced subgraph G of H, (1.3) = (1.2). Greene [19]gave the graph of Fig. 1 to show that not all perfect graphs satisfy (1). It does not satisfy (1) for q = 2. The Dilworth-Greene-Kleitman min-max theorem ([8],[20])says that comparability graphs satisfy (1) for all q. Greene’s min-max theorem [191 and a more general theorem proved independently by Edmonds and Giles [9]say that cocomparability graphs satisfy (1) for all q. Lov6sz [24]proved that the substitution operation preserves perfection: if H and K are disjoint graphs and v is a node of H, then to substitute K for v in H, join each node of K to each neighbour of v, and delete v. Two important special cases of substitution are joined and unjoined duplication: to create m joined duplicates of node v, substitute a clique of m nodes for v ; to create m unjoined duplicates of v, substitute a stable set of m nodes for v. Note that creating 0 joined or unjoined duplicates of node v corresponds to deleting v, and thus taking an induced subgraph is a special case of duplication.

G2

Fig. 2.

17

Min-max relation for partial q-colourings V

Fig. 3.

Creating joined duplicates need not preserve min-max (1): The graphs GI and G2 in Fig. 2 satisfy (1) for all induced subgraphs and all q , but G3 does not satisfy (1) for q = 3. (See Section 6 for more examples and proofs.) Jean Fonlupt pointed out that creating unjoined duplicates need not preserve min-max (1). The graph in Fig. 3 satisfies (1) for all induced subgraphs and all q, but if node v is replaced by two unjoined duplicates, the new graph does not satisfy (1) for q = 2. Let us examine the effect of creating joined or unjoined duplicates in G on the min-max equality (1). For each v E V ( G ) ,let a, be a non-negative integer. Replace each v E V ( G )by a set of a, joined duplicates to get a new graph G’-that is, substitute a clique of size a,, for v. G’ satisfies (1) if and only if

I

c

(2.1) maximum(

I J EV

xu: v clique

c in G ,

xu 6 q ; ueC

(C)

Vv E V ( G ) ,0 =sxu =s a,, xu integer

-

I 1x1+ 2

(2.2) minimum q

veux

a,:

I

1

(2)

x is a set of cliques in c .

Since creating joined duplicates need not preserve min-max (l), equivalently if a graph satisfies (1) for all induced subgraphs and all q , it need not satisfy (2). The graph of Fig. 4 does not satisfy (2) for q = 3 and the aI,’sas shown. 2

2

Fig. 4.

K. Cameron

18

For each v E V ( G ) ,let w,,be a non-negative integer. Replace each v E V ( C )by a set of w,,unjoined duplicates to get a new graph G’-that is, substitute a stable set of size w,,for v. G’ satisfies (1) if and only if: ((3.1) maximum

1

I=

Lc

(3.2) minimum{q

w,,:S E V ( C ) ;V clique C in G, IS r l CJS q ]

2 yc+

cliques C

y , , : V v ~V ( G ) , weV(G)

yc+y,,3w,; U€C

I

V cliques C , y , 3 0, y , integer; Vv

E

V ( G ) ,y,, a 0, y,, integer

(3)

.

Where q = 1, Fulkerson [14] called a graph pluperfect if it satisfies (3) for every non-negative integer-valued w = (w,,: v E V(G)). We now look at a unification of (2) and (3). A graph G is called box perfect if vE for every positive integer q, and all non-negative integer-valued w = (w,,: V ( G ) )and a = (a,,: v E V ( G ) ) ,the following min-max equality holds:

c

((4.1) maximum(

w,,x,: V clique C in G,

V€V(C)

I=

Vv

4

2

cliques C

c

xu s q ;

WE,

E

V ( G ) ,0 S X , ~s a,,, x, integer

yC+

I

(4)

a,,y,: U€V(C;)

Vv

E

V(G),

c y c + y , a w,,;

we,

V cliques C , yc

3 0,

yc integer;

Vv E V ( G ) ,y,, 3 0 , y, integer

Where w,,= 1 V v E V ( G ) , (4) is (2). Where a, = 1 V v E V ( G ) , (4) is (3). Where w, = 1 and a, = 1, V v E V ( C ) ,(4) is (1). In Section 4, we will prove that box perfect graphs are precisely the graphs for which a certain system of linear inequalities is box totally dual integral. Note that if C is box perfect then so is any induced subgraph of G : choose a,, = 0 (or w, = 0) for v not in the induced subgraph.

2. Box perfection and joined and unjoined duplicates

For a fixed w = (w,,: v E V ( G ) ) and a = (a, : v E V ( G ) ) , let G,(w, a) be the graph obtained from G by substituting a stable set S,, of size w,, for each

Min-max relation for partial q-colourings

19

v E V ( G ) ,and then substituting a clique of size a, for each u E S,,. Then:

(4)holds for G e ( 1 ) holds for Gl(w, a)

(5)

If G is box perfect, it turns out that it is not necessary to substitute a clique of the same size for each node of S,, in order to conclude that (1) holds. In Section 5 we will prove:

Theorem 1. Creating unjoined duplicates preserves box perfection. Corollary 1. G is box perfect A n y graph obtained from G by first creating unjoined duplicates and then creating joined duplicates satisfies ( 1 ) . Alternatively, for a fixed w and a, let Gz(w, a) be the graph obtained from G by substituting a clique C, of size a, for each v E V ( G ) , and then substituting a stable set of size w,, for each u E C,,. In general, G,(w,a) # Gz(w, a). However:

(4)holds for G e (1) holds for Gz(w, a).

(6)

Similar to before, if G is box perfect, it turns out not to be necessary to substitute a stable set of the same size for each node of C,, in order to conclude that (1) holds. In Section 5 we will prove;

Theorem 2. Creating joined duplicates preserves box perfection. Corollary 2. G is box perfect. A n y graph obtained from C by first creating joined duplicates and then creating unjoined duplicates satisfies ( 1 ) . Corollary 3. G is box perfect. 9 Any graph obtained from G by creating a series of joined andlor unjoined duplicates satisfies (1).

3. Classes of box perfect graphs Theorem 3. The following classes of graphs are box perfect. (i) Comparability graphs [4]. (ii) Cocomparability graphs [9, 41. (iii) Graphs whose clique-node incidence matrix is totally unimodular. (vi) p-Comparability graphs [4],defined immediately below. A p-comparability graph is a graph which arises in the following way: start with a digraph C that has a set T E V ( G ) , IT1 S p , such that every edge of G is in a

20

K . Cameron

Fig. 5.

dicircuit, and every dicircuit of G intersects T exactly once; add the chords of every dicircuit, delete T, and make all edges undirected. l-comparability graphs are precisely comparability graphs [4]. It was proved in [4] that p-comparability graphs (and thus comparability graphs), and cocomparability graphs are box perfect. These proofs were based on our Coflow Polyhedron Theorem ([4,51) which gives strong min-max properties for any digraph. In particular, this provides new proofs of the Greene-Kleitman Theorem and Greene’s Theorem. It also follows from the Greene-Kleitman Theorem and Corollary 1 that comparability graphs are box perfect since creating joined or unjoined duplicates preserves being a comparability graph. Similarly, it follows from Greene’s Theorem and Corollary 1 that cocomparability graphs are box perfect. The Edmonds-Giles Theorem says cocomparability graphs are box perfect.

Proof of (iii). It is easy to see that if the clique-node incidence matrix A of graph G is totally unimodular, then G is box perfect: For a postive integer q, let q be a vector of all 4’s. Then it is well known [22] that for integer-valued vectorsb and a, the following linear program (7) and its dual have integer-valued optimum solutions. By the linear programming duality theorem, the optimum objective values of a linear program and its dual are equal. This is precisely (4). maximize

wx subject to

Ax sq

Qaxsa.

(7) 0

We comment that if G is box perfect, its complement need not be. The graph of Fig. 5 is the line-graph of a bipartite graph, and hence box perfect, but its complement, the graph of Fig. 1, does not satisfy (1) for q = 2.

4. Total dual integrality Let A be a matrix, d and _c vectors of constants, and x a vector of variables. A system, Ax a d, of linear inequalities with rational A and d is called totally dual


21

integral (TDI) if the dual of the linear program: maximize {ex:Ax s d } has an integer-valued optimum solution for every integer-valued c such that it has an optimum solution [9]. TDI systems are interesting because of the following result.

The TDI Theorem (Edmonds and Giles, [9]). If Ax S d is a totally dual integral system with integer-valued d, then for any c such that maximum{cx:Ax s d } exists, there is an integer-valued optimum solution x*. Thus a TDI system with integer-valued 4 provides the following integer min-max equality for any integer-valued _c for which either the min or the max exists. maximum{cx: Ax S d , x integer-valued} minimum{yd: y- A = c, y

3 0,

y integer-valued} .

A system, Ax S d, of linear inequalities is called box totally dual integral (box TDI), if it together with any upper and lower bounds on the individual variables

u:::{f

is TDI; that is, if for any 1, u E (Q U {kt.})", the system

is TDI.

A x s d is called upper box TDI if it together with any upper bounds on the variables is TDI; that is, if for any u E (Q U { +t.})", the system

{,":Zd

is TDI.

Groflin [21] proved that Ax S d is box TDI if and only if for any subset J of the variables, and any values uj E Q for j E J, the system

is TDI. Also, if Ax S d is TDI, so is any system obtained by changing some of the inequalities to equations (for a proof, see [28], Theorem 22.2). It follows that if Ax s d is upper box TDI, it is also box TDI, and thus box TDI and upper box TDI are equivalent. We will consider the following system of clique inequalities and non-negativity constraints for graph G. (10.1)

v clique c in G , C xu s 1; "€C

(10.2) v v E V ( G ) ,X"

3 0.

For our discussion here it does not matter if we consider only maximal cliques in (10) or all cliques.

Theorem 4. G is perfect.

a The system (10) of

clique inequalities and non-negativity constraints is TDI.

K. Cameron

22

Proof. The “if” part of this theorem follows by the TDI Theorem. The “only if” part is immediate by Lovasz’s theorem that creating unjoined duplicates preserves perfection (and thus for perfect graphs, (3) holds for q = 1). Theorem 4 motivates us to study graphs for which the system (10) is box TDI, which we will now show are the box perfect graphs, defined earlier in Section 1.

Theorem 5. G is box perfect.

e The

system (10) of clique inequalities and non-negativity constraints is box

TDI.

Lemma. If y- is an optimum solution to the dual of the linear program maximizeicx: Ax s d } , and r is a positive rational then y is an optimum solution to the dual of the linear program maximize {cx:Ax S r d } . Thus if Ax s d is TDI, so is Ax s r d . Proof of Theorem 5. Suppose G is box perfect. We will show that (10) is upper box TDI, and hence box TDI. We must show that for an integer-valued _c, the dual of the linear program (11) below has an integer-valued optimum solution.

2

maximize

CJ,

subject to

usV(C)

v clique C , C xu G I ; usc

v v E V ( C ) ,o s x ,

SU,.

We may assume that c, 0 and 0 s u, < cc, Vv. Let q be a positive integer so that for each u,, qu, is an integer. Then for this q , for w,,= c,,, and a,, = qu,, (4) holds. By the lemma, the y of (4.2), which is integer-valued, is an optimum solution to the dual of (11). Now suppose that the system (10) is box TDI. We must show that for a positive integer q, and non-negative integer-valued a and w , (4) holds. The system:

v clique C , C xu G I; usc

Vv E V ( G ) ,O S X , s a , l q ,

is TDI. Thus by the lemma, so is the system:

v clique C ,

x,

=S q ;

usc

[VV

E

v(G), o ~ x ==au. ,

Then the min-max (8) where c = w and Ax

G

d is (12) is the same as (4).


23

5. Proofs of Theorems 1 and 2 Theorem 1. Creating unjoined duplicates preserves box perfection. Proof. Assume G is box perfect. Consider the graph G’ obtained from G by creating an unjoined duplicate t of u E V ( G ) . We will show that G’ is box perfect by showing that (1) holds for G;(w,a ) for all non-negative integer-valued w and a. Fix w and a, and let H’ denote the graph obtained from G’ by substituting for each node u E V ( G ’ ) a stable set S, = { u , : i = 1, . . . , w,} of size w,. It suffices now to show that (2) holds for H’ with the upper bound a,, = a, for each uiE S,. If a, = a , , then (2) holds for H’ since (1) holds for Gl(w‘, a ’ ) where wh = w, + w,,w: = w, for E V ( G )- u, a: = a,, for u E V ( G ) . Thus without loss of generality, 0 < a, < a,. We will use the fact that (2) holds in each of the following instances: H = H’ - S, (i.e. H is obtained from G as H’ was from G’) with the same upper bounds a,, as H’.

(13)

H’ with the upper bounds on the nodes u, E S, lowered from a, to a,, but the other upper bounds unchanged.

(14)

If in some optimum x for (13), xu, 3 a, for some uiE S,, then setting x:, = a, for t, E S,, and x; = xp for p E V ( H ’ )- S,, and taking X ’ = any optimum X for (13), it is clear that x’ and X ’ satisfy (2) for H‘ with the given upper bounds. Thus we may assume that for every optimum x for (13), xu, < a,, for all uiE S,.

(15)

Note that every I feasible for (14) is feasible for If’with the given upper bounds. If for some optimum X for (14), at least w, members of S, U S, are in U X , then we can assume that all members of S, are in UX, and then this X and any optimum x for (14) satisfy (2) for H’ with the given upper bounds. Thus we may assume that for every optimum X for (14) fewer than w, members of S, U S, are in UX . Let X * be an optimum X for (14) such that no node of S, is in U X . Let x* be an optimum x for (14). By complementary slackness, since some node uiof S, is not in U X , x:,=a,. It is easily seen that x* restricted to H and X * are optimum for (13). But this contradicts (15). 0

Theorem 2. Creating joined duplicates preserves box perfection. We comment that Theorem 2 also follows from Edmonds and Giles’ Theorem ([9, 10, 111) that duplicating variables preserves box total dual integrality [4].

Proof of Theorem 2. This proof is similar to the proof of Theorem 1. Assume G is box perfect. Consider the graph G’ obtained from G by creating a joined duplicate t of u E V ( G ) . We will show that G‘ is box perfect by showing

K . Cameron

24

that (1) holds for Gi(w, a) for all non-negative integer-valued w and g. Fix w and g, and let H' denote the graph obtained from G' by substituting for each node v E V ( C ' )with a clique C, = {vi:i = 1, . . . , a,} of size a,. It suffices now to show that (3) holds for H' with the weights w",= w, for each v, E C,. If w , = w , , then (3) holds for H' since (1) holds for Gl(w', g') where a: = a , a,, a: = a , for v E V ( G )- u , wl= w, for v E V ( G ) .Thus without loss of generality, 0 < w,< w,. We will use the fact that (3) holds in each of the following instances:

+

H = H ' - C, (i.e. H is obtained from G as H' was from G') with the same weights w, as H'.

(16) H' with the weights on the nodes u, E C, lowered from w, to w,, but the other weights unchanged. (17)

If in some optimum y- for (16), y,, s w, - w,, for some u, E C,; that is, Cu,Ecyc3 w,; then setting y & = y , for cliques C in H ' - C, with u, E C ; y & = yc for cliques C in H' - C, with ui@ C ; y:, = 0 for t, E C,; and yj, = yp for p E V ( H ' )C,; and letting S' be some optimum S for (16), it is clear that S' and y ' satisfy (3) for H' with the given weights. Thus we may assume that: for every optimum -y for (16), y , > w, - w,> 0 , for all u, E C,.

(18)

If for some optimum S for (17), at least a, members of C, U C, are in S, we can assume all members of C, are in S, and then this S together with any optimum y for (17) with y , increased to y,, + (w, - w,)for each u, E C, satisfy (3) for H' with the given weights. Thus we may assume that for every optimum S for (17), fewer than a, members of C, U C, are in S. Let S* be an optimum S for (17) such that no node of C, is in S*. Let y * be an optimum y for (17). By complementary slackness, since some node u,-of C, is not in S * , y:, = 0. It is easily seen that S* and y- * restricted to H are optimum for (16). But this contradicts (18). 0

6. An infinite class of graphs which satisfy (1) for all induced subgraphs and all q , but which are not box perfect. An rn-trampoline (rn 2 3 ) is the graph C with 2rn nodes

V ( G )= { v i :0 s i srn - 1) U { u i :0 s i srn - l } and E ( G ) = { ( v i , u i ) : O s i s r n - l } U { ( v i~, ~ + , ( ~ ~ ~ , , , ) : O ~ i ~ r n - 1 } U {(ui,u j ) :0 s i < j

s rn - l}.

A trampoline is a graph which is an rn-trampoline for some rn. A trampoline is called odd or even according to whether rn is odd or even. The rn 3-cliques {ui, vi,u ~ + ~ ( 0~ G~i s~rn~-)1,} of , the rn-trampoline are called the outer

25

Min-ma* relation f o r pariial 9-colourings

triangles. The graph of Fig. 1 is the 3-trampoline. The graph G, of Fig. 2 is the 5-trampoline.

Proposition 1 [4]. All the trampolines except the 3-trampoline satisfy ( 1 ) for all induced subgraphs and all q. Proposition 2 [4]. None of the odd trampolines are box perfect: The (2q - 1)trampoline does not satisfy (4) where a , = 1, a,,, = q - 1, and wu,= w,,, = 1, Osism-1. Proof of Proposition 1. In [4], it was shown that any induced subgraph F of an rn-trampoline G, except G itself, is a p-comparability graph, and hence satisfies (1). Assuming rn 3 4, Table 1 give an S and X satisfying ( 1 ) for G, for all values of

q<m. 0 Table 1

Is

9

1 2 9 = 3, . . . ,m - 1

Uu, u1,.

. . , U,-l

uo, u1,. . . I

Urn-!,

uo, u , , . . . ,

uo, u2 uo, u , , . . . , u q - l

the outer triangles the m-clique the m-clique

Proof of Proposition 2. We will display an x feasible for (4.1) except that it is not integer-valued, and a y feasible for (4.2) except that it is not integer-valued, such that the objective values of this x and y are equal, but are not an integer. (=m-1)

Osis2q-2

y, = 4 for each of the 2q

- 1 outer

triangles

y, = 0 for any other clique Y , = 0I} Y,, = 2

o s i s 2q - 2

2 F 2 x u+ a i=O

c

2cl-2 i=O

xv, = (2q - l)(t)

+ (2q - l ) ( q - 1 ) = 2q2 - 2q + 4

=2+2q+;.

0

K. Cameron

26

7. Balanced graphs A 0-1 matrix is called balanced if it has no odd order submatrix with exactly two 1’s in each row and column. A graph is balanced if its incidence matrix of maximal cliques versus nodes is balanced. Fulkerson et al. [16] proved that if matrix A is balanced, and d and a are non-negative integer-valued vectors, then the following linear program (19) and its dual have integer-valued optimum solutions:

maximize 1 & subject to Ax sd OS&SQ

Where A is the incidence matrix of maximal cliques versus nodes of balanced graph G and d is a vector of all q’s, this implies that (2) holds for G. The graph of Fig. 3 is balanced but not box perfect.

Acknowledgement I would like to thank Les Trotter and Jack Edmonds for helpful comments,

References [l] [2] [3] [4]

C. Berge, Balanced matrices, Math. Programming 2 (1972) 19-31. C. Berge, Graphs and Hypergraphs (North-Holland, Amsterdam, 1973). C. Berge, A min-max relation for the partial q-colorings of a graph, Part 1, this volume. K.B. Cameron, Polyhedral and Algorithmic Ramifications of Antichains, Ph.D. thesis, University of Waterloo (1982). [5] K.B. Cameron and J. Edmonds, Coflow polyhedra, to appear. [6] V. Chvatal, On certain polytopes associated with graphs, J. Combin. Theory (B) 18 (1975) 138- 154. [7] G.B. Dantzig, Linear Programming and Extensions (Princeton University Press, Princeton, New Jersey, 1963). [8] R.P. Dilworth, A decomposition theorem for partially ordered sets, Ann. Math. Ser. (2) 51 (1950) 161-166. [9] J. Edmonds and R. Giles, A min-max relation for submodular functions on graphs, Ann. Discrete Math. l(1977) 185-204. [lo] J. Edmonds and R. Giles, Box TDI polyhedra, unpublished manuscript. [ l l ] J. Edmonds and R. Giles, Total dual integrality of linear inequality systems, in Progress in Combinatorial Optimization, W.R. Pulleyblank, ed. (Academic Press, Toronto, Ontario, 1984) 117- 129. [12] M.R. Farber, Application of L.P. Duality to Problems Involving Independence and Domination, Ph.D. thesis, Rutgers University, New Brunswick, New Jersey (1981). [13] M.R. Farber, Characterizations of strongly chordal graphs, Discrete Mathematics 43 (1983) 173-189. [14] D.R. Fulkerson, Blocking and anti-blocking pairs of polyhedra, Math. Programming 1 (1971) 168-174.


27

[15] D.R. Fulkerson, Anti-blocking polyhedra, J. Combin. Theory (B) 12 (1972) 50-71. [16] D.R. Fulkerson, A.J. Hoffman and R. Oppenheim, On balanced matrices, Mathematical Programming Study 1 (1974) 120-132. [17] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs (Academic Press, New York, 1980). [18] C. Greene, Sperner families and partitions of a partially ordered set, M.I.T. Math. Center Tracts 56 (1974) 91-106. [19J C. Greene, Some partitions associated with a partially ordered set, J. Comb. Theory (A) 20 (1976) 69-79. [20] C. Greene and D.J. Kleitman, The structure of Sperner k-families, J . Combin. Theory (A), 20 (1976) 41-68. [21] H. Groflin, On switching paths polyhedra, Combinatorica 7(2) (1987) 193-204. (221 A.J. Hoffman and J.B. Kruskal, Integral boundary points of convex polyhedra, in H.W. Kuhn and A.W. Tucker (eds.), Linear Inequalities and Related Systems, Annals of Mathematics Study No. 38 (Princeton University Press, Princeton, New Jersey, 1956) 233-246. [23] L. LovBsz, A characterization of perfect graphs, J. Combin. Theory (B) 13 (1972) 95-98. [24] L. LovBsz, Normal hypergraphs and the perfect graph conjecture, Discrete Math. 2 (1972) 253-257. [25] L. LovBsz, Perfect graphs, in Selected Topics in Graph Theory 2, L.W. Beineke and R.J. Wilson, eds. (Academic Press, London, 1983) 55-87. [26] A. Lubiw, r-free matrices, Masters’ thesis, University of Waterloo (1982). (271 M.W. Padberg, On the facial structure of set packing polyhedra, Mat. Programming 5 (1973) 199-215. [28] A. Schrijver, Theory of Linear and Integer Programming (Wiley and Sons, Chichester, 1986).

This Page Intentionally Left Blank


29

ON LOCALLY-PERFECT COLORINGS Pierre DUCHET C.N . R.S., Uniuersite'Paris 6, Park, France

Received April 1988 A coloring of a graph is locally-perfecf if for every vertex u, the closed neighborhood of u contains no more than o ( u ) colors, where w ( u ) is the order of a largest clique containing u. Here is constructed, for any 9 a 3, a 9 + 1-chromatic graph, with clique number q, that admits a locally-perfect coloring. This answers a problem of Preissmann [3].

A (proper) coloring of a finite simple graph (G) is perfect if it uses exactly

o(G)colors, where o ( C )denotes the order of a largest clique in G. A coloring is locally-perfect [3] if it induces on the neighborhood of every vertex v a perfect coloring of this neighborhood. A graph G is perfect (resp. locally-perfect) if every induced subgraph admits a perfect (resp. locally-perfect) coloring. Preissmann proved that locally-perfect graphs form a proper subclass of the class of perfect graphs, and she asked for the following:

Problem 1 [3]. Is it true that if a graph C has a locally-perfect coloring then it has a locally perfect coloring in w ( v ) colors? The answer is affirmative for o(C)= 2 since a triangle-free graph that admits a locally-perfect coloring is necessarily bipartite. We show below that the answer is negative when o(G)L 3. In fact we exhibit a graph G admitting a locally-perfect coloring but which is not even o ( C ) colorable. A ( p , q)-coloring [ l ] of a graph is a coloring with p colors such that at most q appear in the neighborhood of every vertex.

Proposition 2. If a graph G admits a ( q + 1, q)-coloring, it is q-colorable. Proof. Let Q, : V ( G ) + (1, . . . , q + l} be (q + 1, q)-coloring of G. A q-coloring Q,' can be defined as follows, If q ( v )# q + 1, then Q,'(v) = Q,(v). If ~ ( v =) q + 1; then Q,'(v)is the smallest color not appearing in the neighborhood of v. 0 A stronger form of (2) is proved in [l]. More generally, [l] gives rather good estimates of the smallest integer f ( s , q) such that there exists a graph admitting a ( f (s, q), q)-coloring but which is not s-colorable. The next theorem yields f ( q , q) = q 2 and answers negatively to Problem 1.

+

0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

30

P . Duchet

+

Theorem 3. For any integer q 2 3, there exists a q 1-chromatic graph, of clique number q, which admits a locally-perfect coloring into q + 2 colors. Clearly, it suffices to construct a non-q-colorable vertex-transitive graph' Gq, with clique number q , which admits a (locally-perfect) ( q 2, q)-coloring. Notice that G9will be q + 1-chromatic, by Proposition 2. Let X be a q 2-element set. We define G, = (V, E) by:

+

+

Vertices ( x , A ) and ( y , B) are joined in G9 whenever x $ B, y $ A and x Z y . Obviously, G, is vertex-transitive and has order

(z)

the respective coordinate mappings. The We denote by cp : V +X and $J : V + clique number of G, is at most q since if {(xl, A , ) , . . . , (x,,,, A,)} is a clique of G,, we necessarily have xi + x i (for 1=z i , j S w ) and { x , , . . . ,x,} nA , = 0. For x E X and A E (f),set:

v, = v n c p - ' ( ~ ) W, = V n $J-'(A)

z,

= {x

E

v; x E $J(tJ)}.

The Vx's are stable sets of order (9 g I ) which partition V. The W,'S are cliques of order q which partition V. Observe also that for any distinct elements A, p, Y of X,the set

v, u (2, n Vp) u { Y , { A , P}} of order (" ;I ) + q + 1= (, ;*).

SLp.v:=

is a stable set It follows that cp is a ( q + 2, 4)coloring of G, and that G, has clique number q and stability number (7' ;):'. It remains to prove that G9 is not q-colorable. We first establish the following lemma:

Lemma 4. In G,, every stable set of order (" g2) is some Slmn. Proof. Let S be a stable set of order (" 12).Put S,:=S observation is immediate:

n V,. The following

If S, and S, are not empty and if x Z y , then we have x all v E S, or y E ~ ( vfor ) all v E S,.

E

~ ( v for ) (1)

'In a vertex-transitive graph, all vertices play the same role, i.e. the automorphism group acts transitively on vertices.

31

Locally -perfect colorings

Because of (l), it is convenient to consider the digraph D whose vertices are the non-empty sets of the form S,, with an arc (S,, S,) each time we have x E v ( v ) for all v E S,. By the definitions, the in-degrees d-(S,) in D are not greater than 2. Moreover, we have: If d-(S,) = 1 then IS,l s q. If d-(S,) = 2 then IS,l = 1. By (l), D is complete; since q 2 3, it follows, by (2) and (3) that if D has order 24 then IS1 < ( 9 t’). The same inequality holds if D has order 1 or 2 or if D contains a directed 3-cycle. Hence D contains the transitive tournament T3 with vertices, say SA,S,, S,,. Henceforth, by (2) and (3), we have S = SAU S, U S,, with

Since IS( = ( 9 t 2 ) ,we have S = SAW,,.0

Proof of theorem (3) (continued). If G9 is q-colorable, it admits by Lemma (4) partition into q sets S ( i ) = SA,,,,,,, 1zs i S q. Necessarily the Aj’s are all distinct hence at least two pi’s coincide, say p , = p2. But this is a contradiction, since the vertex ( p , , {A,, A?}) = ( p 2 , {A2, A,}) belongs to both S(1) and S ( 2 ) . 0 Graph G9 defined above looks like a Kneser graph [ 2 ] . Define, more generally a graph G,,,,, = ( V , E ) by:

v := [( R , T ) E);(

[

x

(f)R

E : = { ( R , T ) , ( R ’ , T ’ ) }E

;

(r)

T)

:R

I

n T’ =R’ nT =0 .

Here 0 S r 6 t S n and X is an n element set. Kneser’s graph is G , , , , and G9 corresponds to G,,2,9+2.

Problem 5. What is the chromatic number of G,,,,,? Remark 6. Graph G9, as defined above, it not perfect. Problem (1) remains open for perfect or locally-perfect graphs [3].

Acknowledgement Our sincere thanks are due to M. Preissmann who kindly pointed [ l ] to our attention.

32

P. Duchet

References [l] P. Erd6s, Z. Furedi, A. Hajnal, P. KomjBth, V. Rod1 and A . Seress, Coloring graphs with locally few colors, Disc. Math. 59 (1986) 21-34. [2] L. Loviisz, Kneser’s conjecture, chromatic number and homotopy, J. Comb. Th. 25 (1978) 319-324. [3] M. Preissmann, Locally-perfect graphs, preprint IMAG, Grenoble, France (1986) submitted.

33


THE SUBCHROMATIC NUMBER OF A GRAPH M.O. ALBERTSON Smith College, U.S.A.

R.E. JAMISON and S.T. HEDETNIEMI* Clemson Uniuersify, U.S.A.

S.C. LOCKE Florida Atlantic University, U.S.A.

The subchromatic number X,5(G)of a graph G = (V, E) is the smallest order k of a partition { V , , V,, . , . , V,} of the vertices V ( G ) such that the subgraph induced by each subset V, consists of a disjoint union of complete subgraphs. By definition, &(G) s X ( G ) , the chromatic number of G . This paper develops properties of this lower bound for the chromatic number.

(v)

1. Introduction An n-coloring of a graph G = ( V , E) is a function f from V onto N = { 1, 2, . . . , n } such that whenever vertices u and 21 are adjacent, then f ( u ) #f(v). An equivalent definition, which provides much of the motivation for this paper, is that an n-coloring of C is a partition {Vl, V2,. . . , V,} of the vertices V ( G )into color classes, such that for every i = 1, 2, . . . , n, the subgraph (V,) induced by V, is totally disconnected, or equivalently, is the disjoint union of K,’s (complete graphs with one vertex). A partition {V,, V,, . . . , V,} of V ( G )is called complete if for every i , j , 1 G i <j S n , there exists a vertex u in V, and a vertex v in V, such that u and 21 are adjacent. The chromatic number X ( G ) and the achromatic number Y ( G ) are the smallest and largest integers n , respectively, for which G has a complete n-coloring. Notice that in the definition of the chromatic number the completeness of the partition is not required but follows easily from the definition, whereas the completeness of the partition used in defining the achromatic number is essential. The chromatic number, of course, is a very well studied parameter, whose history dates back to the famous Four Color Problem and the early work of Kempe in 1879 [25] and Heawood in 1890 [22]. The achromatic number was first studied as a parameter by Harary, Hedetniemi and Prins in 1967 [20], and later named and studied by Harary and Hedetniemi in 1970 [19]. This paper was motivated in part by an interest in developing interesting lower * Research supported in part by Office of Naval Research Contract N00014-86-K-0693. 0012-365X/89/$3.50 01989, Elsevier Science Publishers B.V. (North-Holland)

34

M.O. Alberrson et al.

bounds for the chromatic number, ones that would involve partitions {Vl, V2,. . . , V,} of the vertices of a graph, with certain conditions imposed on the induced subgraphs ( V , ) . In general, let P be an arbitrary property of a graph. We say that a property P is (induced) hereditary if whenever a graph G has P, then so does every (induced) subgraph G’ of G. A subset S of V ( G )is a P-set of C if (S), the subgraph induced by S, has property P. A (generalized) P-coloring of G is an assignment of colors to the vertices of G such that for any given color, the set of vertices having this color is a P-set. A (P, m)-coloring of G is a partition n = {Vl, V2,. . . , V,} of V ( G )such that each subset V, is a P-set. The P-chromatic number, denoted XJG), is the smallest integer m for which G has a (P, m)-coloring. The notion of generalized P-colorings of graphs, or of P-partitions of the vertices of a graph, is certainly not a new one. It seems to have been ‘independently discovered’ by several authors around 1968 and again by several authors around 1985. The definitions involving P-sets and P-colorings given here are those given by Hedetniemi in 1968 [23], whose studied the non-hereditary property PI: that a graph be disconnected or the trivial graph, and the hereditary property P2: that a graph be acyclic, in which case the P2-chromatic number is called the point arboriciq of a graph. Independently in 1968, Chartrand, Kronk and Wall also defined and studied the point arboricity of a graph [lo]. In 1968, Chartrand, Geller and Hedetniemi [8] studied the hereditary property P,(k): that a graph contain no path of length k, for some fixed value of k. Also in 1968, Sachs and Schauble [37] studied the hereditary property P4(k): that a graph contain no complete subgraph on k vertices, for some fixed k. P4(3)-colorings were also studied by Harary and Kainen in 1977 [21], where they showed that the vertices of any planar graph can be partitioned into two sets V, and V2such that neither ( V , ) nor ( V2) contains a KJ. In 1969 Kramer and Kramer [27], [28] and [26] studied colorings in which no two vertices in (V,) whose distance apart in C is 4 receive the same color; this of course is equivalent to studying (normal) colorings of the kth power of a graph. In 1970, Lick and White [32] introduced k-degenerate graphs, which are defined in terms of the hereditary property P,(k), called k-degenerate: that no induced subgraph of a graph have minimum degree exceeding k, for some fixed value of k. K-degenerate graphs have since been studied by Cook [13] and Borowiewcki [4]. Also in 1970, Hedetniemi [24] continued to study disconnected colorings of graphs, and offered this comment: “In spite of the unnaturalness of the concept of disconnected colorings, more than two dozen results have been constructed for D-colorings and X, which are virtually identical to corresponding results that have been established for the traditional colorings and chromatic number X ( G ) of graphs. These new results indicate that the established results reveal much less about properties of colorings than they do about concepts which are much more general; they also reveal that most of the established results on coloring can be proved using little more than purely set theoretic arguments”. Again in 1970, Folkman [16] proved a result

Subchromatic number of a graph

35

which in effect said that for any, nontrivial hereditary property P of a graph, there are graphs whose P-chromatic number is arbitrarily large. Three other studies of generalized P-colorings were offered by Cockayne [ l l ] , and Cockayne and Miller [12], who proved that for any hereditary property P, if a graph G has a complete (P, k)-coloring and a complete (P, n)-coloring (k < n), then for every m, k s m < n, G must also have a complete (P, m)-coloring. In 1977, three other papers appeared along these lines: Lesniak-Foster and Straight [30] studied the hereditary (co-chromatic) property P6: that a graph be either a complete graph or an empty graph; Lesniak-Foster and Roberts [29] studied, among other things, the property P3(2), i.e. ( K ) has no paths of length 2; and Sampathkumar, Neeralagi and Venkatachalam [38] studied the hereditary property P,(k): that no two vertices in ( K ) lie at distance k from one another in G, for some fixed value of k. Around 1985 the subject of generalized colorings of graphs seems to have experienced a “rebirth”. In [18] Harary referred to generalized colorings as conditional colorings and considered, in particular, the general classes of properties PF, that a particular graph F is forbidden to be induced by any of the vertices in a set K. Also Andrews and Jacobson [l], [2], studied the hereditary property P,(k): that the maximum degree of a vertex is less than or equal to k, for some fixed value of k. Mynhardt and Broere [35], [5], also studied the general property PF: that a graph contain no induced subgraph isomorphic to a given graph F; Brown and Corneil [7] studied general P-chromatic numbers for hereditary properties, vertex P k-critical graphs and uniquely P k-colorable graphs; and Domke, Laskar, Hedetniemi and Peters [14] studied the hereditary property Pg: that a graph is isomorphic to a complete r-partite graph. We have not attempted here to complete a comprehensive survey of generalized colorings or point partition numbers of graphs, rather we have attempted to indicate something of the wide variety of properties which have been studied. Table 1 summarizes some of the these properties. Table 1. Some P-colorings which have been studied. Let for:

n=[V,, V,, . . . , V,,,) be a partition of V ( G ) ,where for every i, 1

i G m, (V,) has property P

PI: ( V , ) is disconnected or trivial (231, [24] P,: (V,) is acyclic [lo], [23] P,(k): (V,.) has no path of length k , for some fixed k [8] P,(k): (V,) has no complete subgraph of size k , for some fixed k [37]

P&): (V,) has no induced subgraph whose minimum degree is bk for some fixed k [32] P6: ( V , ) is either a complete graph or a graph without edges [30] P,(k): (V,) has no two vertices at distance k in G, for some fixed k [38] P,(k): (V,) has maximum degree G k , for some fixed k [l], [2] Ps: (V,.) is a complete r-partite graph, for any r [14] PF: (V,) has no induced subgraph isomorphic to F [5], (351, [7]

PF: (V,) has no induced subgraph isomorphic to any graph F i n F [7] P,”: (V,) is a disjoint union of complete subgraphs [35], [this paper] P,,(k): ( V , ) contsins no induced K , , k [13]

36

M.O. Albertson et al.

In this paper we study another generalized coloring of a graph, one which provides both an interesting lower bound for the chromatic number and which has interesting properties in its own right. A subcoloring of order n of a graph G is a partition {Vl, V,, . . . , V,} such that for every i = 1, 2, . . . , n, the subgraph (F)is a disjoint union of complete subgraphs (of various sizes), i.e. = UKj’s. The subchromatic number X,(G) is the smallest n for which G has a complete subcoloring of order n ; while the subachromatic number Y,(G) is the largest integer n for which G has a complete subcoloring of order n. The following chain of inequalities follows immediately from these definitions:

(v)

For any graph G, X,d X d Y d Y,.

(1)

It also follows from the definition of a subcoloring that if 17 = { V,, V,, . . . , V,} is a subcoloring of minimum order, i.e. X,(G) = n, then I7 is in fact a complete partition of V ( G ) . Although we have newly named the subchromatic number here, it was originally studied in a different framework by Mynhardt and Broere, [35] and [5], who referred to X,(G) as the F-chromatic number, F(G), for the graph F = K1,2. That is, if a graph has no induced subgraph isomorphic to the path on three vertices, also denoted K1,*, then it must be a disjoint union of complete subgraphs. Subcolorings are also related to the co-colorings of Lesniak-Foster and Straight [30], in which the induced subgraphs must either be complete graphs or empty graphs. If we define the co-chromatic number X c ( G ) to equal the minimum number of colors in a co-coloring of the vertices of G , then it follows that, in fact X,S X c S X , since every co-coloring is a subcoloring. The remainder of this paper reviews results which have either been proved or can readily be inferred from previous results in the literature, and develops new properties and bounds for the subchromatic number of a graph.

(v)

2. Results from previous studies Because the property Plo: (K) is a disjoint union of complete subgraphs is hereditary, we can immediately infer a number of results about the subchromatic number, which are corollaries of earlier studies. [7] For any induced subgraph H of a graph G, X , ( H ) S X , ( G ) .

(2)

[16, 35, 71 For every k 3 1, there is a graph Gk with x,(Gk) = k.

(3)

[35] For every k 5 1, there is a K3-free graph Gk with X,(C,) = k.

(4) (5)

[30] For any graph G with p

6 8 vertices,

X,(G) S X , ( C ) d 3.

[30] for any graph G with p vertices, minimum degree 6 and maximum degree A, X,(G) S X c ( G )d min{l + A, p - S } . (6) ,...,P n , [30, 351 For the complete multi-partite graph, G = KplPz whereplSp2d.-.~pn,X,(G)=Xc(G)=minl,j,n{n, n-j+pj}. (7)


31

A graph G is vertex k-subcritical if X,(G) = k but X,(G - v ) < k for all v in V ( G ) .

[7] Any k-subchromatic graph contains a vertex I-subcritical sub(8) graph, for all 1 S k. [7] If G is a vertex 2-subcritical graph, V ( G )= [vo, v,, . . . , v,} and H,, H2, . . . , H, are k-subchromatic graphs (k 2 l), then the graph F which results from successively substituting Hifor vi (every vertex in Hi is joined to every vertex adjacent to vi) is not k-subcolorable.

(9)

Note in (9) above, that if Hi= H for all i = 1, 2, . . . , n and IV(H)I = rn then IV(F)(= rnn + 1. Note also that K l , 2is the only vertex 2-subcritical graph.

[35] If X,(G) = k and H = K1,2[G], then X , ( H ) 3 k + 1, where V(G,[G2]) = C(CJ x V(G2)and ( u , , u 2 ) is adjacent to (v,,v2) if and only if either u l v l is in E l or u 1= v1 and u2v2is in E2. (10) [5] For every outerplanar graph G, X,(G)S3, and this bound is tight, i.e. there exists an outerplanar graph H for which X , ( H ) = 3. (11) [5] For every planar graph G, X , ( C ) S 4, and this bound is tight. (12) Next, let X J G ) , the partite chromatic number, equal the minimum order of a partition V,, V2,. . . , V, such that for every i, 1 < i s m , (V;:) is a complete r-partite graph, for some (variable) r.

[14] For any graph G, X,(G)=X,(G'), where G' denotes the complement of G. (13) [14] For any graph G with n vertices, (14) (i) 1<X,(G) *x,(G') s n2/4; (ii) 2 ~ X , ( G ) + X , ( G ' ) s X ( G ) + X ( G ' ) s n+ 1; (iii) X,(G) +X,(G') S 2X(G). Finally, in (111 Cockayne generalized a classic result of Szekeres and Wilf [39], that X ( G )S 1 + maxC, d , then A and B cannot meet. Indeed, suppose they did meet. Note that A meets tc+dn, B meets and neither A nor B meets an earlier interval (cf. Fig. 2). Thus, the left endpoint of B is larger than bc+,,-,. Since A intersect B is not empty, it follows that A contains the left endpoint of B. Thus A meets B and the n intervals Zi for i = c + dn, . . . , c + ( d + 1)n - 1, and the set of vertices of G corresponding to this set of n + 2 intervals fcims an induced K1,,+l, which contradicts our assumption that G contains no induced K I , , + l . Thus, no A in C C + , can meet any B in C,,,, if e > d , and we produce a legitimate n-subcoloring of C .

Corollary 5 . For any indifference graph G , X,v(G)S 2. An indifference graph is a graph G = (V, E) for which a real-valued vertex function f exists satisfying: (u,v ) in E if and only if I f ( u ) -f(v)l< 1. In [36] Roberts showed that G is an indifference graph if and only if C is an interval graph containing no induced K1,3. We close this section by noting the following about 3-subchromatic graphs.

Proposition 6. (i) For any wheel W, ( = C , + K l ) , f o r n 2 5 , XJW,) = 3; (ii) [35]for any graph G with n S 8 vertices, X,(G) s 3, and this bound is tight; (iii) [5] for any maximal outerplanar graph G , X , ( C ) S 3 , and this bound is tight. We suspect that the upper bound in Proposition 6(ii) can be considerably improved. Although we do not know the smallest value of n for which there exists a 4-subchromatic graph on n vertices, a construction presented in the next section can be used to produce a 4-subchromatic graph with 13 vertices. Proposition 6(iii) is a bit disappointing since the chromatic number of any maximal outerplanar graph with n 2 3 vertices is 3; it is easy, however, to construct maximal outerplanar graphs whose subchromatic number also equals 3.

M.O. Albertson

42

et al.

4. General results It is easy to construct graphs whose chromatic number is arbitrarily large. For example, the process of adding a new vertex and joining it to every vertex in a given graph G produces a graph (denoted K, G) whose chromatic number is one greater than the chromatic number of G, i.e. X ( K I G) = 1 + X ( C ) . However, this is not always true for the subchromatic number.

+

+

Proposition 7. For any graph G, X,(K, + G) = 1 + X , ( G ) if and only if there does not exist a X,-subcoloring of C in which one color class V , induces a complete graph As previously mentioned in Section 2, several authors [16], [35] and [7] have established results which are sufficient to show that there are graphs whose subchromatic number is arbitrarily large. The following is a simple proof of this fact.

Theorem 8. [35] For any positive integer n, there exists a K,-free graph G with X J G ) 3 n. Proof. It is well known that there exist K3-free graphs having arbitrarily large chromatic numbers (cf. [34]). Therefore, let G be a K,-free graph for which X ( G )= 2n. Let X J G ) = m and let {V,, V2,. . . , V,} be a subcoloring of G. Since G is K,-free, each is a disjoint union of K,'s and K,'s, and therefore the vertices in each V , can be 2-colored in the normal sense. Therefore, X ( G ) = 2n G 2m = 2Xs(G),or A',((?) 3 n. 0

(v)

Corollary 9. If C denotes the size of a largest clique in a graph G, then

X(G ) 6 C * X,(G ), or X / C6 X,. Theorem 8 establishes the existence of graphs with arbitrarily large subchromatic numbers. Theorems 10 and 13, which follow, provide methods for constructing such graphs. Let C and H be graphs and let C v H denote the graph which results from adding a new vertex v to the disjoint union of G and H and joining v to all vertices in G and H.

Theorem 10. For any graphs G and H with X , ( G ) > k and X , ( H ) > k , X,(G v H ) 3 k + 1. Proof. Assume that X,(G v H) 6 k and let n = { V , , V2,. . . , V,} be a subcoloring of G v H of order k. Then k colors must be used to color the vertices of G and k colors must be used to color the vertices of H. Consider the color assigned to vertex v , call it 'red'. It follows that there must be at least one vertex u in G


x S =1

x

x S= 2

x

S

43

=3 S

=4

Fig. 3. Small k-subchromatic graphs.

and at least one vertex w in H which are also colored ‘red’. That is, there is a u - v - w red path. But since u and w are not adjacent in G v H, this cannot be a legitimate subcoloring of C v H. Therefore, X,(G v H) k + 1. 17 Fig. 3 illustrates the smallest k-subchromatic graphs constructible by the G v H-process, for k = 1,2, 3 and 4. Even though S, is not drawn with a planar embedding, it is easy to see that S, is a planar graph. Thus, as is pointed out in [5],X,(G) = 4 is possible for planar graphs. Notice that these graphs have 2k - 1 vertices and contain the full binary trees with 2k - 1 vertices as spanning subgraphs. These graphs, furthermore, can be seen to be interval graphs and they are also chordal graphs (graphs in which every cycle of length strictly greater than three possesses a chord, i.e. an edge joining two nonconsecutive vertices of the cycle), for which the size of the largest clique, the subchromatic number and the chromatic number are all equal. The next two results illustrate properties of the G v H construction. A perfect elimination ordering of a chordal graph G = (V, E) is an ordering of the vertices in V, say

44

M . 0. Albertson et al.

vl, v2, . . . , v,, such that each vertex ui is simplicia1 (i.e. ( N ( v , ) )is a complete subgraph) in the subgraph induced by the vertices vi, vi+,,. . . , v,. A Hamiltonian elimination ordering (HEO) is a perfect elimination ordering in which ui is adjacent to viClfor 1 6 i 6 n - 1. Lemma 11. If G and H are both chordal (resp. Interval) graphs, then so is G v H , and if G and H have Hamiltonian elimination orders, then so does G v H . Proof. The fact that G v H is chordal is obvious. If C and H are interval graphs, represent G by negative intervals and H by positive intervals, and then represent the vertex v by an interval covering all the others. For the H E 0 for G v H take the H E 0 for G , next take vertex v, and then take the H E 0 for H. 0 Theorem 12. For every integer k > 0, there is a chordal graph G with clique number k and subchromatic number k. Moreover, there is such a graph that is an interval graph with an HEO. Proof. We proceed by induction. Let GI be the graph with one vertex. Clearly GI has all of the required properties for k = 1. Inductively define G,,, = G,, v G,. The result follows from Lemma 11 and the fact that a largest clique in G,,,, consists of a largest clique in G, together with vertex x. 0 The following result, which is a corollary of (6) in Section 2 [35], describes another family of graphs for which the size of the largest clique, the subchromatic number and the chromatic number are all equal.

Theorem l3 [35]. For any positive integer m , X,( Km,m . _ _, m . ) = X ( Km,m,., , m ) =m , is the complete m-partite graph containing m classes of m vertices where Km,m,...,m each. ,

Proof. Trivially, X,(K,,, , _ . m _ ,) S m since X ( K m , ,,_..,m ) = m and X,(G) 6 X ( G )for any graph G. Therefore, assume that Xs(Km,m,...,m) = k < m , and let II= {V,, V,, . . . , V,} be a k-subcoloring of Km,m ,__., m. It follows that at least one color class, say V,, must have more than m vertices. Therefore, (V,.) cannot be a totally disconnected graph and must contain at least one pair, say ( u , v ) , of adjacent vertices. But every other vertex in is adjacent to either u or v. Therefore ( V , ) is a connected graph. But clearly ( K ) cannot be a complete graph since it has more than m vertices. Therefore I7 is not a legitimate subcoloring and x s ( K m , m ,...,m ) 3 m* 0

(v)

In 1941 Brooks [6] proved his classical theorem: for any connected graph G with maximum degree A, X ( G )< A 1, where equality holds if and only if G is a complete graph or an odd cycle. For the subchromatic number we can derive a stronger result.

+


45

Theorem 14. Let G be a k-regular graph, k 3. Then C contains a ( [ k / 2 ]+ 1)colorable, spanning subgraph H such that d H ( v )2 k - 1, for every v in V ( G ) , where d H ( v )denotes the degree of vertex v in subgraph H.

+

Proof. Let H be a ( [ k / 2 ] 1)-colorable, spanning subgraph of G having a maximum number of edges, and let the vertices of H be colored with [ k / 2 ] 1 colors. Suppose d H ( v )< k - 1 for some v in V ( G ) . Let NH(v)and N,(v) denote the set of vertices adjacent to v in H and G, respectively, and let N,[v] = {v} U N,(v). Consider the colors assigned to the vertices in NG(v).Since d H ( v ) < k - 1, there must be at least two vertices, say w1and w2, which are adjacent to v in G but not in H. Let c ( v ) denote the color of v. If c ( v ) # c ( w 1 ) (or (c(v)#c(w2))then we can add the edge between v and w1 (or w2) to H, contradicting the maximality of E ( H ) . Therefore, c(v) = c(w,)= c(w2). Now consider the colors assigned to the vertices in N H [ v ] .If fewer than [ k / 2 ]+ 1 colors are used to color these vertices, then we can recolor v with one of the unused colors and add edges (v, w l ) and (v2,w2)to H , again contradicting the maximality of E ( H ) . Therefore, we assume that [ k / 2 ]+ 1 colors are used to color the vertices in N H [ v ] .It follows that at least one color is used at most once to color the vertices in NH(v)(i.e. [ k / 2 ]colors must be used to color N,(v); if each of these colors were used at least twice, we would have at least 2 [ k / 2 ]2 k - 1 vertices in N,(v), but by assumption INH(v)l< k - 1). Let vertex u be the only vertex in N,(v) assigned this color. Then if we delete edge (v, u ) and recolor v the same as u, we can add edges (v, w l ) and (v2, w2) to H, contradicting the maximality of E ( H ) . Therefore, every vertex v in H must have d H ( v ) b k - 1. 0

+

Corollary 15. For any k-regular graph G , k 0, X,(G) s [ k / 2 ]+ 1. Furthermore, there is a subcoloring of G with [ k / 2 ]+ 1 colors in which every color class is a disjoint union of only KI’s and K2’s. Proof. A 0-regular graph G is simply a disjoint union of K,’s; hence X,(G) = 1= [0/2] + 1. A 1-regular graph G is a disjoint union of K2’s; hence X , ( G ) = 1 = [ 1 / 2 ]+ 1. A 2-regular graph G is a disjoint union of cycles. From Proposition 1 it follows that X,(G) S 2 = [ 2 / 2 ]+ 1. For k 2 3 we apply the proof of Theorem 14 as follows. Let H be a ( [ k / 2 ] 1)-colorable, spanning subgraph of C having a maximum number of edges. If H = G , we are done. Suppose H Z G . By the proof of Theorem 14, d H ( v )b k - 1 for every vertex v of G. Let the vertices of H be colored with [ k / 2 ]+ 1 colors, let v be a vertex with d H ( v ) = k - 1 and let w be the vertex adjacent to v in G but not in H. Then, as in the proof of Theorem 14, it follows that c(v) = c(w). Thus, if we add the edge (v, w ) to H we will produce a subcoloring of a spanning subgraph H‘ in which the set of vertices assigned color c(v) is a disjoint union of only K l ’ s and K 2 k . If H’ = G , we are done; if

+

46

M.0.Albertson et al.

not, we repeat the above argument for some vertex v' of G with d H . ( u ' )= k - 1, and so on. Eventually we obtain a subcoloring of G with [ k / 2 ] 1 colors, in which every color class is a disjoint union of only K,'s and K,'s. 0

+

Corollary 15 enables us to produce another large class of 2-subchromatic graphs.

Corollary 16. For every cubic, i.e. 3-reguIar, graph G , X , ( G ) s 2. Corollary 17. For every graph G with maximum degree A, X,(G) S [A/2]

+ 1.

We conclude this section with the following results which establish another upper bound for the subchromatic number. Let f ( n ) = max{X,(G): IV(G)l = n}.

Lemma 18. f ( n )

i1+f(n - [log,

n]).

Proof. First observe that the Ramsey number r ( k , k ) < 4k, which follows from the facts that r(k, 1) = r(1, k ) = 1, and r(k, 1) < r ( k , I - 1) < r(k, 1 - 1). By induction one can show that r(k, I) < 2k+' (cf. Bondy and Murty, p. 104 [3]). Let G be a graph on n vertices, 4k S n < 4k+1.Then G has an independent set of size k or a clique of size k and k = [log, n ] . Then f ( n ) 6 1 + f ( n - [log, n]). 0 Using Lemma 18 one can prove the following: we omit the details.

Theorem 19.

[fi] S f ( n ) G n/(log,

n

- 1)

+ O(n/(log4 n)').

5. Further directions, open problems An interesting refinement of the concept of a subcoloring is the following. Given a graph G, a partition of the vertices into t sets V , , V,, . . . , V, is called an (I,, I*, . . . , r,)-subcoloring if the induced subgraphs ( V , ) consist of disjoint

Fig. 4. Graphs with no (1,2)-subcoloring.


47

I

Fig. 5 . XJK, X K6) 5.

complete subgraphs, each component of which has cardinality no more than ri. A standard coloring is then a (1, 1, . . . , 1)-subcoloring. It follows from Corollary 15 that every k-regular graph has a ( 2 , 2 , . . . ,2)-subcoloring with [k/2] + 1 colors. Thus, for example, every cubic graph has a (2,2)-subcoloring. This raises the question: which cubic graphs have ( 1 , 2 ) -subcolorings? For example, the Petersen graph has a (1,2)-subcoloring, but the two graphs in Fig. 4 do not. One can show without too much difficulty that if G is toroidal, then X , ( G ) 6 6. However, we believe that X,(G) G 5 holds for all toroidal graphs; in fact, we have not been able to construct a toroidal graph that cannot be 4-subcolored. A number of results can be obtained about the subchromatic number of various products of graphs. For example, for the join of two graphs one can see that max{X,(G), X , ( H ) } GX,(G + H) G X , ( C ) + X , ( H ) . For the Cartesian product G X H, where V(G X H) = V ( G )x V ( H ) and E(G x H) = { ( ( g , , hl), ( g 2 , h 2 ) ) : g l= g 2 and h l h 2 in E ( H ) or h , = h2 and glg2 in E ( G ) } , one can show that: X,(G x H) 6 min{max{X,(G), X ( H ) } , max{X(G) X , ( H ) } 1. The subcoloring in Fig. 5, however, shows that even for complete graphs this bound can be inexact. Finally, we ask, can one bound the subchromatic number of (i) a minimally 2-connected graph? (ii) a chordal graph? or (iii) a perfect graph? Acknowledgements The authors would like to thank the referees for their extremely detailed comments and excellent suggestions which greatly improved the presentation of this paper. References [l] J.A. Andrews and M.S. Jacobson, O n a generalization of chromatic number, Congr. Numer. 47 (1985) 33-48.

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[2] J.A. Andrews and M.S. Jacobson, On a generalization of chromatic number and two kinds of Ramsey numbers, Ars Combin. 23 (1987) 97-102. [3] J.A. Bondy and U.S.R. Murty, Graph theory with Applications (American Elsevier, New York, 1976). [4] M. Borowiecki, Point partition numbers and generalized Nordhaus-Gaddum problems, Colloq. Math. 47 (1984) 279-285. [S] I. Broere and C.M. Mynhardt, Generalized colorings of outerplanar and planar graphs, Graph Theory with Applications to Algorithms and Computer Science (Kalamazoo, Mich., 1984) 151-161 (Wiley, New York, 1985). [6] R.L. Brooks. On colouring the nodes of a network, Proc. Cambridge Philos. SOC. 37 (1941) 194- 197. (71 J.L. Brown and D.G. Corned, On generalized graph colorings, J. Graph Theory 11 (1987) no. 1, 87-99. [8] G. Chartrand, D. Geller and S. Hedetniemi, A generalization of the chromatic number, Proc. Cambridge Philos. SOC.64 (1968) 265-271. [9] G . Chartrand and H. Kronk, The point arboricity of planar graphs, J . London Math. SOC.44 (1969) 612-616. [lo] G. Chartrand, H. Kronk and C. Wall, The point arboricity of a graph, Israel J. Math. 6 (1968) 169- 179. [11] E.J. Cockayne, Colour classes for r-graphs, Canad. Math. Bull. 15 (1972) 349-354. [12] E.J. Cockayne and G.G. Miller, An interpolation theorem for partitions which are complete with respect to hereditary properties, J. Combin. Theory Ser. B 13 (1972) 290-296. [13] R.J. Cook, Point partition numbers and girth, Proc. Amer. Math. SOC.49 (1975) 510-514. [14] G.S. Domke, R. Laskar, S.T. Hedetniemi and K. Peters, The partite-chromatic number of a graph, Congr. Numer. 53 (1986) 235-246. [15] P. Erdos and A. Hajnal, On chromatic number of graphs and set-systems, Acta. Math. Acad. Sci. Hungar. 17 (1966) 61-99. [16] J. Folkman, Graphs with monochromatic complete subgraphs in every edge-colouring, SIAM J. Appl. Math. 18 (1970) 19-24. [17] F. Harary, Graph theory (Addison-Wesley, Reading, Mass., 1969). [18] F. Harary, Conditional colorability in graphs, Graphs and Applications, Proc. First Col. Symp. Graph Theory (Boulder, Colo., 1982) 127-136 (Wiley-Intersci. Publ., Wiley, New York, 1985). [19] F. Harary and S. Hedetniemi, The achromatic number of a graph, J. Combin. Theory 8 (1970) 154-161. [20] F. Harary, S. Hedetniemi and G. Prins, An interpolation theorem for graphical homomorphisms, Portugal. Math. 26 (1967) 453-462. [21] F. Harary and P.C. Kainen, On triangular colorings of a planar graph, Bull. Calcutta Math. SOC. 69 (1977) 393-395. [22] P.J. Heawood, Map colour theorems, Quart. J. Math. 24 (1890) 332-338. [23] S. Hedetniemi, On partitioning planar graphs, Canad. Math. Bull. 11 (1968) no. 2, 203-211. [24] S. Hedetniemi, Disconnected colorings of graphs. Combinatorial Structures and Their Applications, (Proc. Calgary Internat. Conf., Calgary, Aka., 1969) 163-167 (Gordon and Breach, New York, 1970). [25] A.B. Kempe, On the geographical problem of four colors, Amer. J . Math. 2 (1879) 193-204. [26] F. Kramer, Sur le nombre chromatique K ( 9 , G) des graphes, Rev. Francaise Automat. Informat. Recherche Operationnelle 6 (1972) Ser. R-1, 67-70. [27] F. Kramer and H. Kramer, Un probleme de coloration des sommets d’un graphe, C.R. Acad. Sci. Paris Ser. A-B 268 (1969) A46-A48. [28] F. Kramer and H. Kramer, Ein Farbungsproblem der Knotenpunkte eines Graphen bezuglich der Distanz p, Rev. Roumaine Math. Pures Appl. 14 (1969) 1031-1038. [29] L. Lesniak-Foster and J . Roberts, On Ramsey numbers and graphical parameters, Pacific J . Math. 68 (1977) no. 1, 105-114. [30] L. Lesniak-Foster and J . Straight, The cochromatic number of a graph, Ars Combin. 3 (1977) 39-45. [31] D.R. Lick, A class of point partition numbers, Recent Trends in Graph Theory, (Proc. Conf. New York, 1970). Lecture Notes in Mathematics, Vol. 186, 185-190 (Springer, Berlin, 1971).


49

[32] D.R. Lick and A.R.White, k-degenerate graphs, Canad. J. Math. 22 (1970) no. 5, 1082-1096. [33] L. LovBsz, On chromatic number of finite set systems, Acta Math. Acad. Sci. Hungar. 19 (1968) 59-67. [34] J. Mycielski, Sur le coloriage des graphes, Coll. Math. 3 (1965) 161-162. [35] C.M. Mynhardt and I. Broere, Generalized colorings of graphs, Graph Theory with Applications to Algorithms and Computer Science, (Kalamazoo, Mich., 1984) 583-594 (Wiley-Intersci. Publ., Wiley, New York, 1985). [36] F.S. Roberts, Indifference graphs, Proof Techniques in Graph Theory, F. Harary, Ed., 139-146 (Academic Press, New York, 1969). [37] H. Sachs and M. Schauble, Konstruktion von Graphen mit gewissen Farbungseigenschaften, Beitrage zur Graphentheorie, (Kolloquium, Manebach, 1967) 131-136 (Teubner, Leipzig, 1968). 1381 E. Sampathkumar, Prabha S. Neeralagi and C.V. Venkatachalam, A generalization of the chromatic and line chromatic numbers of a graph, J. Karnatak Univ.: Sci. 22 (1977) 44-49. [39] G. Szekeres and H.S. Wilf, An inequality for the chromatic number of a graph, J. Combin. Theory 4 (1968) no. 1, 1-3.



51

CONNECTED SEQUENTIAL COLORINGS A. HERTZ and D. de WERRA Ecole Polytechnique Fidirale de Lausanne, Switzerland

Let us consider a graph G = (V, E). A k-coloring (Sl, . . . , S,) of its nodes is called canonical if any node u E V of any color i is contained in a clique K of size i such that K f l S, # 0 for l s j s i . A connected order on a connected graph G = (V, E ) is any order u I f IV(G)l. and a second to the effect that a regular simple graph G with d ( G ) 3 IV(C)l is 1-factorizable. We set out the evidence for both these conjectures and show that the first implies the second.

1. Introduction We are concerned here with simple graphs, that is finite graphs without loops or multiple edges. An edge-colouring of a graph G is a map 4: E ( G ) + %',where %' is a set of colours, such that no two vertices with the same colour have a common vertex. The chromatic index x ' ( G ) is the least value of l%'l for which an edge-colouring exists. A well-known theorem of Vizing [17] states that A ( G ) s x ' ( G )s A ( C ) + 1, where A ( G ) denotes the maximum degree of G. If ~ ' ( ( 3= ) A ( G ) , then G is said to be Class 1, and otherwise G is Class 2. The question of deciding whether or not a graph is Class 1 was shown by Holyer [14] to be NP-complete. However, for certain types of graph, the problem of classifying Class 2 graphs seems to be tractable. If G satisfies the inequality

then G is overfull. Clearly if G is overfull, then IV(G)l is odd. An overfull graph has to be Class 2, since no colour class of G can have more than 1; JV(G)] edges. In [6], Chetwynd and Hilton made the following conjecture (now slightly modified).

Conjecture 1. Let G be a simple graph with A(G) > f IV(G)l. Then G is Class 2 if and only if G contains an overfull subgraph H with A ( H ) = A(G). The graph C obtained from Petersen's graph by removing one vertex is Class 2, but contains no subgraph H with A ( H ) = A ( G ) ; this shows that the figure 3 in Conjecture 1 cannot be lowered. 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

62

A.J.W . Hilton

At the time of writing, Conjecture 1 has been proved in a number of cases. Plantholt ( [ 1 5 , 161) and Chetwynd and Hilton ([3-61) have between them established the following.

Theorem 1. Conjecture 1 is true

if A(G) 3 I V(C)l- 3.

In [ l ] , Chetwynd and Hilton posed the following conjecture about regular graphs of even order. First note that a Class 1 regular graph is often called l-factorizable, as it is the union of edge-disjoint l-factors. Also note that a regular graph of odd order is overfull, and so is Class 2 . If a graph G is regular, let d ( G ) denote its degree.

Conjecture 2. Let C be a regular simple graph of even order satisfying d ( G ) 3 1 IV(C)l. Then C is l-factorizable. This conjecture seems to have been known however long before being posed by Chetwynd and myself. When I told Dirac of it, he said it was “going around” in the early 1950s. The figure 12 IV(C)l in the conjecture cannot be lowered, as is shown by the example of a graph C consisting of two K,’s, when n is odd. Chetwynd and Hilton ( [ 1 , 7 , 8 ] )have proved this conjecture in a number of special cases.

Theorem 2. Conjecture 2 is true if either d ( G )3 4

( f i - 1 ) IV(G)l

or d ( G ) 3 IV(C)l- 4. The object of this note is to prove the following theorem.

Theorem 3. If Conjecture 1 is true, then Conjecture 2 is true.

2. Proof of Theorem 3 Let C be a regular graph with IV(C)l = 2n and d ( C ) a a . Suppose that Conjecture 1 is true and that G is Class 2. Let H be an overfull subgraph of G with A ( H ) = d ( G ) . Since H is overfull, it follows that IV(H)I is odd, so H # G. Let def(H) =

2 tJtV(H)

( d ( G )- d H ( V ) ) .

Two conjectures on edge-colouring

63

It is shown in [2] that, if H is overfull, then def(H) S A ( H ) - 2 = d ( C ) - 2. It follows that G has an edge-cut S with )SI G d ( G )- 2 such that G\S = H U J , where V ( H )f Vl ( J )= 9. Since A ( H ) = d ( G ) > n , it follows that H has at least n 1 vertices. Consequently J has at most n - 1 vertices. Thus d ( G ) 1> IV(J)(.Since C is regular, the number of edges joining vertices of J to vertices of H is at least ( d ( G )- IV(J)I 1) IV(J)I. For fixed d ( G ) , ( d ( C )- IV(J)(+ 1) IV(J)I is a quadratic in IV(J)(.In the range 1 s IV(J)(S n - 1, it has two minima, one at each end point, with values d ( G ) and ( d ( G ) - n + 2 ) ( n - 1). But d ( G ) > IS[, and ( d ( C )- n 2)(n - 1) 2 2n - 2 3 d ( G ) - 1> IS(, contradicting the definition of S. Thus C has no overfull subgraph H , and so, by Conjecture 1, is Class 1, or in other words is 1-factorizable. Thus Conjecture 2 is true. This proves Theorem 3.

+

+

+

+

3. A final remark Conjecture 1 has many other implications. Some of these are discussed in [ll-131 by Hilton and Johnson. A survey of the main implications is given in [lo]. See also [9].

Note added in proof A.G. Chetwynd and I have recently proved Conjecture 1 in the case when A G ~ = a ( f i - l)(lV(C)l + 1)+ 1 and IE(C)l= A(G)Li IV(G)ll. See [18].

References [l] A.G. Chetwynd and A.J.W. Hilton, Regular graphs of high degree are 1-factorizable, Proc. London Math. SOC.(3), 50 (1985) 193-206. [Z] A.G. Chetwynd and A.J.W. Hilton, The edge-chromatic class of graphs with large maximum degree, where the number of vertices of maximum degree is relatively small, J. Combinatorial Theory (B) to appear. [3] A.G. Chetwynd and A.J.W. Hilton, Partial edge-colourings of complete graphs or of graphs which are nearly complete, Graph Theory and Combinatorics. Vol. in honour of P. Erdos’ 70th birthday (1984) 81-98. [4] 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. [5] A.G. Chetwynd and A.J.W. Hilton, The edge-chromatic class of graphs with maximum degree at least IVI - 3. Proceedings of the conference held in Denmark in memory of G . A . Dirac, to appear. [6] A.G. Chetwynd and A.J.W. Hilton, Star multigraphs with three vertices of maximum degree, Math. Proc. Camb. Phil. SOC.100 (1986) 303-317.

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A.J. W. Hilton

[71 A.G. Chetwynd and A.J.W. Hilton, The edge-chromatic class of regular graphs of degree 4 and their complements, Discrete Applied Math. 16 (1987) 125-134. 181 . . A.G. Chetwynd and A.J.W. Hilton, 1-factorizing regular graphs of high degree -an improved bound, submitted. [9] A.G. Chetwynd and A.J.W. Hilton, A A-subgraph condition for a graph to be Class 1, J. Combinatorial Theory (B) to appear. [lo] A.J.W. Hilton, Recent progress on edge-colouring graphs, Discrete Math. 64 (1987) 303-307. [ l l ] A.J.W. Hilton and P.D. Johnson, Graphs which are vertex-critical with respect to the edge-chromatic number, Math. Proc. Camb. Phil. SOC., 102 (1987) 211-221. [12] A.J.W. Hilton and P.D. Johnson, Graphs which are vertex-critical with respect to the edge-chromatic class, submitted. [13] A.J. Hilton and P.D. Johnson, Reverse class critical multigraphs, Discrete Math. 69 (1988) 309-311. [14] I.J. Holyer, The NP-completeness of edge-colourings, SIAM J. Computing 10 (1980) 718-720. [15] M. Plantholt, The chromatic index of graphs with a spanning star, J. Graph Theory 5 (1981) 5-13. [16] M. Plantholt, On the chromatic index of graphs with large maximum degree, Discrete Math. 47 (1983) 91-96. [17] V.G. Vizing, On an estimate of the chromatic class of a p-graph (in Russian), Diskret. Analiz. 3 (1964) 25-30. [l8] A.G. Chetwynd and A.J.W. Hilton, Star multigraphs with r vertices of maximum degree, submitted.


65

A NEW UPPER BOUND FOR THE LIST CHROMATIC NUMBER B. BOLLOBAS and H.R. HIND? Dept. of Pure Mathematics, Univ. of Cambridge, 16 Mill Lane, Cambridge, U.K . CB2 1SB For large values of A, it is shown that all A-regular finite simple graphs possess a non-trivial vertex partition. This is then used to show that for finite simple graphs of maximal degree A(G) = A, the list chromatic number is bounded by z;(G) s 7A/4 o(A).

+

Given a graph G and for each edge of G a set (list) of colours, we call an assignment of a single colour to each edge of G a list colouring if the colour assigned to an edge is in the list of colours associated with that edge and no two adjacent edges are assigned the same colour. The list chromatic number of the graph C, x;(G), is defined to be the minimal positive integer such that if for each edge of G , the list of colours associated with that edge has size at least x;(G), then there exists a list colouring for G for any choice of lists. The list colouring conjecture, which has been attributed to various sources, states x;(C)s x’(C). Little progress has been made towards establishing this conjecture, even for special classes of graphs. Bollobfis and Harris [2] have shown that for graphs with maximal degree A ( C ) = A, x ; ( G )S 11A/6 + o(A) and Chetwynd and Haggkvist [3] have shown that for triangle free graphs x;(G)s 9A/5. In this paper we combine the approaches used in the above two papers and the existence of a specific partition for regular graphs to obtain an improved upper bound for the list chromatic number. The term graph will be used to mean a finite simple graph. Before proving the intended result, we need a few definitions. Given a graph G, let A: E ( G ) + P(N) be an arbitrary function. Then we define a A-colouring of graph G to be a function 9 such that

where

and

# ( e ) # @ ( e ‘ ) if edges e and e’ are adjacent.

t Supported by ORS grant ORS/84120 and CSIR grant 9/8/1-2019. 0012-365X/89/$3.50 @ 1989, Elsevier Science Publishers B.V. (North-Holland)

66

B. Bollobis, H . R . Hind

The more general name list colouring is applied to such a colouring where the function A is not specifically mentioned. The function A can be thought of as assigning a list of acceptable colours, A(e), for the edge e. We refer to the function A as the list function for the graph. For a more detailed discussion of list colourings, see [2]. For simplicity we let A, denote a function A, :E( G )+ N(').

For a given list function, A, we define a partial list colouring of G to be a function, I) say, which has an associated edge of G, e* say, such that

q : E(G)\{e*}-. N where V(e)E A(e)

Ve

E

E(G)\{e*},

and

q ( e ) # q ( e ' ) if edges e and e' are adjacent. If we think of the function @ as a colouring of the edges of the graph G, then q is a colouring of all but one of the edges of G. Let E ( v ) be the set of edges incident to vertex v and for a partial list colouring q, with associated edge e*, define

3 :V ( G ) +

6 N(j)

i=l

where

$(v)

= { V ( e ) : eE E(v)\{e*}}.

The definitions given below were first given in [3], but are restated here for convenience. Let G be a graph with list function A and a partial list colouring q. Let the associated edge for q be e*. An edge uv (where uv # e * ) is said to be a floppy edge if

IA(uv>\(3(u>u

3(v))la 1

i.e. if there exists a partial list colouring, q* say, with the same associated uncoloured edge, e*, as q, such that

q*(e)= q(e)

Vefuv

or e*

and

q*(uv)E A ( u v ) \ { V ( u v ) } . We call the colour q*(uv) an escape colour for edge uv. Later, with slight abuse of notation, we refer to the escape colour of a floppy edge; here having assigned a partial list colouring to the graph under consideration, we chose one

New upper bound for list chromatic number

67

(of the one or more) escape colours and assign this fixed colour as the escape colour of the edge. By considering partitions of A-regular graphs, we obtain an upper bound on the list chromatic number. We recall the Erdos-Lovfisz Theorem (see [ l ] , page 22).

Theorem 1. Let A l , A 2 , . . . , A,,, be events with dependence graph F. If F has maximal degree A z=3 and P(AJ s

(A - l)A-l AA

then

We use this theorem to show that A-regular graphs are vertex partitionable with vertex disjoint classes Vl, V2,. . . , V, such that for x E V , there are non-trivial upper and lower bounds on the cardinality of r ( x ) r l V, (denoted by d,,(x)) for all j # i. As is customary, we let Sn,pbe the sum of n independent Bernoulli random variables, with value one or zero, where each Bernoulli random variable has value one with probability p . Then we see P(S,,, = k ) = ( ; ) p * ( l - p y k

Definition. For a given A and r, we call a set of ordered pairs D = {(d,j, Ajj)lsj,jsr,i+j} a (A, 0)-acceptable set if (i) O s S , G A j j s A f o r l s i , j s r a n d i Z j , and (ii) there exists a set P = { p l , . . . ,p r } (called the set of probabilities) with O < p i < l , C I = , p j = 1, such that

The theorem below follows from Theorem 1.

Theorem 2. Let D = {(aij,Ajj)lej,jsr,i+j}be a (A, 0)-acceptable set. Then every A-regular graph has a partition, V ( G )= UTZlK with V, n = 0 if i # j and every vertex in V;: is adjacent to at least 6,j and at most A, vertices in V,. Proof. Let G be a A-regular graph and D = {(aij,Aij),ei,jsp,i+j}be a (A, 0)acceptable set. Let P = { p l , . . . ,p , } be the associated set of probabilities for D.

68

B. Bollobh, H . R . Hind

Take a random partition of the vertex set V ( G ) into disjoint classes V,, . . . , V,, by putting a vertex into class V, with probability pi. Let A, be the event that the condition of the theorem is violated for vertex x , i.e. there exist i, j (i # j ) such that x

E

V , and d v l ( x ) < 6,

or dvl(x)> A,.

We note that event A, is independent of the system {A, :dG(x, y) 2 3}, that is to say the set of events A,, such that x and y are not adjacent and have no common neighbours. The event A, is thus independent of the system of all events {A, :y E V ( G ) }from which at most A2 1 events have been omitted. From the choice of event A, we get

+

r

P(A,) =

r

c c

Pi(P(sA,pl < 6i,) + P(sA,pl > Aij))?

i=l j=l,j#i

but D is a (A, 0)-acceptable set with set of probabilities P, so by condition (ii) of the definition,

It now follows from the Erdos-Lovisz Theorem that 0

Definition. For a given A and r, we call a set of ordered pairs D = {(dij, Aij)l 100 log A), in the left-hand side of the proposed inequality are easily seen to be o ( A - ~ ) An . upper bound for P(SA,,, > 100 log A ) is available (see [l], page 14), namely

and with u = 12.5, it follows that

Thus we have that

and (A2)A2 =0(A-3) (A2 1)'

+

+

So for sufficiently large A , the inequality has been established.

0

Before turning to the main result of the paper, we make two preliminary observations.

70


Remark 1. In the proof below which produces a list colouring of a graph G, for a given list function A,, we may assume that a partial list colouring exists. Suppose G does not have a partial list colouring. Let El c E ( G ) be a maximally sized edge set such that for each edge e E El we can choose a colour, O(e) say, with O(e) E A , ( e ) such that e ( e ) # e ( e ’ ) if e and e’ are distinct adjacent edges contained in the set E l . Thus GIEl] is the largest edge induced subgraph of G for which a A,-colouring exists. Then choose an edge e’ E E(G)\EI and define a new list function A;: E(G)-+ 9(N) such that A;(f) = A,(f) for all f E E l U { e ‘ } , and for e E E(G)\(E, U { e ’ } ) , the sets A ; ( e ) are disjoint and do not intersect the set Uf.E,u(e’) 4 ( f )* Clearly G has a partial list colouring for the list function A;. Therefore if we show that a list colouring exists for G with list function A;, this implies that our choice of El was not maximal. Thus without loss of generality we may suppose that G has a partial list colouring for list function A,.

Remark 2. In the proof below when showing the existence of a list colouring for a graph G, we may assume G is A-regular. If G is not A-regular we may create a new A-regular graph G’ where G E G’, by adding vertices and edges to the graph G. We define a new function A; :E(G‘)+ N(‘) such that A;lE(G) = A, and for each e E E(G’)\E(C) we choose 1-sets, A ; ( e ) , such that A ; ( e ) f l A , ( e ’ )= 0 for all e’ E E(G’)\{e}. Then we obtain a A-regular graph G’ such that G is list colourable with list function A, iff G’ is list colourable with list function A;.

Theorem 5. If A is sufficiently large, then for every graph G with maximum degree A(G) = A, we have

Proof. Suppose G is a graph of maximal degree A, that 1 > 7A/4 + 125 log A] and A,: E(G)-, N(‘) is an arbitrary function. We shall show that there exists a A,-colouring for G. From the remarks above, we may assume G is A-regular and there exists a partial list colouring for G. We choose V Oto be a partial list colouring with a maximal number of floppy edges. Let the associated uncoloured edge be denoted aobo. For each vertex x E V ( G ) ,let { y l , y2, . . . , Y , , , ( ~ ) be } the set of those vertices y adjacent to the vertex x such that xy is not a floppy edge of the partial list colouring q0. If m ( x ) 3 A/2 define U ( x ) = { y l , y2, . . . ,yrN21} and if m ( x ) < A/2 let U ( x ) = { y l , . . . y m d u { Z , , , ( ~ ) + I , . . . , q A / 2 1 ) , where z , , , ( ~ ) + ~., . . , 2 [ A / 2 1 are any other vertices adjacent to the vertex x.


71

For a set W c V ( C )and a vertex x , let d,(x) be the number of neighbours of x in the set W. By Corollary 3 and Lemma 4 above, there exists a partition of V ( C ) into V, and V, such that for each x E V,,

1 6 d&)

6

100 log A, and

v, n U ( X )# 0, and for each x E V2, 1 s d,,(x) s A,

and

v,n U ( X )z 0. We distinguish three cases depending on the location of the endvertices of the uncoloured edge sob,. Case (1). a, E V, and b~ E V,.

We create a sequence of partial list colourings I)(,, $,, . . . and their associated uncoloured edges aOb,,,a , b,, . . . . Having defined I),-,and edge a,-, b ,-,, choose q, and the associated uncoloured edge a,b, such that (1) I), is a partial list colouring, (2) a, = b , - , , (3) 6, E VIP (4) I),(e) = I),-,(e) if e E E ( G ) \ { a , - l k l , arb,), and I),(a,-lb,-,) = I),-,(a,h), ( 5 ) v,(a,-,bz-d (2 $,(a,-A (6) $ ~ , ( a , - , b , -is~not ) the escape colour of a floppy edge incident to a , - , , and (7) arbr# a,b, for any j < i. Condition (6) ensures that the number of floppy edges in v, IS not less than that is chosen so that the number of floppy edges is maximal, in I),-,and since equality must hold. Condition (7) implies that for a finite graph the sequence of partial list colourings constructed in this way is finite, say it ends with v,. To simplify the notation let ab = asb, be the associated uncoloured edge. Condition (7) also ensures that if a colour is not in $()(v) but is in $J,(v)for i >0, then it is in $,(v) for all j a i . We note that since I)() has a maximal number of floppy edges, it is sufficient to show that either there is a list colouring of the graph G or there is a partial list Suppose neither case applies. colouring I),with more floppy edges than I)(). Given C with a partial list colouring I),, if e is a floppy edge we let v : ( e ) denote the escape colour assigned to edge e. Then for the partial list colouring v, and a vertex v E V ( G ) ,let F ( I ) , , v ) be the set of colours assigned to floppy edges incident to vertex v, H ( I ) , , v ) be the set of escape colours assigned to floppy edges incident to v, K ( + , , v ) be the set of new colours used to colour edges incident to v , S(q,, v ) be the set of colours always used to colour an edge incident to v, and R ( v , , v ) be the set of colours originally used to colour an edge incident to v , but which have since been removed. Finally let S’(I),, v ) be the set

u,,

H(Vi, v)= {VT(uv):IA(uv)\(3i(v) U $ J i ( u ) ) l a1) and

K(Vi,

21)

= +;(v)\$dv),

S( Vi, v) = 3i(v)n 30(v),

R(Vi, v ) = 3idv)\3i(v), S'(Vi, v) = {Vi(uv): V i ( u v ) E 3 d v ) and Vi(uv)

3idu)).

For simplicity the partial list colouring qi is omitted from the notation for the above sets where no ambiguity can arise. Further we let f ( v ) =f(Vi, v ) = IF(q;, v)l, h(v) = h(Vi, v ) = IH(Vi, v)l and SO on. Since we cannot extend the sequence of partial list colourings beyond vs it follows that all the colours in A/(&) must be in the union of (9 30(a) or 3s(a)9 (ii) the set of colours assigned to edges ajbj with j < s, which are incident to vertex b, (iii) the set of escape colours of floppy edges incident to vertex a, and (iv) the set of colours assigned to edges of the form bx where x E V,. Writing D ( b ) for this last set of colours and d ( b ) for its cardinality (from the choice of the partition, it follows that d ( b ) 6 100 log A), it follows that A/(ab)E ( R ( a )U S ( a ) U K ( a ) ) U ( K ( b )U S'(b))U H ( a ) U D ( b ) , Setting K ' ( b ) = K ( b ) f l A/(ab),we get

Al(ab) E (R(a)U S ( a ) U K ( a ) )U ( K ' ( b )U S ' ( b ) )U H ( u ) U D ( b ) = ( R ( a )U S ( a ) U K ( a ) ) U S ' ( b ) U H ( a ) U D ( 6 ) since K ' ( b ) E (R(a)U $ ( a ) ) = &(a): the edge ab would have been defined as a floppy edge if a colour in Al(ab)was not the colour of an edge incident to either vertex a or vertex b in the partial list colouring V,. We also note that IS'(b)l= IK(b)l= k ( b ) since a colour is added to the set K ( b ) every time the vertex b occurs as the endvertex aj of an edge ajbj in the sequence of uncoloured edges and a colour already used to colour an edge incident to the vertex b is used to colour a new edge incident to vertex b (i.e. added to S ' ( b ) )every time b occurs as the endvertex bj of such an edge ajbj. These events occur in pairs. Thus we get

+

1 6 r ( a ) s(a)

+ k ( a )+ k ( b ) + h ( a )+ d ( 6 )

(5.1)

Clearly

+

r ( a ) s(a) 6 A,

(5.2)


13

and each colour in A,(ab)must be assigned by qsto a non-floppy edge incident to vertex a or vertex b, so

f ( a ) + f ( b )zs 2 4 - 1. Since the number of escape colours assigned to floppy edges incident to vertex a is at most the number of colours assigned to those floppy edges, a crude upper bound for h ( a ) is

h ( a ) S 2 4 - 1.

(5.3)

It remains to obtain bounds for k ( a ) and k ( b ) . With f ( a ) bounded above by 2 4 - 1 < A/2, our choice of partition for the graph G, ensures that there exists a non-floppy edge ac such that c E V,. At most A of the colours in Al(ac) are in qo(c) and further qo(c)= qs(c) so at least 1 - A colours in A,(ac) must be in qn(a). Since these colours must also be in q j ( a ) for each 0 6 j S s (or we obtain a new floppy edge ac), it follows that

s(a)sl- A, and (with a similar argument for vertex b ) k(a)S 2 4 -1

k ( b ) S 2 4 - 1.

(5.4)

Recalling that d ( b ) S 100 log A = o ( A ) and substituting the bounds (5.2), (5.3) and (5.4) into inequality (5.1) we get 16

A

+ ( 2 4 -1) + ( 2 4 - 1) + ( 2 4 -1) + 100 log A

= 7 A - 31

+ 1001Og A

or

74 1 s - + 25 log A, 4 which is a contradiction. Since qowas chosen to be a partial list colouring for G with a maximal number of floppy edges, this contradiction shows that there is a A,-colouring of G for list function A,.

Case 2. a. E V, and bo E V,. We choose an edge a& such that ah = uo, bh E V, and function

q;: E(G)\{ahbb} + N such that qh(e) = q o ( e ) for all e E E(G)\{aobo, ahbb} and q;(aobn) = qo(a;b[)) and such that qh is a partial list colouring with at least as many floppy edges as for partial list colouring qo.This case then reduces to Case 1. exists. We define the sets F ( q , , v ) We need only show that such an edge and H ( q o , v ) and the cardinalities f(qo,v) and h ( q o ,v ) as in Case 1. Suppose


74

such an edge a&, does not exist, then

Ar(aobo) E $"(b") u WVO, bo) u W o )

so 1 A + ~ ( V Obo) , + d(ao) or

h(qo, b,)

3 1 - A - o(A).

Since the number of floppy edges incident to vertex 6,) must be at least as large as the number of escape colour assigned to these edges,

f(Vo, bo) 3 (1 - A - 4 A ) ) . Then for sufficiently large A, at most 2 4 - (1 - A - o(A)) = 3 4 - I - o ( A ) < 1 non-floppy edges are incident to vertices a, and bo, so there exists a colour in A,(aobo)assigned to either only floppy edges incident to vertices a, or b,), or not assigned to an edge incident to vertex a,) or vertex 6,. This gives us an immediate list colouring for G.

Case 3. a. E V2and b, E V,. Here we note that there exists a non-floppy edge from u E V, to some IJ E V, for every u E V, such that the number of floppy edges incident to vertex u is at most A/2. We define a sequence of partial list colourings qO,V1, . . . and associated edges a&,, a l b , , . . . , using the conditions listed below. Having I);-, and ~ ; - , b ; - ~ define 3, and aibi such that 9;is a partial list colouring, aI. = b.1-1, V , ( e ) = V ; - , ( e ) if e E E ( G ) \ { U ~ - ~ a~$~; }-, , and , V;(~;-lb;= - ~V) i - l ( ~ ; b ; ) , ~;(a;-lb;-I) Uj k. Now, observe that no vertex t of A ( x ) can be assigned an integer n > k , otherwise in A ( t ) there is a vertex z with integer k. But then the ordered P3{z, t, x } is of type 1, a contradiction. From the above observation, we may assume that x < y . If S = ( 3 ) then A ( y ) is a clique (the order is simplicial); thus we have F ( x , C) = k' - 1 = IA(y)l = w ( N ( x)) Now, we may assume that S = (2). By our previous observation, in A ( x ) , only integers m < k can appear, and thus by Lemma 4.2 I w ( A ( x ) ) l = k - 1. Next, observe that (i) t sees z whenever t E A @ ) , z E B ( x ) and (ii) B ( x ) is a clique (if (i) fails then G has a e3of type 1, and if (ii) fails then G has a P3 of type 3). Hence, we have w ( N ( x ) )= ( k - 1) + IB(x)l= F ( x , C ) . 0

-

Acknowledgement We would like to thank an anonymous referee for pointing out several errors in our original manuscript.

References [I] C. Berge and P. Duchet, Strongly perfect graphs, in Berge, C. and V. Chvital, Ed., Topics on Perfect Graphs (North-Holland, Amsterdam, 1984) 57-61. [2] C. Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind (Zusammenfassung). Math.-Natur. Reihe 114 (Wiss. Z. Martin-Luther-Univ., HalleWintenberg, 1961). [3] V. Chvital, Perfectly ordered graphs, in C. Berge and V. Chvital, Ed., Topics on Perfect Graphs (North-Holland, Amsterdam, 1984) 63-65. [4] V. Chvital, C.T. Hoing, N.V.R. Mahadev and D. de Werra, Four classes of perfectly orderable graphs, J. Graph Theory 11, 4 (1987) 481-495. [5] R. P. Dilworth, A decomposition theorem for partially ordered sets, Ann. Math. 51 (1950) 161-166.

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C.T. Hobng, N.V.R. Mahadev

[6] G. A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25 (1961) 71-76. [7] P. Duchet, Classical graphs, in C. Berge and V. Chvital, Ed., Topics on Perfect Graphs (North-Holland, Amsterdam, 1984), 67-96. [8] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs (Academic Press, New York, 1980). [9] A. Hertz, Slim graphs, Report ORWP 87/1, Swiss Federal Institute of Technology in Laussane (1987). [ 101 M. Preissmann, Locally perfect graphs, manuscript. [ l l ] E. S. Wolk, The comparability graph of a tree, Proc. Amer. Math. SOC.3 (1962) 789-795.


85

ON THE PENROSE NUMBER OF CUBIC DIAGRAMS Franqois JAEGER Laboratoire de Structures DkcrPtes, IMAG, BP 68, 38402 St Martin d 'HPres Cedex, France

A cubic diagram is a cubic graph G drawn in the plane, possibly with edge-crossings. The drawing defines a sign for each edge-3-coloring of G . The Penrose number of G is the sum of signs of its edge-3-colorings. For plane graphs it coincides with the number of edge-3-colorings. Given a cubic diagram G , we define a sign for every Eulerian orientation of its line-graph L ( C ) and prove that the Penrose number of G is equal to the sum of signs of the Eulerian orientations of L ( G ) . This yields a new recursive scheme for the computation of the Penrose number. Another consequence is a simple formula which gives the number of vertex-4colorings of a loopless plane triangulation in terms of the mappings from the vertex-set to { 1,2,3) which take exactly two distinct values on each triangle.

1. Introduction The Four Color Theorem [l] is equivalent to the statement that every cubic bridgeless plane graph C has an edge-3-coloring. The number T ( G ) of such edge-3-colorings can be computed using a recursive scheme corresponding to the evaluation of the chromatic polynomial of either the dual graph or the line-graph of G. Penrose proposes in [12] another recursive scheme for the computation of T ( G ) . In fact this scheme is not restricted to plane graphs. It works on cubic plane diagrams where edges may cross and computes a signed analogue of T ( C ) (we call this invariant the Penrose number) which coincides with T ( G ) if C is a plane graph. We present Penrose's work in Section 2. Then in Section 3 we establish a simple formula which gives the Penrose number of a cubic diagram in terms of the Eulerian orientations of its line-graph. Finally in Section 4 we show how this formula leads to a new recursive computation scheme for the Penrose number, and we give special results in the case of plane graphs.

2. The Penrose number of a cubic diagram 2.1. General definitions The graphs which we consider here are finite, and may have loops and multiple edges. It will be convenient to view each edge e as subdivided into two half-edges (the two halves of e ) , one incident to each end of e . In particular if e is a loop 0012-365X/89/$3.50 01989, Elsevier Science Publishers B.V. (North-Holland)

86

F. Jaeger

incident to the vertex u , both halves of e are incident to u . The degree of a vertex is the number of half-edges incident to it. A graph is cubic (respectively: 4-regular) if every vertex is of degree three (respectively: four). Let u be a vertex of degree 2 in a graph G , and let h l , h2 be the two half-edges incident to u . For i = 1, 2, hi is a half of some edge ej and we denote the other half of this edge by h i . Assume first that e l f e z . Then deleting u, e l , e2, h l , h2 and creating a new edge with halves h ; , h; yields a graph which is said to be obtained from G by erasing v. If e , = e2 = e (so that e is a loop at v), then h ; = h2 and h; = h l . In this case the erasing process just described replaces the “loop-graph’’ ( { u } , { e } ) by a structure consisting of two half-edges incident to no vertex. By convention we shall consider this as a graph with one edge and no vertex and call it a free loop. For instance a cubic graph or a 4-regular graph may contain connected components consisting of free loops. An orientation of a graph is obtained by choosing for each edge one of its halves as initial and the other as terminal. An orientation is said to be Eulerian if each vertex is incident to an equal number of initial and terminal half-edges. Note that a graph consisting of one loop or one free loop has exactly two Eulerian orientations. A rotation at a vertex u of a graph C = (V, E ) is a cyclic permutation of the half-edges incident to u . A rotation system of C is a family p = (p,,, u E V), where pu is a rotation at u . Rotation systems are the basis of the classical combinatorial description of embeddings of graphs on orientable surfaces known as the “permutation technique” (see [6]). To be more precise, each rotation system p = (p,,, u E V) of the connected graph G = (V, E ) defines an embedding of G on some orientable surface S. If we identify p with the permutation n I v E v p ,and , denote by a the fixed-point-free involution which maps every half-edge to the other half of the same edge, the faces of the embedding correspond to the orbits of pa. The number of these orbits will yield the genus of S by Euler’s formula. Other usual definitions on graphs will be found in [3] or [ 5 ] .

2.2. Diagrams Let G = (V, E ) be a graph. A drawing of G in the plane is obtained by representing each vertex u of C by a point u* of the plane and each edge e of C by a simple Jordan curve e* according to the following rules: - if u,, u2 are distinct vertices then u f # u: - if the edge e has ends v,,u2 (which might be identical) then e* joins uT and v: and contains no u* for u in V - { v l , u 2 } - if e l , e2 are distinct edges then e : n e,,*is finite.

Remarks. (i) Note that if we replace in this last rule “finite” by “contained in V*” (where V * denotes { v * / uE V}) we obtain the usual definition of plane embeddings. (ii) A free loop is drawn as a simple closed Jordan curve disjoint from V*.

Penrose number of cubic diagrams

87

(iii) In graph theory texts (including the present one where we state it explicitly), figures depicting graphs actually depict them as drawn in the plane. Ambiguity is avoided through the use of additional standard conventions which need not be incorporated in our mathematical definition of drawing. The subdivision of an edge e into its halves will be represented in the obvious way by the selection of a point on e* - V * (or two such points in the case of a free loop). To each drawing of the graph G in the plane we associate the rotation system p = ( p u , v E V) of G defined as follows: pv maps each half-edge incident to v to the next half-edge incident to v in the clockwise order around v*. Conversely, it is easy to see that every rotation system of G is associated in this way to some drawing in the plane. Thus plane drawings appear as a convenient description of rotation systems and equivalently of embeddings in orientable surfaces. In the sequel we call diagram a graph drawn in the plane. If the drawing is a plane embedding, the diagram is then called as usual a plane graph. For the sake of simplicity we no longer distinguish between a vertex v (an edge e) and its representative v * ( e * ) .

2.3. Edge-3-coloringsin cubic diagrams An edge-3-coloring of a cubic graph C is a coloring of the half-edges of G with 3 colors such that the two halves of any edge receive the same color (which we call the color of the edge) and the three half-edges incident to any vertex receive three distinct colors. We denote by T ( G ) the number of edge-3-colorings of G. For instance if G has a loop then T ( G )= 0. On the other hand, if G is a free loop, T ( G )= 3. We now define a signed analogue of T ( G )for diagrams which is due to Penrose [12]. Our presentation is slightly different from those given in [8, 91 or [12]. Assume that G is drawn in the plane and consider an edge-3-coloring f of G with the colors 1,2,3. A vertex v of G will be said positive (respectively: negative) with respect to f if the colors of the half-edges incident to v are 1 , 2 , 3 (respectively: 1 , 3 , 2 ) in the clockwise order around v. Let n + ( f ) (respectively: n - ( f ) ) denote the number of vertices which are positive (respectively: negative) with respect t o f . Note that n + ( f ) and n - ( f ) have the same parity since G has an even number of vertices. We write s(f) = 1 if n + ( f )= n - ( f ) mod4, and s(f) = -1 otherwise. The number s(f) is the sign o f f . We define the Penrose number of the diagram G as the sum of the signs of the edge-3-colorings of G , and we denote this number by T ’ ( G ) . Consider now another drawing of G whose associated rotation system is the same as the previous one except at one single vertex v where the rotation is opposite. Let f be any edge-3-coloring of G. The sign of v with respect to f changes, and the sign of the other vertices do not change. Hence the values of n + ( f )- n - ( f ) in the two drawings differ by 2 or -2, and the sign off changes. It

88

F. Jaeger

follows that the sign of T ’ ( C ) changes, but not its absolute value. In other words, IT’(G)l is a graph invariant: it depends only on C , not on the way it is drawn. 2.4. Basic properties of the Penrose number The Penrose number appears as an interesting tool for the study of edge-3colorings of cubic graphs, in particular in the planar case. This stems from the following results (see [8, 9, 121).

Proposition 1. Let G be a cubic plane graph. Then T ( G )= T ’ ( C ) . Proof. All edge-3-colorings of a cubic plane graph have positive sign. This result appeared for the first time (as far as I know) in [13] and was rediscovered by several authors in a number of equivalent forms. An elementary proof is as follows (see also [8, 91). Consider an edge-3-coloring f of C with the colors 1 , 2 , 3 . Replace each edge colored 3 by two parallel edges, one colored a and the other colored b , in order to obtain a plane 4-regular graph. In this new graph the edges colored a or 1 form a family C1of disjoint simple closed curves in the plane, and the other edges form another such family C2. Then it is easy to check that the ends of an edge of C colored 3 have the same signs with respect to f if and only if exactly one of them corresponds to a crossing of a curve of C1with a curve of C 2 . The total number of such crossings is even, thus there is an even number of edges of color 3 whose ends have the same sign, and hence n f ( f ) = n - ( f ) mod 4. 0 Remark. Proposition 1 cannot be extended to non-planar graphs: it is easy to check (see [12] and [7], Section 2.3) that the Penrose number of the Kuratowski graph K3,3is zero. Let G = (V, E) be a cubic graph drawn in the plane with associated rotation system p = (pu, v E V). As before we consider p as a permutation on the set of half-edges. Let e be an edge of G with distinct ends x , y and let h , h’ be the two halves of e . We define two new graphs H’ and H as follows: we first delete the edge e and its halves, and then rearrange the incidences of the half-edges incident to the vertices x and y in the following way. In H‘, ph and p2h‘ are made incident to one of these vertices, and ph’ and p2h are made incident to the other. On the other hand in H”, ph and ph’ are made incident to one vertex of { x , y } while p2h and p2h’ are made incident to the other (see Fig. 1). Note that in the graphs H’, H” all vertices have degree 3 with the exception of x and y which have degree 2. Hence if we erase x and y we obtain cubic graphs (recall that we allow free loops in our definition of such graphs). These graphs will be drawn in the plane, starting with the drawing of G and performing only a local change in the neighbourhood of e as indicated in Fig. 1. Clearly the associated rotation systems can be identified with (pu, v E V - { x , y}). The cubic


89

Fig. 1.

diagram thus obtained from H’ (respectively: H”) will be said to be obtained from G by the non-crossing (respectively: crossing) dissolution of the edge e.

Proposition 2. Let G be a cubic diagram, and e be an edge of G with two distinct ends. Let G’ (respectively: G”) be obtained from G by the non-crossing (respectively:crossing) dissolution of the edge e. Then T ’ ( C )= T’(G’)- T‘(G’’). Proof. Let H be the diagram obtained from G by the contraction of e. Thus all vertices of H have degree 3, with the exception of one vertex v which has degree 4.We may extend to H the definitions of edge-3-coloring and of the sign of such a coloring given in Section 2.3 by replacing everywhere the term “vertex” by the term “vertex of degree 3”. Then edge-3-colorings of G , G’, G” can be identified with special edge-3-colorings of H. For any partition P of the set of half-edges of H incident to v , let S ( P ) be the sum of the signs of the edge-3-colorings of H which induce this partition. Then clearly, using the above-mentioned identification: T’(G’)= S ({{p h ,p2h’), {ph’, p 2 h ) ) )+ S ({{p h ,p2h‘, ph’, p 2 h ) ) )

T’(G‘’)= S ( { { p h , ph’), {p2h,p 2 h ’ ) ) )+ S ({{p h ,p2h’,ph‘, p 2 h ) ) ) . Similarly, a simple analysis of the signs of the edge-3-colorings of G yields: T’(G)= S ( { { p h ,p2h’),{ph‘, p 2 h ) ) )- S ({{p h ,p h ‘), {p2h,p 2 h ’))), and the result follows immediately. 0

Fig. 2.

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Proposition 2, together with the easy fact that the Penrose number of a cubic diagram is equal to the product of the Penrose numbers of its connected components, allows the recursive computation of 7”(G) for any cubic diagram G. An example of such a computation is given in Fig. 2, where each diagram stands for its Penrose number.

3. Penrose number and Eulerian orientations of the line-graph 3.1. The line-graph of a cubic graph Let G = (V, E ) be a cubic graph with no loops or free loops and let v be a vertex of G. An angle of G at v is an unordered pair of distinct half-edges incident to v. The line-graph of G , which we denote by L (G), has vertex-set E and contains one edge with ends e, e’ for each angle of G consisting of one half of e and one half of e ’ . It is easy to see that L ( G ) is a 4-regular graph with no loops or free loops. For each vertex v of G we denote by t, and call triangle at v the set of three edges of L( G ) corresponding to the three angles of G at v. Note that {t,, v E V} is a partition of the edge-set of L ( G ) . If G is a diagram, let p = (p,,, v E V) be the associated rotation system. Consider an edge a of L ( C ) with ends e, e ’ , corresponding to an angle { h , h ’ } of G , where h is a half of e and h’ is a half of e ’ . Let us choose the half of a incident to e as initial and the other half of a (incident to e ’ ) as terminal whenever h’ = p h . This defines an orientation of L ( G ) whose restriction to every triangle t,, (v E V) is Eulerian. Thus this orientation of L ( G ) is Eulerian and we call it the canonical orientation of L(C). Consider the embedding of G in an orientable surface S defined by the rotation system p = (p”, v E V) (see Section 2.1). By.representing each vertex of L ( G ) by an interior point of the corresponding edge of G and by drawing each edge of L ( G ) sufficiently close to the corresponding angle of C it is easy to obtain an embedding of L ( G ) on S. This embedded graph is known as the medial graph of G on S (see [ l l ] p. 47 for the planar case). Then each triangle t,, (v E V ) of L ( G ) bounds a face and the canonical orientation of L ( G ) corresponds to a clockwise orientation around each such face. 3.2. Eulerian orientations and the Penrose number Let H be a 4-regular graph and v be a vertex of H. Let h , , h2, h3, h4 be the four half-edges incident to v. A transition at v is a partition of { h , ,h2, h 3 ,h 4 } into pairs. Thus there are exactly three distinct transitions at v. Consider an Eulerian orientation of H. Then each vertex v will be incident to two initial half-edges and two terminal half-edges. The transition at u consisting of the pair of initial half-edges together with the pair of terminal half-edges will be called anticoherent, and the two other transitions at v will be said coherent.


91

Consider a cubic diagram G. The anticoherent transitions of the canonical orientation of its line-graph L ( G ) will be said to be crossing, and the other transitions to be non-crossing. This terminology is motivated by the topological interpretation of L ( G ) as a medial graph given in the previous section. Let d be an Eulerian orientation of L ( G ) . We write s ( d ) = 1 if the number of non-crossing anticoherent transitions of d is even, and s ( d ) = -1 otherwise. The number s ( d ) is the sign of d .

Lemma 3. An Eulerian orientation of L ( G ) has positive sign if and only if it is obtained from the canonical orientation by reversing an even number of edges. Proof. Let do be the canonical orientation of L ( G ) and d l be an Eulerian orientation of L ( G ) obtained from do by reversing a set X of edges. It is easy to check that every vertex v of L ( G ) is incident to an even number of halves of edges of X, and that if there are exactly two such half-edges, one of them is initial and the other is terminal (this is true in both orientations). Moreover the anticoherent transitions of do and d l at Y are distinct, that is, the anticoherent transition of d l at v is non-crossing, if and only if v is incident to exactly two halves of edges of X. Since 2 =C i where Y; is the set of vertices of L ( G ) incident to exactly i halves of edges of X, the number lY21 of non-crossing Ci anticoherent transitions of d l has the parity of

1x1

1x1,

1x1.

Proposition 4. Let G be a cubic diagram with no loops or free loops. Its Penrose number T ’ ( G ) is equal to the sum of the signs of the Eulerian orientations of L(G). Proof. Consider a vertex v of G and the associated triangle t, of L ( G ) . Let e l , e2,e3 be the three edges of G incident to Y and let d be an Eulerian orientation of L ( G ) . Two cases may occur for the restriction of this orientation to the edges of t,,. (i) Either it is Eulerian. Then we say that d orients t,, as a circuit. (ii) Or we may assume without loss of generality that e l is a source and e3 is a sink. Then we say that d orients t,, from e l to e3, and that d marks the edges e l , e3 at v. Note that d marks an edge e of G at both of its ends, or none. In the first case we shall simply say that d marks e . We denote by C ( d ) the set of edges of G marked by d . For each subset F of the edge-set E of G we denote by Z ( F ) the set of Eulerian orientations d of L ( C ) such that C ( d )= F, and by z ( F ) the sum of their signs. Thus the sum of the signs of the Eulerian orientations of L ( G ) is equal to CFGEz ( F ) . Let F G E be such that z ( F ) # 0 and consider an orientation d in Z ( F ) . Then d marks zero or two edges at each vertex of G , and hence F is partitioned into vertex-disjoint cycles.

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Assume that some vertex v of G is incident to no edge of F. Then every orientation in Z ( F ) orients the corresponding triangle t, of L ( G ) into a circuit. To every orientation d in Z ( F ) we may associate another such orientation, which we shall denote by d’, by reversing this circuit. Clearly (d’)’ = d. Moreover it easily follows from Lemma 3 that d and d’ have opposite signs. This implies that z ( F ) = O , a contradiction. It follows that every vertex of C is incident to two edges of F. In other words, F is a 2-factor of C . Then for every orientation d in Z ( F ) there exists an unique Eulerian orientation q ( d ) of F such that for every vertex v of G , d orients the triangle t, from the edge of F with terminal half-edge incident to v to the edge of F with initial half-edge incident to v. Moreover, every Eulerian orientation of F is of the form q ( d ) for an unique orientation d in Z ( F ) . Let F,, . . . , Fk be the connected components of F. Let d be an orientation in Z ( F ) and let X be a subset of ( 1 , . . . , k } . We denote by r ( X , d) the unique orientation in Z ( F ) such that q ( r ( X , d)) is obtained from q(d) by reversing the edges of Fx = Uicx8.Note that r ( X , d) is obtained from d by reversing 3 lFxl edges of L ( G ) . If one of the connected components of F, for instance F,, is an odd cycle, we may consider the fixed-point-free involution which associates to every orientation d in Z ( F ) the orientation r({l}, d). These two orientations have opposite signs by Lemma 3. This implies that z ( F ) = 0 , a contradiction. It follows that all components of F are even cycles. In other words, F is an even 2-factor of G. Let d be any fixed orientation in Z ( F ) . Then clearly Z ( F ) = { r ( X , d ) / X r (1, . . . , k}}. It follows that lZ(F)I = 2k. Moreover, Lemma 3 implies that all orientations in Z ( F ) have the same sign, which we denote by s ( F ) . Thus z ( F ) = s(F)2k. Let us say that an edge-3-coloring f of G with the colors 1 , 2 , 3 is compatible with the even 2-factor F if F consists of the edges colored with the colors 1 , 2 by f. Let n = 2p be the number of vertices of G. Consider an orientation d , in Z ( F ) and an edge-3-coloring f compatible with F. There are exactly p vertices v of G such that dl orients the triangle f,from an edge colored 1 (respectively: 2) by f to an edge colored 2 (respectively: 1) b y f . If we reverse the 2p edges of L ( G ) which have both ends in F, we obtain an Eulerian orientation d2 which orients every triangle t,, into a circuit. Exactly p of these oriented triangles meet successively edges of G of colors 1 , 2 , 3 in this circular order. By reversing the 3p edges of the p other triangles we obtain an Eulerian orientation d 3 which orients every triangle t, into a circuit meeting the colors in the order 1 , 2 , 3 . Finally we may obtain the canonical orientation by reversing 3n-(f) edges in d,. Then it follows from Lemma 3 that s(dl) = 1 iff p + n - ( f ) is even. O n the other hand, s(f) = 1 iff n + ( f )- n - ( f ) = 2p - 2 n - ( f ) is a multiple of 4. Hence s(dl) =s(f) and all edge-3-colorings of G compatible with F have sign s ( F ) . Since there are exactly 2k such edge-3-colorings, z ( F ) = s ( F ) equals ~ ~ the sum of their signs. It follows that CFGEz ( F ) = T ‘ ( G )as required. 0


93

4. Some consequences 4.1. Another recursive computation of the Penrose number Let H be a 4-regular graph. A transition system of H is a family p = ( p ( v ) ,v E V ( H ) ) ,where p ( v ) is a transition at v which is said to be in p. Let us define the sign (with respect to p ) of an Eulerian orientation d of H as equal to 1 if the number of anticoherent transitions of d not in p is even, and to -1 otherwise. Let S ( H , p ) be the sum of signs (defined with respect to p ) of the Eulerian orientations of H. This concept extends the Penrose number, since Proposition 4 asserts that for a cubic diagram G (with no loops or free loops) T ‘ ( G ) is equal to S ( L ( G ) ,p ) , where, for every vertex v of L ( G ) ,p ( v ) is the crossing transition at this vertex. We now present a further extension in terms of what is known in statistical mechanics as the partition function of an ice-type model (see [ 2 ] , Chapter 8). Let us denote by O(H) the set of Eulerian orientations of H. Consider a mapping W which assigns to every transition t of H a “weight” W ( t ) chosen in some set of numerical variables or constants. The pair (H, W ) will be called a weighted 4-regular graph. For any Eulerian orientation d and vertex v of H , we denote by d, the transition at v anticoherent in d . We associate to (H, W ) the sum S ( H , W) = rIvsv(H) W(d,,). For instance, if p is a transition system of H and W, assigns the weight 1 to the transitions in p and the weight -1 to the others, clearly S ( H , W,) = S ( H , p ) . We now present a recursive scheme for the computation of S ( H , W ) which is based on the “splitting” of transitions. Let v be a vertex of the 4-regular graph H, and let h l , h 2 ,h3,h4 be the four half-edges incident to v. Consider for instance the transition t = { { h l ,h 2 } , {h3,h 4 } }at v. Let us delete from H the vertex v , introduce two new vertices x , y and rearrange the incidences of the half-edges previously incident to v as follows: h , , h2 are made incident to x , and h3,h4 are made incident to y . In the resulting graph, x and y have degree 2 and the other vertices have degree 4. Hence by erasing x and y we obtain a 4-regular graph which we denote by H * t. We now present a result which also appears in a wider context in [8]. Our approach here is different and more direct than in [8].

Proposition 5. Let ( H , W ) be a weighted 4-regular graph and let x be a vertex of H. If t l , t2, t3 denote the three transitions at x , for i = 1, 2, 3 let A ( t , )= - W ( t , )+ (112) C,=1,2.3W(t,). Then: S ( H , W )= r=1,2.3

A(t,)S(H * t,, W l x ) ,

where W l x is the restriction of the weight function W to the transitions at vertices distinct from x .

Proof, For i = 1, 2, 3 let O’(H) denote the set of Eulerian orientations of H whose anticoherent transition at x is t,, and let S,(H, W )= CdEOl(H) IIvEv(H)-(x) W ( & ) . Clearly S W , W )= Cr=1.2.3 W ( W , ( H ,W ) . O n the

F. Jaeger

94

other hand, for i = 1, 2, 3 we may identify O(H * t i ) with the set of Eulerian orientations of H whose anticoherent transition at x is not ti. It easily follows that

Hence

Since

zjE(l,2,3)-(jJ A ( t i ) = W ( t j )for j = 1, 2, 3, the result follows immediately.

0

Proposition 5, together with the obvious fact that a graph consisting of n free loops has 2" Eulerian orientations, yields a recursive computation method for S(H, W ) . The following result is just a special case of Proposition 5.

Proposition 6. Let H be a 4-regular graph and let p be a transition system of H . Let x be a vertex of H . I f t l , t 2 , t3 denote the three transitions at x , with p ( x ) = t l , then : S(H,p ) = (i)(S(H *

fz,

P I X )+ S ( H * t3r P / X )

- 3S(H

* t i , pix)),

where p / x is the restriction of the transition system p to the transitions at vertices distinct from x.

Let us call a transition t of a 4-regular graph H separating whenever H * t has more connected components than H . It is easy to show that if p is a transition system of H and there exists a separating transition not in p , then S ( H , p ) = 0 . This generalizes via Proposition 4 the well-known fact that if a cubic diagram has a bridge, it has no edge-3-coloring and hence its Penrose number is zero. But the converse is false. Fig. 3 depicts a 4-regular plane graph H with no separating transitions (this graph also appears in Fig. 4 of [lo]). It is easy to check, either directly or by repetitive application of Proposition 6 (an interesting exercise), that if p denotes the transition system consisting of the crossing transitions, S ( H , p ) = 0.

Fig. 3.

Penrose number of cubic diagram

95

4.2. Special results in the planar case It is well known (see for instance [2], Section 8.13 and [4], Chapter 3) that the Eulerian orientations of a connected 4-regular plane graph H are in 1-to-1 correspondence with the colorings of its faces with 3 colors 1 , 2 , 3 such that adjacent faces receive different colors (these colorings will be called face-3colorings) and such that the infinite face is colored 1. Indeed, let us consider the colors as elements of Z,.A face-3-coloring c being given, for any oriented edge e denote by lc(e) (respectively: rc(e)) the color of the face lying on its left (respectively: right) side. If we orient each edge e in such a way that rc(e)- lc(e) = 1, it is easy to check that we obtain an Eulerian orientation which we denote by d ( c ) . Conversely, every Eulerian orientation corresponds in this way to a unique face-3-coloring such that the infinite face is colored 1. This can be shown for instance by considering the Eulerian orientation as a tension in the dual graph and then viewing this tension as a potential difference. Note that the face-3-colorings such that the infinite face is colored x ( x in Z,)are obtained from the face-3-colorings such that the infinite face is colored 3 by adding x to all colors. We observe that for a given vertex v, the face-3-coloring c uses 3 colors on the faces incident to v (we shall then say 'that c tricolors v ) if and only if the anticoherent transition at v of the Eulerian orientation d ( c ) is non-crossing. Let us denote by t ( c ) the number of vertices tricolored by c. Then s ( d ( c ) ) ,the sign of d ( c ) (defined with respect to the crossing transitions), equals (-l)'('). We call this number the sign of c and we denote it by s(c). Let C ( H ) be the set of face-3-colorings of H. We have proved the following result.

Proposition 7 . Let H be a connected plane 4-regular graph. The sum of signs of the Eulerian orientations of H is equal to (4) CceC(H) s(c). Now let G be a connected cubic plane graph with no loops or free loops. It follows from Propositions 1 and 4 that T ( G ) , the number of edge-3-colorings of G , is equal to the sum of signs of the Eulerian orientations of L ( G ) , the medial graph of G , which is a connected 4-regular plane graph. By Proposition 7, this is also equal to (4) CcsC(L(G)) s(c). The faces of L ( G ) belong to two different types: those bounded by the triangles t, (v E V ( G ) ) which can be identified with the vertices of G , and the others which can be identified with the faces of G. Thus we may identify the face-3-colorings of L( G ) with the mappings from the set of vertices and faces of G to {1,2,3} such that the value of every vertex is different from the values of the three incident faces. We shall call such a mapping a full 3-valuation of G . For each mapping f from the set of faces of C to {1,2,3} (we shall call such a mapping a face-3-valuation of G), let y ( f ) be the sum of signs of the full 3-valuations of G which extend this mapping. Clearly iff assigns different values to the three faces incident to a vertex v,f cannot be extended into a full

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96

Fig. 4(i).

Fig. 4(ii).

3-valuation and hence y ( f ) = 0. On the other hand, iff assigns the same value to the three faces incident to v , the full 3-valuations extending f can be partitioned into pairs, two valuations of one pair being identical except for the value of v . Then it is easy to see that two valuations of one pair have opposite signs and hence y ( f ) = 0 also in this case. Finally, if, for every vertex v of G, f assigns exactly two distinct values to the three faces incident to v , we shall say that f is correct. Then we call sign o f f and denote by s(f) the sign of the unique full 3-valuation which extends f . Thus, denoting the set of correct face-3-valuations of G by F ( G ) , we have proved the following result.

Proposition 8. Let G be a connected cubic plane graph with no loops or free loops. The number of edge-3-colorings of C is equal to (4) Cf,F(c,s(f). Now let us make this result more precise. For f in F ( G ) and distinct elements i, j of { 1,2,3} we shall say that the vertex v has type iij in f whenever f assigns the value i to two of the faces incident to v and assigns the value j to the remaining one. On Fig. 4 (i) (respectively: '(ii)) we depict a triangle t,, in L ( G ) when v has type 112 (respectively: 221) in f . We have indicated the face-3-coloring of L ( C ) extending f and the associated Eulerian orientation. We observe that the oriented triangle t,, differs from the canonical orientation on two edges (respectively: one edge) in case (i) (respectively: (ii)). Moreover, if v has type 223 or 331 (respectively: 332 or 113) the situation is similar to the one in case (i) (respectively: (ii)). It then follows from Lemma 3 that the sign o f f is positive if and only if the number of vertices of type 221, 332 or 113 in f is even. Finally, using planar duality and the classical correspondence between edge-3colorings and face-4-colorings in cubic plane graphs, it is easy to reformulate Proposition 8 as follows (the relevant definitions have been dualized in the obvious way).

Proposition 9. Let K be a loopless plane triangulation. The number of vertex-4colorings of K is equal to ($)Cf,F(K*)s(f),where F ( K * ) is the set of correct vertex-3-valuations of K and the sign s( f ) of such a valuation f is equal to 1 if the number of triangles of types 221, 332 or 113 is even, and equal to -1 otherwise. References [l] K. Appel and W. Haken, Every planar graph is four colorable, Part I; W. Haken, K. Appel and J. Koch, Every planar graph is four colorable, Part 11, Illinois J . Math. 21 (1977) 429-567.


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[2] R.J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, 1982). (31 C. Berge, Graphes et Hypergraphes (Dunod, Paris, 1974). 141 N.L. Biggs, Interaction models, London Math. SOC. Lecture Note 30 (Cambridge University Press, 1977). [S] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (MacMillan, London, 1976). [6] J. Edmonds, A combinatorial representation for polyhedral surfaces, Notices Am. Math. SOC.7 (1960) 646. [7] F. Jaeger, On edge-colorings of cubic graphs and a formula of Roger Penrose, in Graph Theory in Memory of G.A. Dirac, L.D. Andersen, editor, to appear. [8] F. Jaeger, On transition polynomials of 4-regular graphs, preprint (1987). [9] L.H. Kauffman, Map coloring and the vector cross product, preprint (1987). 101 M. Las Vergnas, Le polyn6me de Martin d’un graphe EulCrien, Annals of Discrete Mathematics 17 (North-Holland, Amsterdam, 1983) 397-411. [ l l ] 0. Ore, The Four-Color Problem (Academic Press, New York, 1967). [12] R. Penrose, Applications of negative dimensional tensors, in: Combinatorial Mathematics and its Applications, Proceedings of the Conference held in Oxford in 1969 (Academic Press, London, 1971) 221-244. [13] L. Vigneron, Remarques sur les rCseaux cubiques de classe 3 associks au probltme des quatre couleurs, C.R. Acad. Sc. Paris, t. 223 (1946) 770-772.



99

ON THE EDGE ACHROMATIC NUMBERS OF COMPLETE GRAPHS Robert E. JAMISON Mathematical Sciences Department, Clemson University, Clemson, South Carolina 29634-1907, U.S.A.

Suppose we wish to color the edges of the complete graph K , with as many colors as possible so that (1) no two edges with a common node get the same color, and (2) for any two colors c , and c2, there are two edges with a common node, one colored c , and the other colored c2. What is the maximum number A ( n ) of colors possible in such a coloring? Coloring problems are notoriously hard and this problem is no exception. In fact, a remarkable theorem of AndrC Bouchet implies that an exact determination of A(n) for all odd n would yield as a corollary all odd orders for which projective planes exist. Thus such a determination is clearly beyond the hopes of this study. The goals here are more modest: (1) to give a careful study of the best available upper bound on A(n), (2) to add to the constructions which give reasonable lower bounds for A @ ) , and (3) to contribute a few more values of n for which A(n) is known exactly.

1. The achromatic index Let G be a simple graph. A proper (vertex) k-coloring of G is a map of the vertices of G onto a set of k “colors” so that any two adjacent vertices of C receive different colors. Moreover, if for each pair of colors c1 and c2 there are adjacent vertices v1 and v 2 so that vi is colored c,, then the coloring is complete. The smallest number k for which a coloring of G exists is the chromatic number x ( G ) of C. Any coloring with x(G) colors is necessarily complete since completeness means that it is impossible to merge any two color classes and still have a proper coloring. The largest k so that there exists a complete k-coloring of (the vertices of) G is the achromptic number v ( G ) of G introduced by Harary and Hedetniemi [8]. An old result of Harary, Hedetniemi, and Prins [9] says that for any k between x(C) and v ( G ) , a complete k-coloring of G exists. Thus the extreme values x(G) and v ( G ) determine the range of possible complete colorings. The achromatic number and the computational complexity of its determination have been studied by various authors. In general it appears that the exact determination of the achromatic number, even for simple structures such as trees, is quite difficult (cf. Lopez-Bracho [lo] and Farber et al. [ 5 ] ) . However, an easy upper bound on +(G) may be obtained as follows. If G has a complete k-coloring, then since there is an edge between each pair of color classes, G must have at least !El 1k ( k - 1)/2 edges. Hence (k - 1)2 G k ( k - 1) < 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

R.E. Jambon

100

The purpose here is to investigate the achromatic number A ( n ) of the line graph L(K,,) of the complete graph K,,. Notationally, A ( n ) = V(L(K,,)). As is well-known, the chromatic index x(L(K,,))is n if n is odd and n - 1 if n is even. That the precise determination of the achromatic indices V ( L ( K , ) ) will be much more difficult is evident from a remarkable result of Bouchet [3] stated below. A complete edge-coloring of K,, with the maximum number A ( n ) of colors will be called an optimal coloring. If r is any color class in an edge-coloring, then the nodes covered by the edges in r will be called the support of r.

Theorem 1.2 (Bouchet). Suppose q is odd and n = q2 + q + 1. Then A ( n ) = qn if and only if a projective plane of order q exists. Indeed, if A ( n ) = qn, then the supports of the color classes in any optimal coloring form the lines of a projective plane with the nodes of K,, as points. Aside from the values given by Bouchet’s theorem, the exact value of A ( n ) is now known only for n 6 11 and n = 25. The best current estimates on A ( n ) for n s 100 are summarized in the last section. In general, since L(K,,) is regular of degree 2(n - 2), inequality (1.1) yields the following upper bound:

A ( n ) 6 v n ( n - l)(n

- 2)

+ 1 6 (n - l)* + 1.

(1.3)

Although this bound can be improved slightly, it is of the right order of magnitude. The proof uses the monotonicity A ( n + 1)Z=A(n) which is a consequence of the following simple lemma.

Lemma 1.4. For any graph G , v ( G )3 Ilt(H)*

if H is an induced subgraph of

G , then

Proof. It suffices to show this if H i s obtained from G by deleting a single node u. Given a complete coloring C of H, extend this to a complete coloring of G by either (1) coloring u with a new color if all the colors of C appear on neighbors of v in G, or (2) coloring v with a color of C not on any neighbor of v otherwise. 0

Theorem 1.5. A ( n ) / n * + 1 as n + w. Proof. The proof depends on a strengthened version, due to Tchebychev, of Bertrand’s “Postulate” which follows from the Prime Number Theorem (cf. Gioia [6]): For any E > 0, there is an N, such that for any real x N,, there is a prime p between x and (1 + E ) X . Now let E > 0 be given, and suppose n > ( N , + 1)*(1 +

Edge achromatic numbers

101

E ) ~ . Set x = [ f i / ( l + E ) ] - 1, so x 2 N,. We may then select a prime p with x s p s (1 E ) X . Note that p 2 p 1 < ( x 1)2( 1 E ) = ~ n. Since projective planes of all prime orders exist (cf. Hall [7]), it follows from Bouchet’s theorem and Lemma 1 . 4 that

+ +

+

+

+

~ ( n ) 3 ~ ( p ~ + p + i ) = p ( +p i~) > + pp 3 a x 3 = ( f i - i - E)3/(1+E)3. Since

E

was arbitrarily small, the result follows.

0

+

Since A(n) grows asymptotically like ng, one might expect to have A(n 1) A(n) =O(fi), but this remains unproved. The best known result on the difference A(n 1) - A(n) is the trivial inequality A ( n ) 3 A(n 1) - n obtained

+

+

by deleting any vertex and the n incident color classes from any optimal coloring of K,,,,. It is also characteristic of the quirks of this problem that no proof of the strict inequality A(n) < A(n 1) is known in general, although this is almost surely the case. The constructions given in Section 3 do confirm this strict inequality for an inflnite class of n, and the result below establishes a two-step strict monotonicity.

+

Theorem 1.6. A(n + 2) 3 A ( n ) + 2 if n > 4. Proof. Consider an optimal coloring of K,. Select a maximal collection r of disjoint edges of different colors, which implies that r m e e t s every color class. Let st be an edge of r. The n - 2 other edges at f all have distinct colors. Since r is a matching, it contains at most n/2 edges. Hence there is an edge tu whose color does not occur in r. Starting with tu, select a maximal collection A of disjoint edges colored with colors not used for r. The subgraph G generated by r U A is bipartite because the bipartition {r,A} is a proper 2-coloring of its edge set. So the vertices of G may be properly colored black and white. Now add two new nodes b and w. Let xy be an edge of G where x is black and y is white. If xy E r, color the edges bx and wy both with the color of xy. If xy E A, color the edges wx and wf both with the color of xy. Since there are at most two edges (one from r and one from A) at each node of G, this is a consistent coloring. Moreover, since all the colors in r U A are different, no color appears twice at either b or w. Now erase the old colors on the edges of r and make r a new color class. Notice for any edge xy of r, its old color class still has x and y as well as b and w in its support in the new coloring. Now erase the old colors on the edges of A and make A* = A U {bw} a new color class. This is a proper coloring by the remarks above. Since the supports of old colors are either left the same or enlarged by b and w , it follows that any two old color classes still meet. Moreover, r meets every old color and A* meets every old color not originally on an edge of r. But edge bw meets all these colors. Finally, r and A* meet on the special edges st and tu. Hence the coloring is complete. 0

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From the computational viewpoint, Yannakakis and Gavril [ 141 showed the following problem to be NP-complete: Given a graph G and an integer n, is v ( G )an? But for fixed n, Farber et al. [ 5 ] proved there is an algorithm which, for an abritrary graph G, determines in O(IE(G)()time whether v ( G )3 n. Their proof was nonconstructive and they were able to exhibit such an algorithm only for n s 4. Since by Bouchet’s theorem I J J ( L ( K ~ 2~ 3615 ~ ) ) is equivalent to the existence of a projective plane of order 15, one may expect that the constant involved is quite large.

2. Upper bounds

In this section a refinement of the bound in (1.3) is presented and studied. Except for a few values where an improvement by 1 is possible, the result is the best upper bound known for A(n). The following functions play a crucial role: g(x, y ) = 2y(x - y - 1) and h ( x , y ) = x ( x - 1)/(2y).

(2.1)

Note (say, by differentiation) that for fixed x , the quadratic g ( x , y ) in y is increasing for y < ( x - 1)/2.

Lemma 2.2. For any t < ( n - 1)/2, A ( n )s max{g(n, t) + 1, h(n, t + 1)). Proof. Consider any complete k-coloring of L(K,,). Suppose first that there is a color class r with exactly s S t edges of K,, in it. Let S be the set of 2s nodes of K,, covered by the s edges in An edge of K,, is adjacent to an edge of T i n L(K,,) iff it has an endnode in S. There are n - 1 edges of K,, incident with each point of S; there are s(2s - 1) edges of K,, incident with two points of S. Hence the number of edges of K,, not in r but incident with a point of S is 2s(n - 1) - s(2s

- 1) - s = g ( n , s).

Since r m u s t be adjacent to at least one edge of every other color class, it follows that

k o if t > O

if t > 0 if

t22

Now let t 2 2 be fixed. First let us investigate the range between 4t2 - t and 4t2 + 3t - 1. Differentiating, we find D,q(x, t) = 2x - (4t2 4t 1). For x 2 4t2 - I , this is positive, so q(x, t) is increasing. Since q(4t2 + 3t - 1, t) < 0 by (2.4) it follows that q(n, t) < O for all n in the range 4t2 - t S n s 4 t 2 3t - 1. Thus for such an n, = g ( n , t ) + 1. h(n, t 1) < g ( n , t ) + 1, so Now if t < u < ( n - 1)/2, then g(n, u ) + 1 a g ( n , t ) + 1 > h(n, t 1) 2 h(n, u + 1) since g ( x , y) is increasing in y (for y < ( x - 1)/2) and h(x, y ) is decreasing in y. It follows that P U ( n )2 P,(n). Now consider s < t. Differentiating, we find D,p(x, t) = 2x - (4t2 + 1). For x 2 4t2 - t, this is positive, so p(x, t) is increasing. Since p(4t2 - t, t) 2 0 by (2.4),

+

+ + +

+

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it follows that p ( n , t) 3 0 for all n 3 4t2 - t. Thus for such an n, Lh(n, t ) ] 3 g(n, t) + 1. Hence for s < t, we have &(n) 2 Lh(n, s + 111 3 Lh(n, t ) ] 3 g(n, t ) + 1= Pr(n). It follows that B ( n ) = &(n) = g(n, t) + 1 as desired.

+

+

Now let us consider the range from 4t’ 3t to 4(t 1)’ - f - 2. As noted above, q ( x , t ) is increasing for x 3 4t’ - t. Since q(4t’ 3t, t) > 0 by (2.4), it follows that q(n, t) > 0 and hence Lh(n, t l)] 3 g ( n , t) 1 for n 3 4t’ 3t. Thus for such n , we have &(n) = [h(n, t l)]. Now if s < t, then Ps(n)3 Lh(n, s l)] 3 Lh(n, t l)] = P l ( n ) . To finish the proof, it suffices to show P U ( n )3 P l ( n ) for t 1S u < (n - 1)/2. To this end, consider the derivative D,p(x, t 1) = 211 - 4(t 1)’ - 1 which is positive if x 3 4t’ 3t. Since p(4(t 1)’- (t 1) - 1, t 1) < 0 by (2.4), it follows that p ( n , t 1) < 0 for all n in the range 4t’ 3t S n s 4(t 1)’ - t - 2. Hence for n in this range, we have P l ( n ) = Lh(n, t l)]s g ( n , t 1) 1S g ( n , u ) 1 s Pl,(n). Thus B ( n ) = P r ( n )= Lh(n, t l)] as desired. 0

+

+ +

+

+

+

+

+ +

+

+

+

+

+

+

+

+

+

+ + +

+

Theorem 2.5 Zf t > 1 and n = 4t’ - t, then A ( n ) G B ( n ) - 1. Proof. Let n = 4t2 - t, and suppose there is a complete coloring of L ( K , ) with B ( n ) colors. By Lemma 2.3, B ( n ) = g ( n , t) + 1 in this case. If there were a color class with s < t colors, it would meet only g(n, s) (t’ + t ) / 2 , at least one of these edges, say ab, must come from a class A with exactly t edges. Let au and bv be the edges in I‘incident with ab. But then au and bv are distinct edges having the same color and incident with the t-edge class A. This contradicts the fact that all edges incident with A must have different colors. 0 Bouchet’s Theorem in conjunction with the Bruck-Ryser Theorem (cf. Hall [7], p. 175) also yields (a much deeper!) improvement of the bound B ( n ) for certain n. Indeed, if q is odd and n = q’ q + 1, then setting t = (q + 1)/2 it is easy to see that B ( n ) = h(n, t) = qn. Thus we have

+

Theorem 2.6 (Bouchet, Bruck-Ryser). Suppose q = 1 mod 4 and n = q’ Zfq is not a sum ofsquares, then A ( n ) S B ( n ) - 1.

+ q + 1.


105

There remains one other case in which it is known that A ( n ) s B ( n ) - 1. One can easily verify that B(6) =9. However, Bories [2] and later independently Turner [12] both realized that A ( 6 ) =8. As this result was not previously published, it is given here for completeness.

Theorem 2.7 (Bories, Turner). A ( 6 ) S 8. Proof. Suppose on the contrary that L(K,) has a complete 9-coloring. For convenience, call an edge which forms a color class by itself a singleton edge. If there were at most two singleton edges, then in all there would be at least 2 ( 7 ) + 2 = 16 edges, a contradiction. Hence there are at least three singleton edges. Since an edge meets exactly 8 other edges, it follows that the 8 edges incident with a singleton edge must all receive different colors. Moreover, the singleton edges must meet each other. Hence they form either a star or a triangle. Suppose first that the singleton edges are ab, ac, ad and that they are colored 1, 2 , 3. Since bc, cd, db form a triangle, they must receive distinct colors. Since colors 1, 2 , 3 are used just once, say that bc, cd, db are colored 4, 5, 6, respectively. Let e and f be the remaining vertices of K,. Now ae cannot be colored 4 or 6 since ab is already adjacent to these colors. Similarly, ae cannot be colored 5 since ac is already adjacent to color 5. Hence ae (and similarly af) must be colored 7 or 8. Say, ae receives 7 and af receives 8. (See Fig. la). Note that ab is now adjacent with all colors except 5 and 9. Thus the colors assigned be and bf must be 5 and 9, or vice versa. Similarly, ce and cf must receive 6 and 9, and de and df receive 4 and 9. But this forces two edges colored 9 to be incident with one of the vertices e or f, a contradiction. Now suppose the singleton edges are ab, bc, ca and that they are again colored 1, 2, 3. Since all edges incident with ab receive different colors, we may assume the coloring is as shown in Fig. lb. The as yet uncolored edges adjacent to ac are

C

a. Star Case

b. Triangle Case

Fig. 1 . Impossibility of 9-coloring K,. a. Star case. b. Triangle case.

R . E. Jamison

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cd, ce, cf and these must be colored 5 , 7, 9 (in some order) so that ac is adjacent to all colors. Similarly, the same edges cd, ce, cf are the uncolored edges adjacent to bc, and these must be colored 4, 6, 8, a contradiction. 0

3. Constructions from projective planes

+

Suppose q is the order of a projective plane P. Then P has n = q2 q + 1 points and n lines. We may regard the points of P as the nodes of a complete graph K,,. Each edge lies in a unique line which is a copy of K,,,. Now if q is odd, then K,,, has a l-factorization. Each line may thus be divided into q color classes, each of size ( q 1)/2. All colors of a line are incident with all points of that line. Hence since any two lines meet, the coloring is complete. This yields the easy part of Bouchet’s Theorem: A ( n ) = qn if q is odd and the order of a projective plane. If q is even, then K,,, is not l-factorable, so the argument collapses. However, by adjoining some additional points, it is possible to obtain some good (if not exact) lower bounds.

+

Theorem 3.1. Suppose q is even and the order of a projective plane. Let n = q2 + 2q + 2. Then A ( n ) 3 nq + 1 and A ( n + 1)2 nq q + 2.

+

Proof. Let P be a projective plane of order q regarded as a complete graph. Each line L of P has an odd number q + 1 of points. Hence L may be edge-colored with q + 1 colors. In any such coloring, at each point v of L, there will be exactly one color missing. Moreover, since each color class in L must contain q / 2 edges, different points of L will be missing different colors in L. Using a different set of q + 1 colors for each line L of P, we obtain an edge-coloring of P with ( q + l ) ( q 2 + q 1) = q n + 1 colors. It is a proper coloring, but because of the “missing” colors, it fails to be complete. For each incident point-line pair (v, L) in P, let c(v, L) denote the missing color at v in the line L. These colors are all different and thus comprise the full set of qn + 1 colors. Now arbitrarily order the lines through each point u as L,,(v), L , ( v ) , . . . , L,(v). Let U = { u ~u,l , . . . , u,} be a set of q + 1 new points. Then K,, may be represented with the n points of P U U as node set. Color each edge vui with the missing color c(v, L , ( v ) ) . Since the colors c(v, L ) are all distinct, this remains a proper coloring. Now for each v E L, we have all colors of L occurring on edges at v. Since any two lines of P have a point in common, it follows that any two color classes are incident, so the coloring is complete. The edges between the ui’s have not yet been colored. However, we do have a complete (qn 1)-coloring of a subgraph of L(K,,). Hence by Lemma 1.4, we have A ( n ) 2 qn + 1. Now continue the above construction by adding another new point u * . Set

+

+


107

+ + + + + +

U* = U U { u * } . Then U* has an even number q 2 of points and hence is 1-factorable. That is, U* can be edge-colored with q 1 colors, each color occurring at each point of U*. Since all previous qn 1 colors are incident at the points of U , it follows that we may choose these q 1 colors to be new and still have a complete coloring. Thus A ( n 1) 3 nq q 2. 0

+

With n as above, taking t = q / 2 in Lemma 2.3, we see that B(n - 1)= nq - q, B(n) = nq q and B(n + 1) = nq + 3q + [6/(q + 2)1. Thus by the above constructions, we have A ( n - 1) s B(n - 1)< A ( n ) s B ( n ) < A(n + l), so the strict monotonicity of A is established for infinitely many values of n.

+

Theorem 3.2. Let q be a power of 2, and let s be an integer such that $4 + 1 ss s q. Then A(q2 q s) L q2(q + 1) + min{(2s - q - l ) ( q + l ) , A(s)}.

+ +

Proof. As in the above construction, let P be a projective plane of order q and edge-color each line of P with q + 1 colors. Now fix a point p of P and denote the lines of P through p by M,, MI, . . . , M4. (We will call these M-lines and the lines not through p will be called L-lines.) Arbitrarily order the points #p on Mk as w(k, j), j = 1, . . . , q. Now delete p and remove all colors from the M-lines. This leaves q2(q 1) “old” colors. For each point v # p , order the lines through v as L,(v), L,(v), . . . , LJv) where L,(v) is the M-line through v and p. Let U = {ul, . . . , u s }be a set of new points. For each v # p and i with 1s i G s , color edge vu, with the “missing” color c(v, Li(v))at v on line Li(v). There then remain q - s “missing” colors at each point v. Say, v = w(k, j). For each i with s < i G q , color the edge from w ( k , j ) to w(k, j i - 1) with the color c(v, Li(v)). (The sum j + i - 1 is modulo q . ) Now every color on an L-line occurs at each point of that L-line. Since any two L-lines intersect at a point #p, the coloring so far is complete. Morever, the edges in U and certain edges on the M-lines are still available to be colored. Consider some k f k and some i with f q + 1G i s s . The edges w ( k , j ) to w(k, j i - 1) on Mk are as yet uncolored. If i = f q 1, these edges from a 1-factor. Otherwise, these edges form a collection of cycles of length A where A is the least integer such that q divides A(i - 1). Since q is a power of 2, A must be even so this collection of edges may be split into two 1-factors. Thus on each Mk, we have available 2(s - f q - 1) 1 matchings, each of which covers all points of Mk. Since each L-line meets each M-line at a point # p , it follows that if one of these matchings is taken as a new color class, then it will meet all old colors. Therefore the difficulty in adding new colors lies only in making new color classes from different M-lines meet each other. This can be accomplished by repeating the new colors in a complete edge-coloring of U . Cl

+

+

+

+

+

Suppose we fix a constant c 3 1 and take s = $4

+ y where

1s y s c. As q

R. E. Jamison

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Table 1. A residual edge-coloring of K,.

grows, A(s) grows like a constant times qj by ( l S ) , so A(s) eventually dominates (2r - q - l)(q 1) = (27 - l)(q + 1). Hence the above result yields A(q2 $q + y ) 2 q3 q2 + (2y - l)(q + 1) for large q. This compares favorably with the upper bound B-in fact, well enough to imply strict monotonicity in the range under consideration. It is possible to improve the bound in Theorem 3.2 by taking advantage of the fact that new colors on the same M-line already meet each other. This is illustrated by the following special constructions.

+

+

+

Theorem 3.3. A(24) 3 89 and A(78) 2 591.

Proof. First, take s = q = 4 and proceed with the construction in (3.2) up to the place where the new colors are to be chosen. The edges to receive new colors form six disjoint K4's-the five M-lines and U.Each K4 factors into three pairs of edges. Table 1 shows a method of coloring these edges with 9 new colors A, A', B, B ' , C , C', D, D', and E. A new color standing alone indicates that both edges of a 1-factor are to receive that color. A pair of colors in parentheses indicates that the listed colors are to be distributed over the edges of a 1-factor. Since each new color stands alone on some M-line (and hence has that M-line in its support), each new color meets every old color. It is easily verified that the new colors are pairwise incident. In the same spirit, Table 2 shows a scheme for the new colors in the case q = 8, s = 6. Graphically, the edges to be colored on each M-line form an 8-cycle with Table 2. A residual edge-coloring of K,,

M,: M,: M,: M3: M4: M,:

M6: M,: Me:

.

A B C D E F G H I

A' ( B , B ' , C , C') B' (C, C ' , C, D ' ) C' ( D , D',E , E') D' ( E , E ' , F, F ' ) E' (F, F ' , A , A ' ) F' ( A , A ' , B ,B ' ) ( A , D, A', D ' , A , C, F, C ' ) ( B , E , B ' , E ' , B , C , F', C ' ) (C, C', D , D ' , E , E ' , F, F ' )

'1

' 2

'3

'4

' 5

'6

H

D' t

U

109


i$J 2

K 4

K

1

~~~

0

1

2

3

4

0

Fig. 2. A residual edge-coloring of Ku,.

its four principal diagonals. For Mo through M,, imagine the edges split into three 1-factors, each of the first two receiving a single color and the third receiving four different colors as indicated. On M6, M,, and M8,the diagonals receive colors G , H, and I , respectively, and the 8-cycles are colored as indicated. Finally, a special color table is given for the edges of U.The verification that the resulting coloring is proper and complete is easy and left to the reader. 0 To conclude this section, we give two special constructions based on removing a point from an odd order projective plane

Theorem 3.4 A(12) 3 31 and A(30) 3 136.

Proof. The projective plane PG(2,5) over GF(5) may be regarded as consisting of (1) the 25 points of the affine plane AG(2,5)-represented in Fig. 2 by the 5 x 5 square grid, (2) the 5 points on the line at infinity corresponding to “slopes” of nonvertical lines-represented in Fig. 2 by the vertices of K5,the vertex labels being slopes, and (3) one “vertical” point on the line at infinity. Take an optimal coloring of K31 based on the plane PG(2,5). This has qn = 155 colors. Remove the vertical point, the 30 edges and the 30 colors incident with it. This leaves 30 points with 125 “old” colors intact and 60 edges which must be recolored. Geometrically, the uncolored edges lie on the 5 vertical lines of AG(2,5) and the “punctured” line at inflnity. Graph theoretically they from 6 disjoint copies of K,. Fig. 2 provides a scheme for recoloring these edges with 11 “new” colors A, B, . . . , J, K. The colors of the 10 edges on the infinite line are as indicated. The colors on the other vertical lines are assigned as follows. Color “X” in grid position (i, j)

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0

1

2

Fig. 3. A residual edge-coloring of K , 2 ,

+

means: on the vertical line x = i, color edges (i, j - l ) , (i, j 1) and (i, j - 2), (i, j + 2 ) with color X. The ambiguity in the (0,4) position is to be resolved by coloring edge (0,3), (0,O) with J and edge (0,2), ( 0 , l ) with K. The result is an edge coloring of KW with 136 colors, 125 old and 11 new. Let us check that it is proper. First note that the coloring on the infinite line is a complete (in fact, optimal) edge coloring of K s with 7 colors. That the coloring is proper on the remaining 5 vertical lines follows from the fact that no color appears more than once in any column of the grid representing AG(2,5). The check that this coloring is complete is more tedious and involves verifying that the support of each new color meets every nonvertical line. The details, which are routine, are left to the reader. The data in Fig. 3 may be used in a similar way to define a complete 31 coloring of the edges of KI2. 0 The same procedure may be applied to the coloring induced by any odd order plane. But as the order grows, the recoloring becomes every more tedious. Moreover, the requirement that the new colors “block” all the nonvertical lines, together with results of Aiden Bruen [ l ] on blocking sets, suggests that the new color classes must be too large to yield an efficient coloring in general.

4. Group divisible colorings

In this section we shall further exploit a construction technique introduced by Bouchet [3]. Let G be a group of order n. A Bouchet diagram (over G ) is a (simple) graph D such that

(1) D has a 1-factorization, (2) the nodes of D are elements of G, (3) for any edges [ x , y ] and [u,u] of D , x - l y = u-lu if and only if x = u and y=u, (4) for any g in G, there are nodes x and y of D with xy-I = g. If D has rn nodes and is regular of degree d , then we shall call it an

111


0

l

o 5

n

=

9, d = 2

G=Z,

Oo----

415

lo----

4

S*---

-09

2

2

n = 25, d = 4 G=%5

graph is complement of edges shown

14

18

n=19,d=3

n = 23, d = 3

G

G

1

Z,,

= 223

Fig. 4. Bouchet diagrams.

(n,m , d)-Bouchet diagram. Let D* = {x-'y: x and y are adjacent in D}.Note that D * = ( D * ) - l since adjacency is symmetric, that D * does not contain the identity since D is loopless, and D * does not contain any involution by (3). It is often convenient to think of D as a labelled graph: by (2) the nodes of D are labelled by elements of G , and each edge [ x , y ] is labelled by a pair of inverse elements { x - ' y , y - ' x } . Fig. 4 shows four Bouchet diagrams. For n = 9, 19 these are due to Bouchet [3]; for n = 23, 25 they are new. The Cayley graph Cay(G, D * ) has the elements of G as nodes and an edge [g, h] whenever g-'h is in D*.The diagram D induces a complete edge coloring of Cay(G, D*)as follows. For each g E G, let g D denote the translate of D by g-that is, the nodes of gD are of the form gx where x E D and two nodes gx and gy are adjacent in g D iff x and y are adjacent in D.The translates then cover the edges of Cay(G, D*).Condition (3) guarantees that the translates gD are edge disjoint and condition (4) guarantees that any two translates of D have at least one node in common. Select a 1-factorization of each gD and regard this as an edge-coloring with d colors of gD, using disjoint sets of colors for different

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translates. Since all colors used in any g D occur at all nodes of g D and since any two translates have a node in common, it follows that this coloring is complete. Since dn colors are used and since Cay(G, D*)is an edge-subgraph of the complete graph on G, we have the following result.

Theorem 4.1 (Bouchet). If an (n, m , d)-Bouchet diagram exists, than A ( n ) 3 dn. For n = 9 and n = 25, the colorings generated by the diagrams in Fig. 2 are optimal since B(9) = 18 and B(25) = 100. It is possible that these may be the first in a series for n an odd square but no general constructions are known yet. For n = 19, the coloring is maximal among all colorings with class sizes 3 or more. But it may be possible to obtain more colors by allowing some color classes of two edges. For n = 23, a better coloring was obtained using projective planes (Theorem 3.2 with q = 4 and s = 3). The following result provides a method for augmenting a group divisible coloring.

Theorem 4.2. If an (n, m, d)-Bouchet diagram exists, then for all k, A ( n + k m ) (d + k)n.

Proof. As before, take the induced coloring on Cay(G, D*).Introduce km new points u&) where i = 1, . . . , k and x E D, and introduce kn new colors c,(g) where i = 1, . . . , k and g E G. Now color the edge [g, ui(x)] with color c,(gx-'). For fixed i and x , as g ranges through G, the products gx-', and hence the colors c,(gx-'), are all distinct. Similarly, when g is fixed and x ranges through D, the colors c,(gx-') are all distinct. Thus the coloring is proper. Now any new color c,(g) is assigned to the edge from gx to ui(x)for all x in D. Thus c,(g) occurs at all nodes of gD. Now any old color occurs at all nodes of some translate hD, which meets g D in at least one node. Thus every new color meets every old color. Now suppose c,(g) and cj(h) are two new colors. By condition (4), select x , y E D so that xy-' = g-'h. Then gx = hy. Letting z = gx, we then have zx-' = g and zy-' = h. Thus the edges [z, ui(x)]and [z, u j ( y ) ]are colored c,(g) and cj(h), respectively. Hence all new colors also meet one another, so the coloring is complete. 0 A well-known theorem of Singer [ l l ] says that if q is a prime power and n = q2+ q + 1, then there is a subset D of Z, (the cyclic group of order n ) such that every nonzero element of 2, is expressible uniquely as x - y for x , y in D . This difference set arises from the projective plane PG(2, q) over the Galois field GF(q). Of course, D has q 1 elements, so if q is odd, the complete graph on D is 1-factorable and D is an (n, q + 1, q)-Bouchet diagram. Thus from Theorem 4.2, we obtain the following corollary.

+


113

Corollary 4.3. Let q be a power of an odd prime and set n = q2 + q + 1. Then for 0, A ( n + k(q 1)) 3 qn + kn.

+

all k

5. Summary An edge coloring of K,, may be represented by a color table: an n x n matrix whose i, j entry is the color on edge [i, j ] . Of course, such a table is symmetric. The coloring is proper iff each row (and hence each column) contains distinct entries. The coloring is complete iff each pair of colors occurs together in at least one row (column). Color tables for optimal colorings of K 6 , K , , K 8 , Klo, and K I 1 are given in Fig. 5 . (An optimal coloring of K s appears in Fig. 2; the (9,4,2)-Bouchet diagram from Fig. 4 yields an optimal coloring of K 9 . ) The values of A ( n ) for n S 7 were known to Bories [2]. Using differencing methods, Bouchet [3] found A ( 8 ) and A(9). Optimal colorings of Klo and K I 1 were found by Ray Rowley and Craig Turner, respectively, using ad hoc methods described in Turner et al. [13]. Table 3 shows the currently best upper and lower bounds on A ( n ) for n 6 100. The upper bound is B ( n ) except as noted: (a) Theorem 2.5 * 1 2 3 4 5

1 * 5 6 7 3

2 5 ' 7 6 1

3 6 7 * 2 4

4 5 7 3 6 1 2 4 ' 8 8 ;

n=6, k = 8 Bones

1 1 2 3 4 5 6 7 8 9 10

* 7 11 12 13 14 2 15 16 17

2 3 7 11 * 18 18 * 19 24 20 16 21 23 1 8 22 6 23 13 24 19

* 1 2 3 4 5 6

I 2 ' 7 7 ' 5 8 5 6 1 0 9 4 2 8

' 1 2 3 4 5 6 7 1 * 5 8 9 1 0 1 1 4 2 5 * 6 1 2 1 3 8 1 0 3 8 6 ' 7 9 1 4 1 2 4 9 1 2 7 * 1 1 3 1 1 51013 9 1 * 2 1 4 611 8 1 4 1 3 2 * 3 7 4 10 12 11 14 3 *

3 4 5 6 2 8 6 9 1 0 4 8 * 9 1 10 9 * 7 3 1 7 * I1 10 3 1 1 *

n=7, k = l l Bories

4 12 19 24

5 13 20 16 * 9 9 * 15 22 14 4 17 21 5 11 18 23

6 7 8 9 14 2 15 21 1 22 23 8 6 15 14 17 22 4 21 * 12 20 12 * 25 20 25 * 10 26 27 3 27 26

1 0 16 17 23 24 13 19 5 18 I1 23 10 3 26 27 27 26 * 25 25 *

n=8, k=14 Bouchet

$

1

1

'

2 3 4 5 6 7 8 9

5 10 11 12 13 14 15 16

2 3 5 10 * 16 16 * 8 6 17 19 18 4 19 17 20 13 21 22

4 5 6 7 11 12 13 14 8 17 18 19 6 19 4 17 * 22 3 20 22 * 15 1 3 15 * 21 20 1 21 * 14 9 10 5 12 I 1 7 18

n = 10, k = 22 Rowley n = 11, k = 27 Turner Fig. 5. Color tables of optimal k-colorings of L ( K , ) .

8 15 20 13 14 9

9 16 21 22 12 11 10 7 5 18 * 2 2 *

R.E. Jamison

114

Table 3. Best current bounds on A ( n ) for 1 n 100. Lower bounds arising from A ( n + 1) 3 A ( n ) are omitted. n

Upper Lower n

Upper Lower n

Upper Lower n

Upper Lower n

Upper Lower

1 -

2 1

1

n

n

Upper Lower n

n

Upper Lower

18 61

19 65 57 d

20 69

27 117 110 P

28 126

29 135 112

30 145 136

m

S

36 I93

37 199 186 e

38 205

39 21 1 188 m

40 217

45 247 22 1 m

46 258

47 270 223

48 282

49 294 250 e

50 306

55 37 1 288

56 385 343 t

57 399 399 P

58 413

59 427 40 1

60 440

m

a

68 505

69 513 460

70 521

79 616

P

a

m

22 77

23 84 83

24 92 89

25 100 100 d

31 155 155 P

32 165

33 174 157

41 223 190

42 229

C

S

21 73 65 e

P

51 318 252 61 449 403 71 529 462 81 648

91 819 819 P

14

S

34 181

52 331

35 187 159

6 8 8 b, c

7 11 11

8 14 14

C

C

16 53

17 57 52 e

26 108 105 P

m

a, m

43 235 219 e

44 24 1

53 344 254

54 357

m

t

m

62 457

63 465 405 m

64 473

65 481 456 e

66 489

67 497 458

72 539

73 545 513 e

74 553

75 561 515 m

76 570

77 585 583 P

78 600 591

82 664 657 P

83 680 666 9

84 697

85 714 668

86 731

87 748 670

88 765

92 837

93 855 821

94 874

m Upper Lower

10 22 22

15 49 41

m

n

9 18 18 d

5 7 7

44

12 33 31

m

Upper Lower

4 3 3

13 39 39

11 27 27

m

Upper Lower

3 3 3

m

m

m

95 890 823 a, m

m

97 91 1 825 m

98 92 1

80 632 593 m

S

m

96 901

C

89 783 672

90 801 728

m

I

99 93 I 827 m

100

94 1


115

(b) Bories-Turner Theorem 2.7 The lower bounds come from these sources: (c) Color tables in Fig. 5 (d) Bouchet diagrams in Fig. 4 (e) Extension of Bouchet diagrams in Corollary 4.3 (m) Monotonicity Theorem 1.6 (A(n 2) 2 A(n) + 2) (p) Projective planes (Theorems 1.2, 3.1, and 3.2) (s) Special constructions from Section 3 (t) The trivial bound A(n) 2 A(n 1) - n Lower bounds which arise simply from the monotonicity of A ( n ) are omitted from the table.

+

+

Note added in proof Improvements on some of the lower bounds in Table 3 using a modification of Bouchet diagrams have come to light since submission of this paper. In particular, A(12) 3 32 and improvements for 46 S n d 49 are known. Details will be reported elsewhere.

References [l] Aiden A. Bruen, Blocking sets in finite projective planes, SIAM J. Appl. Math 21 (1971) 380-392. (21 F. Bories, Sur quelques problCmes de colorations complCtes de sommets et d’argtes de graphes et d’hypergraphes, Thtse de 3tme cycle (Pans, 1975). [3] A. Bouchet, Indice achromatique des graphes multiparti complets et reguliers, Cahiers Centre d’Etudes et Recherche Operationnelle 20 (3-4) (1978) 331-340. [4] R.H. Bruck and H.J. Ryser, The nonexistence of certain finite projective planes, Canadian J. Math. 1 (1949) 88-93. [5] M. Farber, G. Hahn, P. Hell and D. Miller, Concerning the achromatic number of graphs, J. Combinatorial Theory, Ser. B 40 (1986) 21-39. (61 Anthony Gioia, The Theory of Numbers (Markham, Chicago, 1970) 156. (71 Marshall Hall, Combinatorial Theory (Blaisdell, Toronto, 1967). [8] F. Harary and S.T. Hedetniemi, The achromatic number of a graph, J. Combinatorial Theory 8 (1970) 154-161. [9] F. Harary, S.T. Hedetniemi and G. Prins, An interpolation theorem for graphical homomorphisms, Port. Math. 26 (1967) 453-462. (101 R. Lopez-Bracho, Le nombre achromatique d’une Ctoile, Ars combinatoria 18 (1984) 187-194. 1111 James Singer, A theorem in finite projective geometry and some applications to number theory, Trans. Amer. Math. SOC.43 (1938) 377-385. (121 C. Turner, On the edge achromatic number of small complete graphs, Master’s Thesis, Clemson University (1986). (131 C.A. Turner, Ray Rowley, R.E. Jamison and R. Laskar, The edge acrhomatic number of small complete graphs, Congressus Numerantium (1988). [14] M.Yannakakis and F. Gavril, Edge dominating sets in graphs, SIAM J. Appl. Math. 38 No. 3 (1980) 364-372.



117

APPLICATIONS OF EDGE COLORING OF MULTIGRAPHS TO VERTEX COLORING OF GRAPHS H.A. KIERSTEAD* Deparimeni of Mathemaiics, University of South Carolina, Columbia, SC 29208, U.S.A. It is shown that if G is a graph which induces neither K , . 3 nor K I S + , - e and w ( G ) is sufficiently large then z(G) G w ( G ) + s. This result is established by first demonstrating a correspondence between vertex coloring G and edge coloring a certain multigraph and then applying a known result on edge coloring.

1. Introduction Perhaps the two most important results on edge coloring are Vizing’s Theorem [16], which states that the chromatic index x ‘ ( M ) of a multigraph M with maximum degree A ( M ) and maximum multiplicity p ( M ) satisfies A ( M ) s f ( M ) s A ( M ) p(M), and Holyer’s Theorem [8], which states that the problem of determining the chromatic index of even a simple graph is NP-complete. In some sense these two results solve the edge coloring problem for simple graphs. However, the upper bound is quite loose for multigraphs. Unfortunately much of the work on edge coloring has been restricted to simple graphs. One goal of this article is to stimulate research on edge coloring multigraphs by demonstrating how refined bounds on the chromatic index of multigraphs can be used to derive interesting results on the chromatic number of graphs. We will also take the opportunity to pose specific questions about chromatic index. Let S be a set of finite subgraphs and let C ( S ) be the class of graphs which do not induce any subgraph in S. Many vertex coloring theorems give an upper bound on the chromatic number x ( G ) of any graph G in C ( S ) in terms of the clique size w ( G ) of G. GyBrfiis [7] and Sumner [14] independently conjectured that if S = { T}, where T is a tree, then such a bound exists. For stars this follows easily from Ramsey Theory and Brooks’ Theorem [3]. Gyrirfris [7] has shown that the conjecture is true for paths. If S does not have a tree for an element such a statement cannot be true since Erdos [5] has shown that there are graphs with arbitrary high girth and chromatic number and maximum clique size 2. We shall consider classes of graphs which induce neither K1,3nor K2S+3- e, for various choices of s. Vizing’s Theorem can be stated in the above form. Beineke [2] characterized line graphs as those graphs which do not induce any graph in a set S of nine forbidden subgraphs, including K 1 , 3and K s - e. By Vizing’s Theorem x ( G )s

+

* Partially supported by ONR contract NOOO14-85K-0494. 0012-365) k 2 A ( M ) + s, then p ( M ) 2 [ ( m - 1)s + k - S ( M ) 2]/m, for some even m, such that 2 6 m s ( S ( M ) - 2)/s. In particular, i f X ' ( M ) > A ( M ) s, x ' ( M ) > 7s - 2, and p ( M ) s 2, then S ( M ) 3 4s.

+

+

Proof. Let x o be a vertex such that S(xo) = S ( M ) , eo be an edge incident to x o , and f be a ( x ' ( M ) - 1)-edge coloring of M - e,. For a vertex v let f (v) be the set of colors not used by f on edges incident to v. In [lo] a path P was defined to be f-acceptable if P = x o , eo, x l re l , . . . , e n - , , x, where e, is uncolored and f ( e i ) E f(xj), for O s j < i < n . It is implicit in the results of [lo] that, since M is edge critical and f ( M ) > A ( M ) , M has a maximal f -acceptable path P of positive even length (number of edges) such that f ( x i ) f l f (xi)= 0, for i # j. By the maximality of P, if color (Y E f (xi) and i < n - 1 then there is an edge colored (Y from x, to xi for some j < n. Let N = U {f ( x i ):i < n } . Then

(k - 6(xo) + 1) + (s + 1) + ( n - 3)s s IN1 S(x,)

6 A(M).

Thus setting m = n - 1 yields 2 s m s ( S ( M ) - 2)/s, where m is even. By the pigeon hole principle, there exists i < n such that

[(m- 1)s + k - S ( M ) + 2]/m

S

INl/m d p ( q , x,)

6p(M),

which proves the first statement of the lemma. For the second statement, suppose for a contradiction that S ( M ) < 4s. Then m = 2 and 2s < [s

+ (7s - 2) - (4s - 1)+ 2 ] / 2 s p ( M ) .

0

Next we prove a preliminary proposition.

Proposition 6. Let G be a graph which induces neither K 1 , , nor K,,, - e. (i) If X is a maximal clique and w E V - X , then IN(w) r l XI d 2s. (ii) If X is a maximal clique such that 1x1 3 4s, v E X and y , z E N ( v ) - X , then y -2. Proof. (i) Since X is maximal, there exists x E X such that x + w . Thus ( N ( w )nXI s 2, since otherwise Ka+,- e is induced by a subset of { w , x } u ( N ( w )nX),which is a contradiction.

H .A . Kierstead

120

Nl ( z ) n XI s 4s - 1. So there (ii) By (i), I ( N ( y ) U N ( z ) ) f l XI G 4s - I N ( y ) f exists x E X,such that x +y and x + z . Thus y z , since otherwise K1,3is induced by {v, x , y, z } , which is a contradiction. 0

-

Proof of Theorem 3. The result is trivial if s = 0; so assume s 2 1. We will obtain a contradiction by showing that if there existed a graph C = (V, E) which induced neither K 1 , 3nor K2F+3- e, such that R ( 3 , 4s - 1) < x ( C ) and w ( G ) + s < x(G), then there would exist a multigraph M, such that M contains no s-triple and A ( M ) + s < x ' ( M ) . But this is impossible by Lemma 4. So suppose that G is such a graph and that v ( C ) is minimal over all such graphs. Thus x(G) = max(w(G) + s, R(3, 4s - 1))+ 1, G is critical, and max(w(G) + s, R(3, 4s - 1))zs S(G). It suffices to show that G is the line graph of a multigraph M such that A ( M ) = w ( G ) and M contains no s-triple, for then A ( M ) + s = w ( G ) + s < x ( G )= x ' ( M ) . We begin by proving the following: (1) Every v E V is in some 4s-clique. Since 6 ( v ) s R ( 3 ,4s - 1) and C does not induce K 1 , 3 , N ( v ) contains a 4s - 1-clique K. Thus K U {v} is a 4s-clique. (2) For each v E V there is a unique pair of maximal cliques, X and Y , such that v E X n Y and N [ v ]= X U Y. First we prove the existence of X and Y. Let X be a maximal clique such that v E X and 1 x12 4s. This is possible by (1). By Proposition 6(ii) and the fact that o ( G )< 6(G), N ( v ) - X is a non-empty clique. Let Y be any maximal clique extending N [ v ]- X.Then N [ v ] = X U Y. Now we prove the uniqueness of the pair {X,Y} constructed above. Suppose { W , Z } is a pair of distinct maximal cliques such that v E W n 2 and N [ v ] = W u Z . Then

W u Z 3 X and 4s G I X n WI

+ IXn ZI - IXn W n Z J .

Thus I X n W l 3 2 s + l or IXr7ZIs2s+l. So by Proposition 6(i), X E { W , Z } . Say X = W . Then N [ v ]= X U Z = X U Y. We show that Z = Y by showing that Y U Z is a clique. Suppose y E Y and z E Z . Since Y - X = Z - X , if either y or z is not in X, then y 2. Otherwise both y and z are in X and y z . We will call the elements of the unique pair of maximal cliques {X,Y} for which v E X f l Y and N [ v ]= X U Y, the covering cliques of v. The proof of (2) has actually shown the following is true. (3) If v EX, where X is a maximal clique such that (XI 5 4s, then X is a covering clique of v. Call a covering clique X large if 3 4 s ; otherwise call it small. We next prove: (4) If x -y, then x and y have a common covering clique. Suppose that x and y do not have a common covering clique. We shall show that G is not critical. By (1) and (3) both x and y are in at least one large covering clique; neither is in two large covering cliques, since otherwise the covering clique

-

-

1x1

Edge coloring of multigraphs

121

of (say) x , which contains y, is large, and thus, by (2), a covering clique of y. Say X and Y are large covering cliques of x and y, respectively. We define a bipartite graph B = (B,, By,E x y ) as follows:

B, = { v

E X:v

is in exactly one large covering clique}

BY = {v E Y: v is in exactly one large covering clique} Ex, = { x ’ y ‘ E E : x ‘ E B, and y‘ E BY}. Suppose x ’ E B,, y ’ E Y - X, and x ’ - y ’ . Since Y is large, Y is one covering clique of y’; let Y’ be the other. Since Y is large, x ’ f$ Y. So x ‘ E Y‘, and thus Y’ is small. We conclude that y‘ E By.Similarly if y‘ E By,x ’ E X - Y, and x ’ y’, then x ’ E B,. Also by Proposition 6(ii), if x ’ - y ’ in B then N [ x ’ ]- ( X U Y ) = N [ y ‘ ]- (X U Y). Let C = (C,, C,) be the component of b containing x and y. We first show that if v E C then N [ v ]- ( X U Y) = 0. This follows easily by induction on d&, v) once we show it for v = x . So suppose w E N [ x ] - (X U Y). Let X‘ and Y’ be the small covering cliques of x and y. Then w , x , and y are in X‘ and Y’. Since X’ # Y’there exist u E X ’ and v E Y’ such that u + v. Clearly u w v. Since u f$ Y’, u E Y, and similarly v E X. By the remark above, u and v must be in B,U By. Let W and W’ be the covering cliques of w , where W is large. Since neither u nor v is in W, u and v are both in W’. But this contradicts u -/-v. Assume that IC,l S ICvl. We show that there is a matching in B‘ which misses at most s vertices of C,. Otherwise there exists a subset T of C , such that IT1 > ICY n Z(T)I s. Let Z = C yn Z(T) and N = (Cy- Z(T)). First suppose there exist u E N and v E C, such that u + v. Since 6(G) 2 w ( G ) s, IN(v) n NI 2 s - (ZI 3 2s 1- JTI. Thus some subset of { u , v } U T U ( N ( v ) n N) induces K2F+3-e, which is contradiction. Now suppose that every element of N is adjacent to every element of C,. As before, since x and y do not have a common covering clique, there exist v E C, and u E C y such that u v. Since 6(G) 3 w ( G ) +s, ( N ( u )flC,)l 3s. Since IC,l s ICyl, IN1 3 s 1. Thus some subset of {u, v } U N U ( N ( u )n C,) induces K,,, - e, which is a contradiction. Now we are prepared to obtain a contradiction by constructing a ( x ( G )- 1)coloring of G. Partition C, into S and T and partition C yinto S’ and T’ such that (SI s s and there exists a one-to-one function f : T + T’ such that x +f(x). Let qo by a (%(G)- 1)-coloring of G - (T U Cy).The only colored vertices adjacent to vertices in Tare in X.Thus we can fix a set A of IT1 colors that are available for coloring any vertex in T. The only colored vertices adjacent to the vertices in S’ are in S U ( Y - Cy).Thus we can extend qo to a ( x ( G ) - 1)-coloring q1 of G - (T U T’), which does not use any of the colors from A on S‘. The only colored vertices adjacent to vertices in T’ are in ( Y - T‘) U S. Thus there exists a set A‘ of (T’I colors that are available for coloring any vertex in T’. Extend q1to a ( x ( G )- 1)-coloring W of G as follows. For each color /3 E A t l A’ choose an uncolored x E T and color both x and f(x) with @. Color the remaining vertices of T and T’ with the remaining colors in A and A‘.

-

- -

+

+

+

+

+

H.A. Kierstead

122

We now construct M. Let the vertices of M be the covering cliques. Corresponding to each x E V, there is an edge ex of M which joins X and Y, the covering cliques of x . Using (2) and (4), it is easy to check that L ( M ) = G . Clearly A ( M ) = w ( G ) . Finally we show that M does not contain an s-triple. Since G is critical, M is edge critical. By Proposition 6(i), p ( M ) S 2s. Thus by Lemma 5 and the fact that 7s - 2 S R(3,4s - l ) , 6 ( M ) 3 4s. Thus every covering clique is large. Suppose X , Y,and Z are distinct covering cliques such that IX fl YI + IX n ZI 3 2 s 1 and Y fl Z # 0. Since X, Y, and Z are large covering cliques, X f l Y f l Z = 0. Say w E Y f l Z. Then there exists x E X such that w + x . Then some subset of (X n Y) U ( X fl Z ) U { w , x } induces K2F+3- e, which is a contradiction. 0

+

+

We remark that the theorem cannot be improved when o ( G ) s 3 R(3,4s 1). Let M be the multigraph formed by replacing every edge of K2k+,by s parallel edges, where k > 1. Then f ( M ) 3 2 4 M ) / v ( M ) - 1) = 2sk s = A ( M ) s. If C is the line graph of M, then G induces neither K 1 , 3nor K2F+3 - e, w ( C ) = A ( M ) , and x(C) = x ' ( M) . Thus x ( G )= o ( G )+ s. Whether the upper bound holds for o(G) s < R(3,4s - 1) is not known.

+

+

+

3. Problems The most important open question concerning edge coloring is the following conjecture of Goldberg [6]. Equivalent conjectures were posed independently by Andersen [l] and Seymour [ 151, but Goldberg's formulation is particularly attractive. Define the density W ( M ) of a multigraph M by

where M Z I H and v(H) is odd. Since no color class can have more than ( v ( H )- 1)/2 edges in H, W ( M )6 f ( M ) .

Conjecture 7. If M is a multigraph such that x ' ( M ) > A ( M ) + 1 then x ' ( M ) = W(M).

+

The Petersen graph shows that the hypothesis x ' ( M ) > A ( M ) 1 is necessary. This conjecture has many exciting consequences. For instance, if M is critical and f ( M ) > A ( M ) 1, then v ( M ) is odd and less than A ( M ) . It also implies Theorem 4 and that if p ( M ) = 2 and f ( M ) > A ( M ) + 1 then M contains a 5-sided triangle. It was confidence in the conjecture together with this fact that lead Kierstead and Schmerl to the proof of Theorem 1. Goldberg [6] has proved the conjecture for A ( M ) S 9 . The proof provides a polynomial time algorithm that calculates W ( M ) and colors M with W ( M ) colors, if x ' ( M ) > A ( M ) + 1. It is

+

Edge coloring of multigraphs

123

likely that a proof of the whole conjecture would provide a similar algorithm. A reasonable way of attacking the general conjecture is to try to prove interesting results that are implied by it.

Problem 8. Show that if M is a multigraph with p ( M ) = 2 such that x ’ ( M ) > A(M) + 1, then M contains a 5-sided triangle. The next problem was left open in [12]. At first it may seem rather technical, but it has interesting consequences.

Problem 9. Show that if M is a multigraph with no 4-sided triangles such that p(M)s 2 and S ( x ) + S ( y ) S 2A(M) + p(x, y) - 3 whenever x - y , then x ’ ( M ) = A(M). It is shown in [12] that this problem is equivalent to the statement that if G is a linear graph such that A(G) S 2w(G) - 4, then x(C) = o ( G ) and that this would be an optimal result. In particular it would be an improvement on Theorem 2. Theorem 1 is easily seen to be a special case of this formulation. Finally we state the remaining open problem from this article. A positive solution would imply that if G is a graph which does not induce K 1 , 3 , then x(G) s 3w(G)/2, which would be a nice generalization of Shannon’s Theorem

Problem 10. Show that if G is a graph which induces neither K , , 3 nor K,,, - e, then x(G) s o ( G ) s (even when w ( G ) s < R(3,4s - 1)).

+

+

References [l] L.D. Andersen, O n edge colorings of graphs, Math. Scand. 40 (1977) 161-175. [2] L.W. Beineke, Derived graphs and digraphs, Beitrage zur Graphentheorie (Teubner, Leipzig, 1968) 17-33. [3] R.L. Brooks, O n coloring nodes of a network, Proc. Cambridge Phil. SOC.37 (1941) 194-197. [4] S.A. Choudum, Chromatic bounds for a class of graphs, Quart. J. Math. 28 (1977) 257-270. [5] P. Erdos, Graph theory and probability, Canad. J. Math. 11 (1959) 24-38. [6] M.K. Goldberg, Edge coloring of multigraphs: recoloring technique, J. Graph Theory 8 (1984) 123- 127. [7] A. Gyirfils, On Ramsey covering numbers, Coll. Math. SOC.Jinos Bolyai 10, Infinite and Finite Sets, 801-816. [8] 1. Holyer, The NP-completeness of edge coloring, SIAM J. Comput. 10 (1981) 718-720. 19) M. Javdekar, Note on Choudum’s “Chromatic bound for a class of graphs”, J. Graph Theory 4, 3-12. [lo] H.A. Kierstead, On the chromatic index of multigraphs without large triangles, J. Combin. Theory Ser. B 36 (1984) 156-160. [Ill H.A. Kierstead, and J.H. Schmerl, Some applications of Vizing’s theorem to vertex colorings of graphs, Discrete Math. 45 (1983) 277-285.

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[12] H.A. Kierstead, and J.H. Schmerl, The chromatic number of graphs which induce neither K , , 3 nor K , - e, Discrete Math. 58 (1986) 253-262. [13] C.E. Shannon, A theorem on coloring the lines of a network, J. Math. Phys. 28 (1949) 148-151. 1141 D.P. Sumner, Subtrees of a graph and the chromatic number, The Theory of Applications of Graphs (G. Chartrand, ed.) (1981) 557-576. [15] P.D. Seymour, On multi-colorings of cubic graphs, and conjectures of Fulkerson and Tutte, Proc. London Math. SOC.38 (1979) 423-460. [16] V.G. Vizing, The chromatic class of a multigraph, Cybernetics 3 (1965) 32-41.


125

INTERVAL VERTEX-COLORING OF A GRAPH WITH FORBIDDEN COLORS Marek KUBALE Institute of Informatics, Technical Universiry of Gdaiisk, 80-952 Cdaiisk, Poland We consider a problem of interval coloring the vertices of a graph under the stipulation that certain colors cannot be used for some vertices. We give lower and upper bounds on the minimum number of colors required for such a coloring. Since the general problem is NP-complete, we investigate its complexity in some special cases with a particular reference to those that can be solved by a polynomial-time algorithm.

1. Introduction The classical model of coloring the vertices of a graph with single colors so that no adjacent vertices have the same color is too limited to be useful in many practical applications. A good illustration of this is the following school timetabling problem. Suppose that we have to arrange the times at which certain lectures are to be given knowing that some particular lectures cannot be held at the same time, since there may be students who wish to attend both of them. This scheduling problem can be represented by a graph in which vertices correspond to lectures and edges correspond to pairs of lectures that cannot be given simultaneously. Thus the timetabling problem is equivalent to the vertexcoloring problem stated in its standard form. However, in practice there are usually more restrictions generated by student and staff requirements which have to be taken into consideration in finding a satisfactory timetable. For instance, we may have to take into account the fact that certain lectures require at least two consecutive hours and that some teachers are not available at certain hours. Due to such restrictions the mathematical models are not simple coloring problems anymore. For this reason we must consider more general notions of graph coloring, and this paper is devoted to one of such generalizations. More formally, let G = (V, E) be a simple graph with the vertex set V = { v l , . . . , v,} and the edge set E = { e , , . . . ,em}. We define a vertexweighting function W :V + N , where N is the set of all positive integers. The pair (G, W) is said to be a weighted graph. By an interval k-coloring of (G, W ) we mean a function C: V + {S G (1, . . . , k}} whose values are sets of consecutive integers satisfying IC(v)l= W ( v ) and C ( u ) fl C ( v ) = 0 whenever {u, v } E E. The interval chromatic number x(G, W ) is the least k for which there is an interval k-coloring of (G, W). Moreover, we assume that with each vertex v E V there is associated zero or more (but no more than a bounded number of) intervals of 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V.(North-Holland)

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forbidden colors, i.e. the colors which cannot be assigned to v in any interval coloring of the graph G. Colors that are not forbidden form intervals of permissible colors. Since our coloring has to be an interval coloring without forbidden colors, it follows that C ( v ) must be consecutive permissible colors for all v. In other words, we are given a forbidding function F: V - , { S c (1, . . . . k}} with the property C ( v ) r l F(v) = 0, which must be satisfied in any interval k-coloring of a weighted graph (G, W). Consequently, an interval k-coloring of (G, W) that avoids F is an interval function C: V - {S (1, . . . , k}} fulfilling IC(v)l= W ( v ) and C ( v )fl F ( v ) = 0 for each v E V, and C ( u )r l C ( v )= 0 whenever ( u , v } E E. By the generalized (interval) chromatic number x(G, W, F) we mean the least number of colors needed to map the vertices of G to appropriate coloring intervals of size given by W in the presence of forbidden colors specified by F. A chromatic coloring is one that achieves the generalized chromatic number. It is not always easy to determine the value of x ( G , W, F). The general decision problem: “Given G, W, F and an integer k, is there an interval k-coloring of (G, W) that avoids F” is NP-complete, since it is already NP-complete to determine the chromatic number of a graph [5]. Hence it is unlikely that the generalized chromatic number can be calculated by a polynomial time algorithm. For this reason in Section 2 we give lower and upper bounds on x(G, W, F) that can be computed efficiently. In the subsequent two sections we investigate how the complexity of the problem is affected by restricting the problem’s domain. We consider some simplified subproblems caused by restrictions involving the form of functions W and F and the structure of a graph G. In this way we arrive at a number of negative results in Section 3 and positive results in Section 4. All the special-case results are then summarized in Section 5. We conclude with some remarks on the complexity of a relevant problem of interval edge-coloring of a graph with forbidden colors.

2. Bounds on the generalized chromatic number Let p,, be the first color forbidden for vertex Y E V and q,, the last color forbidden for v. Obviously, p,, q,, and p,, = q,, = 0 if F(v) = 0. If p,, = 1 then by s, we denote the size of the first interval of forbidden colors for v, i.e. (1, . . . , s,} E F(v). Otherwise we set s,, = 0. Next by W+(U), U 5 V we mean the quantity W+(U)= W ( U ) + m i n { s , : v ~U } ,

(1)

where W(U) = C U E uW ( v ) is the total weight of subset U. Let S ( C ) be the family of all 1-,2-, . . . ,1-element subsets of V that induce complete subgraphs of C. Then by o ( C , W, F) = max(W+(U): U E S ( G ) } (2)

Interval vertex-coloring of a graph

127

we mean the weight of the “heaviest appended complete subgraph” of G. Now we are ready to prove

Theorem 1. For any graph G and functions W, F w ( G , w , F ) s x(G, w , F ) .

(3)

Proof. In any interval coloring of graph G the vertices of any complete subgraph induced by U c V must have different colors greater than min{s, : v E U } . By choosing the heaviest of such subgraphs of C we get the lower bound. 0 A straightforward way of finding the value of (2) requires finding of all cliques of G. The number of cliques in an n-vertex graph can be as large as 3n’3 and the best known algorithm for generating all the cliques has the complexity of O(1.44”) [9]. Nevertheless, there are graph (e.g. planar graphs, line graphs) for which w ( G , W, F) can be obtained in polynomial time. If this is not the case we can estimate the value of w ( G , W, F) from below by restricting the calculations to all vertices and edges. We then have m a x { W ( v ) + s , : v E V } s x ( G , W, F) v

max{ W(u) + W(v) + min{su, s,} : {u, v } E E } s x(G, W, F) u.u

(4)

(5)

Now bounds (4) and (5) can be found in time O(m + n ) . In order to give an upper bound on x(G, W, F) by 6(G, W, F) we define the quantity

6(G, W, F ) = m a x { W ( N , ) + q , : v ~ V } ,

(6)

where Nv is the neighborhood of vertex v in graph C, i.e. the set of all vertices adjacent to v plus vertex v. Using the previous terminology we can say that 6(G, W, F) is the weight of the “heaviest appended neighborhood” of any vertex of G.

Theorem 2. For any graph G and functions W, F

Proof. To every vertex vi of (G, W) we link one new vertex ui of weight W‘(ui) = qvj. Thus (C, W) is a subgraph of a weighted graph (G’, W’), where W‘ is the extension of function W to set V U { u l , . . . , u,}. Now we apply the following coloring procedure to the list of vertices arranged in order u l , . . . , u,, vl, . . . , v,. First, each vertex ui gets colors C(ui)= { 1, . . . , W‘(u,)}. This makes it possible to avoid all forbidden colors for the vertices of G. Then for every c = 1,2, . . . we check if there is a yet uncolored vertex v i , i = 1, . . . , n that can

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be assigned the set {c, . . . , c + W ’ ( v i )- 1). If so, we color such a vertex vi and continue checking on the list starting from v ; + ~If. not, c is increased by one, and so on until all vertices have been colored. Now observe that when coloring vertex vi with set C ( v , ) all colors preceding C ( v J are occupied by neighbors of vi. For lack of gaps between intervals and because the obtained coloring of (G’, W ’ ) omits forbidden colors of (G, W ) , coloring of vi can be postponed no longer than we get to color W’(N,,,- {v;}) + 1. Thus the last color assigned to vi is less than or equal to W’(N,,,). Since W’(N,,,)= W(N,,,)+ W’(ui), so x(G, W , F ) 6 max{ W(N,,,) q,,,}, i = 1, . . . , n, and the upper bound follows. 0

+

In contrast to the lower bound (2) the upper bound (6) can be obtained in time proportional to the size of G, i.e. O ( m + n). Finally note that the upper bound is tight in the sense that there are graphs for which the generalized chromatic number is equal to S(G, W , F). A simple example is an arbitrary weighted complete graph with a common set of forbidden colors (1, . . . ,q } = F (v ) for all v. Nevertheless, bound (6) can be improved in some special cases of unweighted graphs. For example, if is a complete bipartite graph with unit weights and arbitrary forbidden colors then X ( K ~ - , ,W~ ,,F ) 6 max{g,, :

E V},

(8)

where g,, is the gth permissible color at vertex v E V. Also, every planar graph is properly colorable within the first six permissible colors at each vertex [l].

3. NP-completeness results Let CN1, CN2, CN3 stand for the decision problems for the chromatic number x(G), interval chromatic number x(C, W ) , and generalized chromatic number x(G, W, F), respectively. In what follows, in order to distinguish those features

that make CN3 an intractable problem, we consider the complexity of this problem in some of its more interesting special cases. For brevity, we say that the weighting function is binomial if W : V + { 1, L}, where L is any integer greater than 1. The function W : V - . {L} is said to be uninomial if L > 1, and unary if L = 1. As we know the general problem CN3 is NP-complete. This implies the NP-completeness of all special cases of CN3 where general graphs are allowed, since all of these include the CN1 problem as a subproblem. It turns out, however, that CN3 remains an NP-complete problem even if every pair of vertices is joined by an edge.

Theorem 3. CN3 is NP-complete even when restricted to complete graphs with at most one forbidden color per vertex.


129

Proof. Our reduction uses the well-known partition problem [ 5 ] :“Given a set of positive integers A = { a , , . . . , a,) such that al + a , = 2b; is there a partition P of A such that C i p pa; = C i e pai = b?”.

+- -

For a given set A we construct a complete graph K,, on n vertices v l , . . . , v, with weights W ( v i )= a i , i = 1, . . . , n. Additionally, we assume F(v,)= { b 1) for all i. Suppose that there is an interval (26 1)-coloring of (K,, W) in which color b + 1 is not used. This means that some of the vertices are colored completely within 1, . . . ,b and the others within b 2, . . . , 2b + 1. Conversely, the existence of partition P implies that there is (2b 1)-coloring of K, such that b + 1$ C ( v i )for i = 1, . . . , n. Thus x ( K , , W, F) S 2b 1 if and only if there is a partition P of A, and the claim of theorem follows. 0

+

+

+ +

+

It is worth mentioning that the above subproblem becomes polynomially solvable if function W is binomial (cf. [ 4 ] ) .If, however, there are at most three intervals of forbidden colors per vertex then CN3 is NP-complete even if W is uninomial.

Theorem 4. CN3 is NP-complete even when G is a complete graph, W is uninomial, and F is such that there are at most three intervals of forbidden colors per vertex.

Proof. We transform to CN3 the following multiple choice scheduling within intervals (MCSWI): “Given m processors, a set of n tasks T = { t , , . . . , t , } , a common task length I , and for each task ti E T a collection { [ q ( i ) ,d j ( i ) ]:1s j s k ( i ) } of permissible scheduling intervals with integer release time/deadline endpoints and such that 0 S q ( i )S d j ( i )- 1 for j = 1, . . . . , k ( i ) . The question is whether there exists an rn-processor nonpreemptive schedule for T that places each task i in one of its permissible intervals”. MCSWI was shown to be NP-complete even if m = 1, 1 = 2 and there are at most two intervals (one of length 4 ) per collection [S]. Without loss of generality we assume that in a given instance of MCSWI each task t, has exactly two permissible intervals: [ d , ( i )- 2, d l ( i ) ] , [ d 2 ( i )- 4,d 2 ( i ) ] . Let d be the latest deadline of all collections of the intervals. Then the corresponding instance of CN3 is graph K,, function W such that W ( v , )= 2 for i = 1, . . . , n , and function F such that there are three intervals of forbidden colors per vertex, namely F(vi)= (1, . . . , d , ( i )- 2 ) U { d l ( i )+ 1, . . . , d2(i)- 4 ) U { d 2 ( i )+ 1, . . . , d}. Now it is easy to see that there exists a solution to the single-processor case of MCSWI if and only if x ( K , , W, F) S d . 0 The MCSWI problem remains NP-complete even if m = 1, three task lengths

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are allowed and there is just one permissible interval per task (cf. [ 4 ] ) .Hence, taking advantage of the transformation used in the proof of Theorem 4 one can prove that CN3 is an NP-complete problem if G is a complete graph, W is a 3-valued function, and there are at most two intervals of forbidden colors per vertex. Our last negative result deals with unary bipartite graphs.

Theorem 5 . CN3 is NP-complete even when restricted to unary bipartite graphs with at most two intervals of forbidden colors per vertex. Proof. Our reduction uses the well-known 3-satisjiability (3SAT) problem: “Given a collection C = { c , , . . . , c,} of clauses on a set X = { x , , . . . ,x,} of Boolean variables such that Icj(= 3 for j = 1, . . . , m . Is there a truth assignment for X that satisfies all the clauses in C?”. Assume that we have an instance of 3-SAT with n variables and m clauses. We must construct a bipartite graph G and a positive integer k such that x(G, W, F) s k if and only if there is a satisfying truth assignment for X.For each variable xi E X we create three vertices: vertex xi and two other vertices x & - , and x; adjacent to xi and called literals. We think of x + as unnegated and x - as negated form of x for all x . In addition, every triple of vertices has two permissible colors in common, namely 2i - 1 and 2i for i = 1, . . . , n. Then for each clause cj E C we add one vertex cj adjacent to three literals according to the corresponding variables that occur in cj in their unnegated or negated form. Next, to each cj, j = 1, . . . , m we link 2n pendant vertices named p i , , . . . ,pjZn.There are no forbidden colors for vertices cj. Also vertex pjh, h = 1, . . . , 2n has no forbidden colors if cj is adjacent to x l or x i . Otherwise it has just one permissible color h. The construction of our instance of CN3 is completed by setting k = 2n. It is easy to see that G is bipartite and all the vertices have at most two intervals of forbidden colors. Suppose that the Boolean formula is satisfiable and let a , , . . . , a, be any assignment of variables that evaluates the problem instance to true. With each value a, we associate an integer b, such that b, = 2i if a, =false and b, = 2i - 1 if a,=frue. Each vertex xi can be colored with b,. Also, each pair of literals adjacent to xi can be assigned the other permissible color, i.e. 4i - b, - 1. Next, by the construction, each vertex cj can get color b, corresponding to value a, of xi which evaluates that clause to true. Finally, each pendant vertex pjh, 1 < j =sm , 1 s h s 2n can be colored with h or any other color different from that of cj. Thus the satisfiability of the formula implies that graph G is 2n-colorable. Conversely, suppose that there exists 2n-coloring of G that avoids F. From the way in which 2n - 3 of the pendant vertices pjh on cj are preassigned forbidden sets of colors it follows that each of them must be colored with value h equal to the index of a literal not adjacent to vertex cj. Thus each cj is colored with some color 6, in common with vertex x, whose literal is adjacent to cj. If b, is odd we

lnterual vertex-coloring of a graph

131

may regard variable xi as being assigned ai = true, otherwise ai =false. Now consider the whole formula of the 3-SAT instance. By the construction, each clause cj is satisfied by assignment ail, aj2, aj3 for variables xi,, x,,, xi, contained (either negated or unnegated) in it. Therefore conjunction of all the clauses in C is also satisfiable. Thus the 2n-colorability of G implies the satisfiability of the problem instance, which completes the proof. 0 The proof of Theorem 5 can be probably simplified and strengthened.

4. Polynomial solvability results In this section we consider the cases where chromatic colorings can be found efficiently. Our first positive result involves unary complete graphs. In its original formulation it deals with scheduling unit-length tasks with multiple release time/deadline intervals [8], but can be easily transformed into a graph coloring problem.

Theorem 6. If all vertices of a complete graph K,, have the common weight 1 and an arbitrary number of forbidden colors then a chromatic coloring can be found in time O(n4). Proof. See Simons and Sipser [8]. 0 Note that the complexity of this special case depends on the maximum number of prohibited intervals per vertex. For instance, if the number of permissible intervals is one then it is possible to achieve a chromatic coloring of K,, in time (cf. 131). The remaining special-case results deal with bipartite graphs.

Theorem 7. If G is a bipartite graph, W is unary, and F is such that left endpoints of prohibited intervals are odd and right endpoints are even, then G can be optimally colored in time O ( m + n ) . Proof. Let B = B(V,, V2)be a unary bipartite graph with two independent sets V, and V,, V, U V2= V. A chromatic coloring of B can be obtained in two phases. In the first phase each vertex u E V, is colored greedily, i.e. with s, + 1. In the second one, each vertex v E V2is colored greedily. If the number of colors used is odd, say 2; + 1, its optimality follows from the fact that the greedy algorithm first attempts to assign odd values to all vertices. If the number is 2; + 2 then there exists an edge { u , v } E E such that s, = s, = 2;. Hence u must be colored with 2; + 1 and v with 2; + 2. Thus x ( B , W, F) = 2; + 2, as desired.

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'6

'8

'7

'9

'10

'11

'12

"13

Fig. 1. A caterpillar with 5-vertex body and 8 hairs.

Since the determining of sets V, and V, as well as the greedy coloring can be done in time proportional to the size of B, the theorem is proved. 0 The following theorem involves a caterpillar, i.e. a tree in which the removal of all pendant vertices results in a path. These pendant vertices can be thought of as hairs attached to the body of the caterpillar, i.e. a path of non-pendant vertices (see Fig. 1).

Theorem 8. Zf G is a unary caterpillar and each vertex v E V has at most one interval of forbidden colors such that F ( v ) = (1, . . . , s"}, then G can be optimally colored in linear time.

Proof. Let s be the maximum color among all colors forbidden for the vertices of G. If G contains two adjacent vertices with s forbidden colors each (s-vertices, for short) then G is (s + 2)-chromatic. Thus all we need to do is to divide V into independent sets V, and V,, and color elements of V, with s + 1 and V, with s + 2. So suppose that no two s-vertices are joined by an edge. In this case we apply the following limited backtracking algorithm. Step 1. Color all s-vertices with s + 1. Step 2. Color uncolored segments of the body starting from its left endpoint v,. Namely, succeeding vertices are colored greedily until an (s - 1)vertex requiring color s + 2 is encountered; otherwise we go to Step 3. If vj is such a vertex then we move sequentially back in search of the first vertex vi that can be recolored with the color of value one greater than the previous color (but less than s + 2), however not farther than to the last resumption point (initially, vertex v, is called the last resumption point). Next we increase the color of vi and perform the forward step once more. If the backward step has made it possible to color vj with s then vertex vj is stored as the last resumption point and _ -

-U

even

n

Fig. 2. Illustration for Theorem 8, Case 1; white nodes denote (s - 1)-vertices, black nodes denote s-vertices.


133

& Fig. 3. Illustration for Theorem 8, Case 2; white nodes denote (s - 1)-vertices, black nodes denote s-vertices.

the sequential coloring of the next body segment is realized. Otherwise, two following cases are possible. Case 1. No such vertex vi has been encountered. This case is illustrated in Fig. 2, where the remaining body and hair vertices are left out. Case 2. Such a vertex vi has been encountered but the next forward step failed to reach vj. This case is depicted in Fig. 3. Observe that both the cases imply that G has two s-vertices, say u and w, such that the path between u and w consists of even number of (s - 1)-vertices. Conversely, if there are no vertices u and w as described above then the forward step will reach vj. In both cases the path between and including L(, w requires s 2 colors. Hence the whole graph G is (s 2)-chromatic and can be colored by the method described at the beginning. Step 3. Color the remaining pendant vertices in the greedy way. Since any vertex of G is colored at most three times (by two methods) within constant time, the coloring of all n vertices can be done in time O ( n ) . 0

+

+

Theorem 9. If G is an n-vertex star and functions W , F are arbitrary except that each vertex has O ( n ) intervals of forbidden colors then a chromatic coloring can be found in time O(n2log q ) , where q = max{q, :v E V } . Proof, Let S,, be a star on n vertices numbered as in Fig. 4. Our algorithm consists of two phases. In the first phase we color the vertices in order v l , . . . , v,, in the greedy way. If the number of colors thus used is equal to the lower bound (4) or (5) then the algorithm terminates. Otherwise we proceed to the second

v47k "3

"2

Fig. 4. The star S,.

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phase in which we apply a binary search to try and improve the best interval coloring found so far. Initially, let k be equal to a half of the number of colors used in the first phase. Starting from k downwards we color u, with the highest permissible interval while the leaves are colored greedily in order v,, . . . , v,-, starting with color 1. If for some i vertex vi cannot be colored within the limit then v, is assigned the next permissible interval, i.e. one that makes it possible to , presumed limit k color vi as well. However, if there is no such interval for u , ~the is increased appropriately and the process is repeated with the new value of k. Otherwise, with the same k, the greedy coloring is verified for those leaves among v l , . . . , vi-l that have not been assigned colors greater than the current coloring C(v,) and after that it is continued for vi+,,. . . , v,-,. Finally, if for a prespecified coloring of v, all its neighbors have been colored with nonoverlapping intervals, the interval k-coloring of S, that avoids F is stored, the limit k is diminished appropriately, and the process is repeated with the new limit, etc. The correctness of the method follows from the exhaustness of the binary search for a minimal k, so let us estimate the time complexity of the algorithm. In the first phase each prohibited interval is processed at worst once. Since there are O(nz) many of them, the initial phase can be implemented to run in O(n’) time. In the second phase, for a fixed k the central vertex is colored at most n times and the time spent for any leaf is also O(n). Hence in this stage the algorithm performs O(n2) steps. Since the verifying of k-colorability is executed [log, q1 times, the overall running time is O(n2log q ) , as claimed. 0 Observe that we have ignored forbidden intervals in our analysis. Surely these have to be inputs to the problem. Since there are O ( n z )many of them, the input size is at least O(n210gq). Thus our algorithm can be regarded, in the wide sense, as a linear-time algorithm. Moreover, given the intervals of prohibited colors as part of the input, it is possible to remove all restrictions on the number of forbidden intervals. Theorem 9 implies the following positive result concerning bipartite graphs.

Theorem 10. If G is a bipartite graph B(V,, V,) with a linear number of forbidden i = 1, 2 such intervals per vertex and there is a vertex u of maximum weight in that for every v E V, we have F (v ) c F(u) and N ( v ) c N ( u ) U {v}, then a chromatic coloring of G can be found in time O ( n 2log q ) .

v,

Proof. Assume, without loss of generality and to simplify the notation, that B(Vl, V,) is connected and that the vertex u belongs to V , . We first color the vertices of the star induced by Nu by means of the method described in the proof of Theorem 9. As we know, this can be done in O(lV,12log q ) time. Let C ( u ) be the interval of colors assigned to u in the optimal coloring of the star. Then we color the remaining uncolored vertices in Vl with colors belonging to C ( u ) in time O(lV,l).Thus the time complexity is O(n2log q ) . 0


135

Finally, note that Theorem 10 implies polynomial solvability of all special cases of complete bipartite graphs I?(&, V,) in which all vertices in V, (or V,) have identical forbidden colors, since the heaviest of such vertices can be regarded as vertex u.

5. Complexity classification

In the previous sections we have considered the complexity of CN3 assuming restrictions imposed on vertex weights, the number of prohibited intervals per vertex, and the structure of a graph. In this way we have identified several classes of highly structured graphs for which chromatic colorings can be obtained in polynomial time. The main results of this investigation are summarized in Table 1. Entries in the table are either “NPC” for NP-complete, “?” for “open”, or O ( . ) for an upper bound on the complexity derived from the best polynomial optimization algorithm known for the corresponding subproblem. In addition, in the column CN3 we have placed signs “ - ” to indicate that there is known a negative result (NP-completeness proof) for a special case of the corresponding to indicate that there is a positive result (polynomialsubproblem, and signs time algorithm) for a special case of that subproblem. We conclude with some remarks on the complexity of an analogous problem of interval edge-coloring of a graph with forbidden colors. First of all note that interval edge-coloring is harder than interval vertex-coloring even when there are no forbidden colors at all (cf. [6]). If, however, G is an edge-weighted graph with forbidden colors on the edges then interval edge-coloring is NP-complete even if G is a star with just one forbidden color per edge. This follows from the fact that coloring the edges of a graph G is equivalent to coloring the vertices of a graph L ( G ) , the line graph of G. Using Theorem 3 and by the fact that L(S,) = K , - , we get the desired result. Another interesting result is due to Even et al. [2] and states that it is NP-complete to decide the edge 3-colorability of an unweighted bipartite graph with single forbidden colors. Although there are also some positive results concerning bipartite graphs (e.g. those arising from completing

“+”

Table 1. Complexity classification for interval vertex-coloring ~~

Graphs

CN 1

General graphs Complete graphs Bipartite graphs Trees Stars +

CN2

CN3

References

NPC

NPC

O(n) O(m n ) O(n) O(n)

a n ) O(m n ) a n ) O(n)

NPC NPC-‘+ NPC-++

~ a r 151 p Theorems 3 , 4 , 6 Theorems 5 , 7 , 1 0 Theorem 8 Theorem 9

+

NP-completeness proof. Polynomial-time algorithm.

+

?+ O(n2log 9 )

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partial latin squares), the interval edge-coloring of a graph with forbidden colors is NP-complete for all classes of graphs for which the corresponding problem CN3 was open or polynomially solvable (in Table 1). More details on the subject can be found in [7].

Acknowledgments The author is indebted to Dr. W. Kubiak for his helpful discussions and to the referees whose suggestions have improved the presentation of this paper.

References [l] P. Erdos, A. L. Rubin and H. Taylor, Choosability in graphs, in: Proc. West Coast Conf. on Combinatorics, Graph Theory and Computing (Humboldt State University, 1979) 125-157. (2) S. Even, A. Itai and A. Shamir, On the complexity of timetable and multicommodity flow problems, SIAM J. Comput. 5 (1976) 691-703. [3] G.N. Frederickson, Scheduling unit-time tasks with integer release times and deadlines, Inf. Process. Lett. 16 (1983) 171-173. [4] D.S. Johnson, The NP-completeness column: An ongoing guide, J . Algorithms 4 (1983) 189-203. (51 R.M. Karp, Reducibility among combinatorial problems, in: R.E. Miller, J.W. Thather, eds., Complexity of Computer computations (Plenum Press, New York, 1972) 85-103. [6] M. Kubale, The complexity of scheduling independent two-processor tasks on dedicated processors, Inf. Process. Lett. 24 (1987) 141-147. [7] M. Kubale, Graph coloring, in: A. Kent, J.G. Williams, eds., Encyclopedia of Microcomputers (Dekker, New York) (to appear). [8] B. Simons and M. Sipser, On scheduling unit-length jobs with multiple release time/deadline intervals, Oper. Res. 32 (1984) 80-88. [9] E. Tomita, A. Tanaka and H. Takahashi, The worst-case time complexity for finding all the cliques, Technical Report VEC-TR-CS (University of Electro-Communications, Tokyo, 1988).


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HADWIGER'S CONJECTURE (k = 6): NEIGHBOUR CONFIGURATIONS OF 6-VERTICES IN CONTRACTIONCRITICAL GRAPHS Jean MAYER Universite' Paul Vale'ry, Montpellier, France

1. Definitions; purpose of the study The graphs considered here are simple (without loops or multiple edges). A vertex colouring is such that two neighbour vertices (joined by an edge) are of different colours; colours are designated by numbers 1 , 2 , 3 . . . A k-coloration of a graph G is a colouring using k different colours; when it is possible, G is said to be k-colourable. If G is k-colourable, but not (& - 1)-colourable, it is said to be k-chromatic; k is the chromatic number of G. To contract a graph G consists in deleting the vertices and edges of a connected subgraph H of G which is replaced with a new vertex h ; the edges of G having one end in H are replaced with edges joining G - H with h, the other end remaining unchanged; multiple edges or loops occasionally resulting from this operation are eliminated. It is possible to carry out a contraction as a sequence of elementary contractions bearing upon one edge at once. Thus contraction is a transitive operation. Let us now suppose G connected. It is possible to continue its contraction until K , , the trivial graph, is obtained. Previously a graph r will be reached, such that its chromatic number is k, but every graph contracted from r has a chromatic number strictly lower than k:r is said to be k-chromatic contraction-critical (shortly &-c.c.); this type of graphs was introduced and studied by Dirac [4,5,7,81. Let us call K , the complete graph of n vertices. According to Hadwiger's conjecture [9], every connected &-chromatic graph can be contracted to Kk.In other words: The only k-c.c. graphs are the complete graphs (Kk).This conjecture which is obvious for k 6 3 was proved by Dirac [4] for k = 4. For k = 5, Wagner [14] showed that it is equivalent to the four colour conjecture which was demonstrated by Appel and Haken, with the help of Koch [l, 21. As for k 3 6, it is undecided. However, Dirac [7] showed that every 6-chromatic graph is contractible to K, - E (K, with one edge deleted). Analogous results were found for k z-7 by Jakobsen [ 101 and Mader [12]. 0012-365X/89/$3.50@ 1989, Elsevier Science Publishers B.V. (North-Holland)

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We shall consider here the case k = 6. Let r be a 6-C.C.graph, not isomorphic to K6. We shall prove that, if rexists: 1. the configuration induced by the neighbours of a vertex of degree 6 (configuration designated here by &) can only be one of the eight types presented in Fig. 2 (type H being excluded, as reducible); 2. if an edge joins two v6, these are of the same type (namely A, C or J ) . Notations used

A graph or subgraph is designated by a capital letter, a point or vertex by a small letter (italics are reserved for numbers or indices named by letters). Usual signs for inclusion or belonging to a set are used. Moreover we shall denote: vj:vertex of degree (or &vertex). (a, b ) : edge joining a and 6. P(a, b): path connecting .a and b. K,: complete graph of 0 vertices. SF): K , with one edge subdivided into a path of length m. C > HIC is contractible to H. r:6-chromatic contraction-critical graph, not isomorphic to K 6 . r(a): subgraph of r induced by the neighbours of the vertex a (or neighbour configuration of a ) . If a is a v,, one may write K 6 , by (3.5). (2) 21 is of type G, with xo E K5 - E c v U T ( v ) . But in this case xo is a marked vertex in T ( v ) and cannot be a 216: contradiction. (3) v is of type J: this case is excluded by (3.6.) 0 A (v,, v,)-edge joins two vertices of the same type, namely A , G or J . (3.8)

Proof. Let xI and x, be the two 6-vertices. They can have only two or three common neighbours. If they have two common neighbours joined by an edge, both x, and x, are of type A by (3.7); if the two common neighbours are not joined by an edge, both x1 and x, are of type G; if x I and x, have three common neighbours, both of them are of type J (and their neighbouring configurations have the same centre). In strong contrast with type J , the presence in the cliques K4 extremely rare, so we can state:

r of a v6 of type C or F makes

If there is in T a v, of type C or F, I’does not contain any other 216. (3.9) Proof. Let xo be a v6 of type C with T ( x O )consisting of two triangles (a, b, c) and ( a ‘ , b’, c’) joined by two edges (a, a ’ ) and (b, b’). By ( 3 . 3 , a K4 disjoint from xg necessarily contains a or a’ and b or b‘, i.e. a and b or, symmetrically, a’ and b’. If it contains a and b and not c, (a, b) belongs to four triangles and T(xo,a ’ , b ’ , c’) is a K4 disjoint from { a , b}, a case which is excluded by (3.4). Therefore the K, must contain c and is, say, r(a,b, c, d); r - x , ] can include only one more K4, namely T ( a ’ ,b’, c’, d’). But under these conditions d (or d ’ ) cannot be a v,, by (3.5). Since xo has no 6-neighbours by (3.7), xo is the only vh in r. For xo of type F, i.e. with (c, c’) E r, the proof is strictly the same. 0

4. G-vertices

If r contains non-isolated 6-vertices of type G, they can only form K,- or K,-components (Figs 6 , 7 , 8 , left). In the present section we shall prove that: (1) A K, made of three G-vertices is a reducible configuration. (2) If two G-vertices form a K,, r d o e s not contain any other 6-vertex.

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Fig. 6.

By construction, if two G-vertices x1 and x 2 of r f o r m a K 2 , r(xl) U r ( x 2 ) = X is necessarily isomorphic to Fig. 6 (left) or to Fig. 7 (left). A third G- vertex can be added to Fig. 6 (in position e ( = x 3 ) ) to give the Fig. 8 (left) with X = T ( x l )U r ( x 2 ) U r ( x g ) . In each of these three cases let us designate by U the complement of X in r, i.e. U = r - X. Every vertex of X - x1 - x 2 (resp. X - x l - x 2 - x 3 ) has neighbours in U , by (2.1). These vertices are the attachments of U in X. The attaching sets of U are represented in Figs 6, 7, 8 (right). U can be connected or can consist of connected components U,, U2,. . . , each of which has at least six attachments in X,because of (2.3).

The configuration of Fig. 6 is reducible by r>K 6 , unless U = U l + U2, the connected components U, and U2 being respectively attached at {a, b , , d l , e, d 2 , b 2 } and at {a, c1, d l , el d 2 , c2). (4.1)

Proof. There are three cases, the first two of which are reducible: (1) U, is attached at b l , c l , b2 (or equivalently c2) and at least three vertices

Fig. 7.

Hadwiger's conjecture (k = 6 )

145

a.I

bl

b3

83

a2

Fig. 8. E P(a, c2, d2, e, d,). Let us contract P to an edge ( a , d , ) so that a and d , remain (or become) attachments of Ul; let us then contract U, U b, Ux 2 to a vertex: we get the result r > K 6 . This case includes the case of U connected. (2) U, is attached at a, b,, c,, d , , e, d2. By exclusion of Case 1, U2 isattached at a and d,: therefore the contraction of U, provides the edge ( a , d , ) ; then contract U, U d2 U x , to a vertex: you get a K6 again. (3) U, is attached at a, b , , d , , e, d 2 , b2; U, is attached at a , c , , d , , e , d,, c,. This is the irreducible case of the statement (other irreducible cases differ from that only by the notations, e.g. by the permutation of bl and cI). Of course, if there exists a component U3 providing the edge ( a , d , ) , U, U (b,, c,) U U, can be treated as a connected component of I'- q x , ) and we get a K 6 again. 0

The configuration of Fig. 7 is reducible by r >K 6 , unless U = U , + U2, the connected components U, and U, being respectively attached at {al, b l , d l , a2, b,, d,} and at { a , , C I , d l , a21 c2, d2). (4.2) The proof is quite similar to that of (4.1). A triangle of G-vertices is reducible.

(4.3)

Proof (see Fig. 8). By (4.1), there remains only one case to be considered, in which U = U1 + U,, the attaching sets being respectively { a , , b , , a2, b,, a3, b3} and {al, c,, a,, c2, a3, c3}. Then contract P ( a , , x 3 , a3, c2, u2) to an edge ( a , , a2); contract U, U c3 to a vertex to obtain the edge (c,, c3);contract P ( x l , x,, b2)and P(b3, c3, c,) each to a vertex; finally contract U, to a vertex f : you get a K 6 (see Fig. 9). 0 The irreducible cases of (4.1) and (4.2) are concerned by the following

J . Mayer

146

Fig. 9.

statement:

If r contains two G-vertices x , , x2 joined by an edge, r does not contain any K4 except those included in T ( x , )U T(x2). By consequence, there are no other 6-vertices in r than x , and x2. (4.4) Proof. We shall treat only the case of Fig. 6. The case of Fig. 7 is very similar. U consists of two connected components UI and U2 with respective attaching sets { a , b , , d l , e, d 2 , 62) and { a , cl, d , , e, d 2 , c2}. Let us suppose that rcontains a 4-clique K included in (say) U , U a U e. There are three cases: (1) K contains neither a nor e. By ( 2 . 3 ) , r - a - e is 4-connected, so there are four paths from the vertices of K to { b l , d l , d 2 , b 2 } :contract each of them to a vertex; contract ( x , , x 2 ) ; contract a U U2U e to a vertex: you get a K 6 , against the assumption made about r. (2) K contains a ; it does not contain e, because ( a , e ) F#I'.Since r - b2- e is 4-connected, there are three paths from K - a to { b , , d , , d 2 } : contract each of them to a vertex; contract (a, b2) and ( x l ,x 2 ) ; contract ( b l , cl) and (b2,c2); finally contract U2 U e to a vertex; you get a K6 again.

Hadwiger's conjecture

( b = 6)

147

(3) K contains e (and not a ) . Since r - d2 - a is 4-connected, there are three paths from K - e to { b , ,d l , b 2 } : contract each of them to a vertex; contract (e, d2) and (xl, x 2 ) ; contract ( b l , c l ) and ( b2 ,c2); finally contract U,U a to a vertex: again, you get a K 6 . Since a v6 of r belongs to a clique K4, a v6 different from x1 or x 2 cannot exist in r. 0

5. Concluding remarks It follows from the two preceding sections that, except when it consists of a K 2 made of two G-vertices, the 6-vertex subgraph of r can consist only of: (1) connected components, all made of A-vertices of J-vertices (or is exclusive here, by (3.4)) and (2) isolated vertices. It is important to remark that r may contain no vertex of degree 6. Mader [12] has proved that if a graph has p vertices and q edges, p 2 6 and q 2 4p - 9, then it can be contracted to K6. T h i s for r q G 4 i - 10 and r must contain a v6 or a v,; if it contains no 6-vertex, it must have at kast twenty 7-vertices. But, after our examination, & can have more than a hundred different types. We are greatly indebted to the referee for precise remarks on the paper, and to Mrs Florence Depraz for carefully reading over the English text.

Note added in proof Types F and J are reducible.

References [I] K. Appel and W. Haken, Every planar map is four colorat : Part 1, Disc arging, Illinois J. Math. 21 (1977) 429-490. [2] K. Appel, W. Haken and J. Koch, Every planar map is four colorable: Part 2, Reducibility, Illinois J. Math. 21 (1977) 491-567. [3] C. Berge, Graphes et Hypergraphes (Paris, Dunod, 1973, 28me Cd.), 516 pp. (English translation: North-Holland, Amsterdam, 1973). [4] G.A. Dirac, A property of 4-chromatic graphs and some remarks on critical graphs, J. London Math. Soc. 27 (1952) 85-92. [5] G.A. Dirac, The structure of chromatic graphs, Fund. Math. 40 (1953) 42-55. [6] G.A. Dirac, Extensions of Menger's Theorem, J . London Math. SOC.38 (1963) 148-161. [7] G.A. Dirac, On the structure of 5- and &chromatic abstract graphs, J. Reine Angew. Math. 214 (1964) 43-52. [8] G.A. Dirac, Homomorphism theorems for graphs, Math. Ann. 153 (1964) 69-80. [9] H. Hadwiger, Uber eine Klassifikation der Streckenkomplexe, Vierteljahresschr. Naturforsch. Ges. Zurich 88 (1943) 133-142.

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[lo] I.T. Jakobsen, A homomorphism theorem with an application to the conjecture of Hadwiger, Studia Sci. Math. Hungar. 6 (1971) 151-160. [ l l ] W. Mader, Uber trennende Eckenmengen in homomorphiekritischen Graphen, Math. Ann. 175 (1968) 243-252. 1121 W. Mader, Homomorphiesatze fur Graphen, Math. Ann. 178 (1968) 154-168. [ 131 J. Mayer, Conjecture de Hadwiger: Un graphe k-chromatique contraction-critique n’est pas k-rkgulier, in Graph Theory in Memory of G.A. Dirac, Annals of Discrete Mathematics. 1141 K. Wagner, Uber eine Eigenschaft der ebenen Komplexe, Math. Ann. 114 (1937) 570-590.


149

ABOUT COLORINGS, STABILITY AND PATHS IN DIRECTED GRAPHS Henry MEYNIEL C.N.R.S., Paris, France

In [2] Berge asked: In a digraph, does there exist an optimal coloring and a path meeting each color class exactly once? We give a negative answer to this question when the chromatic number is >5. H e also asked: Does every directed graph G have a maximum stable set (with IS1 = a ( G ) , the stability number of the digraph) and a partition of vertex-set into paths p,, p,, . . . , pn(c) such that lpi f l SI = 1 for all i? We give here a negative answer to this question.

1. Introduction: definitions and notations Definitions and notations are classical. See [l]for instance. In [2], Berge asked whether every directed graph has an optimal coloring and a path meeting each color exactly once. He proved this for perfect graphs, symmetric graphs and graphs with chromatic number 3. We show here that the answer is negative for graphs with chromatic number 2 5 and we suggest some new problems. Also he asked the following question: In every 1-graph, does there exist a maximum stable set S (a set with IS1 = a ( G ) ,the stability number of the digraph) .and a partition of the vertex-set into paths p , , p2, . . . , such that [pin SI = 1 for all i? We give here an example where the answer is negative.

2. The examples (a)Example of the first problem It is not difficult to see that there is a unique optimal coloring in 5 colors (as indicated on Fig. 1) and that every path of length 5 (hence containing either (a, b, c ) or (a', b ' , c ' ) ) meets the same color twice. (By adding new vertices connected to all the preceding, we can obtain examples for every value of y with y 3 5).

Further questions Problem 1. Does the same property hold true for graphs with chromatic number 4? 0012-365X/89/$3.50 @ 1989, Elsevier Science Publishers B.V. (North-Holland)

H.Meyniel

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a l l arcs going from Ci t o C2 except ( d , d ’ )

Fig. 1.

( p ) Example of the second problem Consider a collection of ten antidirected cycles of length five (by antidirected cycle of length five, we mean a cycle of length five directed in such a way that there exists a unique path of length 3) and let C, = (a,, b,, c,, d,, e l ) , 1s i < 10 be these antidirected cycles. For each cycle we will choose an orientation such that (a,, b,, cJ, 1s i < 10, is the unique path of length 3. Furthermore, we add all directed edges with initial vertex on the C,’s, 1G i s 5 and terminal vertex on the C,’s 6 sj s 10, except for the edges (a,, a,), (al, c,), 6 s 1 s 10. It is easy to check that there is a unique maximum independent set S with IS1 = n ( G )= 11 that is the set of vertices {ul, a6, c h , a 7 , c7, a8, c8, a9, c9, a,,,, Cm>.

Now, we prove that there is no partition of V ( G ) into 11 paths meeting S only once. Suppose the contrary. The paths of such a partition cannot contain the subpaths (u,, b,, c,), 6 S i S 10, otherwise these paths would intersect S twice and so the partition must contain at most 5 paths of length at most 5 (containing (a,, b,, c,), 1s i s 5), the remaining paths being of length at most 4. Hence the partition can cover at most 5 X 5 + 6 X 4 = 49 vertices, while the graph has 5 x 10 = 50 vertices. Please note that in every graph there exists a maximal (for inclusion) stable set S’ and a partition of the vertex-set into paths p,, p 2 , . . . , pk such that lp,n S’I = 1, 1 s i s k. This is a trivial consequence of Theorem 11, Corollary 3 in [l], see also [2]. We are pleased to thank Professors Berge, Duchet and Hamidoune for their helpful comments on this paper.

References [ 11 C. Berge, Graphs (North-Holland Publishing Company, second revised edition, 1985). [2] C. Berge, Diperfect graphs, Combinatorica 2 (3) (1982) 213-222. [3] N. Linial, Covering digraphs by paths, Discrete Math. 23 (1978) 257-272.

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ON THE HARMONIOUS CHROMATIC NUMBER OF A GRAPH John MITCHEM Mathematics and Computer Science Department, San Jose State University, San Jose, California 95192, U.S . A.

The harmonious chromatic number of a graph C , denoted by h ( C ) , is the least number of colors which can be assigned to the vertices of G such that adjacent vertices are colored differently and any two distinct edges have different color pairs. This is a slight variation of a definition given independently by Hopcroft and Krishnamoorthy and by Frank, Harary, and Plantholt. D. Johnson has shown that determining h ( G ) is an NP-complete problem. In this paper we give various other theorems on harmonious chromatic number and discuss various open questions.

The harmonious chromatic number of a graph C , denoted by h ( G ) , is the least number of colors which can be assigned to the vertices of G such that each vertex has exactly one color, adjacent vertices have different colors, and any two edges have different color pairs. Here a color pair for edge e is the set of colors on the vertices of e. This parameter was introduced by Miller and Pritikin [6]. It is, however, only a slight variation of the parameter h '(G) introduced independently by Frank, Harary, and Plantholt [ l ] and Hopcroft and Krishnamoorthy [2]. The definition of h'(G) is the same as h ( G ) except that adjacent vertices are not required to have different colors. Thus it is clear that h'(G)s h ( G ) for each graph G. Since the edges must have different color pairs it is apparent that if h ( G ) = h, then the binomial coefficient C(h, 2) 2 IE(G)(.Unless stated otherwise in this paper we let k be the smallest integer such that C ( k , 2) 2 IE(C)l, where G is the graph currently under consideration. Finding the harmonious chromatic number of a graph is apparently quite difficult. In an appendix to [2] David S. Johnson gives an elegant proof that determining h ( C ) (or h '(G)) is NP-complete. However, the harmonious chromatic number of P,,, the path with n vertices has been determined.

Theorem 1 [5]. If k is odd or if k is even and n - 1 = C ( k , 2) - j , j = (k k J 2 , . . . , k - 2, then h(P,) = k. Otherwise, h(P,,) = k + 1.

- 2)/2,

An obvious next problem is to find the harmonious chromatic number of trees. In fact one might guess that for any tree T, h ( T ) is close to k. We show that this is not true, rather a tree T of order p can have h ( T ) be any value between k and 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V. (North-Holland)

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p inclusive. Before formally stating this fact as a theorem we state an obvious proposition and prove a lemma.

Proposition 2. For every graph G h ( C )2 D ( G ) + 1, where D ( G ) denotes the maximum degree of G.

Lemma 3. Let k be the least integer such that C ( k , 2) 3 p - 1. Then C ( k - 1, 2) 3 p - k, and k - 1 is the smallest integer t such that C(t, 2) 2 p - k unless C ( k - 1,2) = p - 2. In that case k - 2 is the smallest such t. Proof. Since k ( k - 1)/2 2 p - 1, we have k ( k - 1)/2 - ( k - 1) 3 (p - 1) - ( k - 1) and thus C ( k - 1,2) 2 p - k. Note that p and k are fixed integers, and we suppose that k - 1 is not the smallest integer t such that C(t, 2) 3 p - k. Thus C ( k - 2,2) 2 p - k. However, from our choice of k we have that p - 1> C(k - 2 , 2 ) + ( k - 2) 2 ( p - k ) + ( k 2). Thus p - 1> C ( k - 1, 2) 3 p - 2. I t follows that C ( k - 1, 2) = p - 2, and furthermore C ( k - 2,2) = C ( k - 1,2) - ( k - 2) = ( p - 2) - ( k - 2) = p - k , which completes the proof of the lemma. 0

Theorem 4. Let k be the least integer such that C ( k , 2) 2 p - 1. Then for each t, k G t G p , there is a tree T of order p such that h( T ) = t. Proof. Let T be the tree of order p which consists of a path vo, u & - ~v,k , . . . , vpp2, together with vertices vl,v2, v 3 , ,. . , vk-2 each of which is adjacent to vo. We note that by Proposition 2, h ( T )3 k. Now for each i, 0 c i s k - 1, color vi with i. Since k is the least integer such that C ( k , 2) > p - 1 we have, by Lemma 3, that k - 1 is the least integer such that C ( k - 1, 2) 3 p - k unless C ( k - 1, 2) = p - 2. The subgraph T’ of T induced by v & - v~k ,, . . . , is p p - k + l . According to Theorem 1 and Lemma 3, h ( T ’ ) = k - 2 or k - 1 or k. If h ( T ’ )S k - 1, then we color T’ with colors 1, 2, . . . , k - 1, where vkPl receives color k - 1. It follows that h ( T ) = k . Similarly if h ( T ’ ) = k, then we obtain h ( T ) = k 1. Now by removing the vertex farthest from vo and joining it to uo we successively create trees with harmonious chromatic number k + 1, k + 2, . . . ,p. The proof is thus = k , we have not constructed a tree T” complete except when h(T‘) = h(Pp--k+l) of order p with h(T”)= k. In this case k - 1 was the least integer t such that C(t, 2) 2 p - k. From Theorem 1 we have that k - 1 is even and thus k is odd. Hence h(Pp)= k and for each t, k s t S p , we have constructed a tree Tp with h(T,)=k.

+

Corollary 5. Let p S q S C ( p , 2). Let k be the least integer such that C ( k , 2) 2 q. Then for any t, k S t S p there exists a graph G with p vertices and q edges such that h ( G ) = t.

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Harmonious chromatic number of a graph

Proof. Let T be the tree of order p given in the theorem such that h ( T ) = t. Now C(t, 2) 3 C ( k , 2) 2 q and thus C(t, 2) - ( p - 1) a q - ( p - 1). That is the number of pairs of non-adjacent colors is at least q - p + 1, which is the number of edges we need to add to T to form a graph with q edges. Hence we can insert q - p + 1 edges joining non-adjacent colors of T. The resulting graph G has p vertices, q edges, and h ( G ) = t. 0 We now determine the harmonious chromatic number of 2-regular graphs with at most 2 components.

Theorem 6. Let r + s = p where 3 C r G s, and k be the least integer such that C(k, 2) " p . If k is odd and p # C(k, 2 ) - i , i = 1 , 2 , then h(C,) = h(C, U C,) = k . q k is even andp # C ( k , 2) - i, i = 0, 1, . . . , k / 2 - 1, then h(C,) = h(C, U C,) = k. Otherwise, h(C,) = h(C, U C,) = k + 1. Proof. Suppose that C, can be harmoniously colored with 1 , 2 , . . . ,t. Then that coloring generates an Eulerian circuit which is a subgraph with p edges of k,. Conversely if K , has as Eulerian subgraph H with p edges, then any Eulerian circuit with these p edges corresponds to a t-coloring of C,. We use these observations to first show that the appropriate k or k + 1 is the value of h(C,,). Case i. k is odd. If p = C ( k , 2), then K k , which I will call El, is the required Eulerian subgraph. If p = C ( k , 2 ) - i , i = 3, 4, . . . , k - 2, then the required Eulerian graph, denoted EZ,is Kk - C1. If p = C ( k , 2) - i , i = 1 or 2, then Kk has no Eulerian subgraph with p edges. Then we consider subgraphs E3 and E4 of K k + , . Let E3 be the graph formed by removing the edges of a P4 from Kk and joining its endvertices to a vertex u which is not in K k . So E, is an Eulerian subgraph of K k + , with exactly C ( k , 2) - 1 edges. Thus h(C,) = k 1 when p = C ( k , 2) - 1. Similarly if p = C(k, 2 ) - 2, we see that h(C,) = k 1 by forming E4 by removing the edges of Ps from Kk and joining the endvertices to a new vertex u.

+

+

Case ii. k is even. In this case Kk has all vertices of odd degree, so the removal of i lines i = 0, 1, . . . , (k/2) - 1, cannot result in an Eulerian graph. Thus if p = C ( k , 2)-i, i = O , 1 , . . . , ( k / 2 )- 1, h ( C , ) > k + 1. In order to see that equality holds, consider the Eulerian graph K k + l . Let t = C(k + 1, 2 ) - p . Now if t s k 1, remove C, from K k + l . If t > k + 1, remove t edges which form two edge-disjoint cycles using all k + 1 vertices of & + l a In either case the result is an Eulerian graph, which we call E5, so that h(C,) = k 1.

+

+

+

If p = C(k, 2) - i, i = k / 2 , . . . , k - 2, then let i = (k/2) t. We remove i edges from Kk as follows. Remove the edges of the complete bipartite graph Kl,zt+l together with (k/2) - f - 1 independent edges which are also independent from

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the edges of K1,2r+l.Thus ( k / 2 )- t - 1+ 2t + 1= ( k / 2 )+ t = i edges have been removed from Kk creating an Eulerian subgraph E6 with C ( k , 2 ) - i edges. It follows that h(C,) = k. We now use the various Eulerian graphs El described above to show h(C, U C,) = h(C,+,). First we observe that it is routine to verify that h(C, U C,) =h(C,+,) for k c 8 . Thus assume k > 8 . In each Ei, a maximum degree vertex q,is non-adjacent to at most 2 other vertices. Find an Eulerian subgraph H consisting of r edges of El - vo. Then consider the graph H , consisting of all of Ei - H except its isolated vertices. Then H I is Eulerian with s edges. Hence h(C, U C,) = h(C,+,), and the theorem is proved. 0 It is obvious that the proof technique used above can be used to show that many graphs which are the union of vertex disjoint cycles have harmonious chromatic number k or k + 1. It is simply necessary to show that the appropriate Ei can be partitioned into appropriate Eulerian subgraphs. A number of researchers have investigated the partitioning of E ( K 2 , + l )into s edge disjoint copies of C,. It follows, for example, that whenever such a partition exists then the graph which consists of s vertex disjoint copies of C,, has harmonious chromatic number k = 2t + 1. The most recent work in partitioning complete graphs into cycles is by Jackson [3]. In that paper it is shown that if 2t + 1= qr where q and r are odd or t = q r , then E ( K 2 , + l )can be partitioned into disjoint copies of C,. These results improve earlier results due to Kotzig [4] and Rosa [7]. We now give another easy lower bound for h ( G ) , where G is regular.

Proposition 7 . Let h ( G ) = h where G is r-regular of order p . Then h a [ p l h l r + 1. Proof. Harmoniously color G with h colors. Now the average color class size is p l h , so one color class has size at least [ p l h ] . But that color occurs on [ p / h l vertices each of degree r. Thus that color is adjacent with at least [ p / h l r other colors, and the total number of colors h a [ p / h l r 1.

+

Corollary 8. Let G be r-regular, k be the least integer such that C ( k , 2 ) 2 IE(G)I, and h be the least integer such that h 2 [ l V ( G ) l / h ] r+ 1. Then h ( G ) 2 max{h, k } . In [6] Miller and Pritikin showed that if B, is the complete binary tree on n levels, then h(B,) = O(2,"). Later, in [ 5 ] Lee and Mitchem proved Theorem 9, which trivially implies that h(B,) = 0(2n'2).

Theorem 9. [4]. For any graph G , h ( G ) 6 ( D 2+ 1) [GI. We give a significantly more efficient coloring of B, in the two theorems below. Before proving the theorems, we state the following easily verified lemma.


155

Lemma 10. Let k be the least integer such that C ( k , 2) 3 IE(B2r+l)l, where r 3 2, then k = Y+' 1.

+

Theorem 11. For each r 2 5 , h(BZr+,)6 2('+') + 2(r-4) + (r - 2) = k (r - 3) where k is the smallest integer such that C ( k , 2) 3 lE(B2r+l)l.

+ 2('-') +

Proof. We first color B7, use that coloring to color B,, and use B9 to show how to color B2r+3given a coloring of B2r+l,r 3 4. Color B7 with 24 3 colors as shown in Fig. 1. The colors of the form C, are called special colors. Note that the root is colored with special color C3 and no two special colors C, and C, are adjacent. We now construct B9 with a coloring with 25 + 4 colors. Take two copies, H1 and H2, of B7.Color H1 as shown in Fig. 1. For each vertex of Hl colored with a number n we color the analogous vertex of H2 with n 24. Furthermore, in H2 interchange the use of colors C2 and C3 from H I . Now construct B9 by adding a root v adjacent to the roots of H1 and H 2 , and color v with C4. Also join 2 new vertices to each endvertex of HIU H2. Now the endvertices of B, which are adjacent to vertices of H1 will be colored with 17,. . . ,32 and the other endvertices will be colored with 1, . . . , 16. Specifically assign colors 17, . . . ,24 to the 8 endvertices adjacent to the vertices of each color i , 1 s i G 8 in H,. Also assign colors 25,26,. . . ,32 to the 8 endvertices adjacent to each color j , 9 < j s 16. Similarly use colors 1, . . . , 8 on the 8 endvertices adjacent to each i, 25 s i G32, in H2 and colors 9 , . . . ,16 on the endvertices adjacent to each i, 17 s i G 24. Thus we have a proper coloring of B9 with Z5 + 4 colors. In general suppose we have a coloring of B2r+lwith 2' 4 colors if r = 4, and 2('+') 2(r-4) ( r - 2) colors if r 3 5. We now color B2r+3.Let H , and H2 be copies of BZr+,.Color H1 as &+, was originally colored. If vertex 2) of H1 is colored with number n, color the corresponding vertex of H2 with color n + Yfl.

+

+

+

+

+

The vertices on the tap level o f this tree are labeled consecutlvely: 1 ; - ~ 4 1 ; - 3 4 1 ~ 3 4 i i 3 4 5 6 7 8 ~ ~ 7 8 ~ ~ 7 a 5 ~ 7 ~ 3 i o i i i 1 1 1;- 9 1 0 1 1 1- 7 1 0 1 1 1s 13 1 4 15 16 13 14 15 1 € 13 14 15 16 13 14 15 16

Fig. 1. Binary tree 8,.

1. Mitchem

156

If a vertex of H1 is colored with Cilet the corresponding vertex of H, also have color C,. Now form BZr+3 from HI and H,, a new vertex which is adjacent to the two roots of HI and H,, and 2"+' new endvertices such that 2 are adjacent to each endvertex of H I U H,. We now color the endvertices of BZr+3. Note that the endvertices of HI are colored with 1 , 2 , . . . , 2 ' + l and the endvertices of H, with 2'+' 1, . . . , 2'+'. Furthermore each color is used exactly T-'times on the endvertices of HI U H2. Assign colors 2'+' 1, . . . ,2'+l+ 2' to the 2' endvertices adjacent to each color i, 1s i S 2' in Hl. Also assign colors 2'+l+ 2' + 1, . . . , 2'+' to the 2' endvertices adjacent to each of the colors i, 2'+ 1Si ST+' in Hi.Similarly use colors 1, . . . , 2' on the 2' endvertices adjacent to each color i, 2'+' 2' + 1 s i G T+', and use colors 2'+ 1 , . . . ,T+' on the endvertices adjacent to each color i, y+' + 1 s j s 2'+' + 2'. Finally we color the root v of BZr+3 and recolor certain vertices of H,. If r = 4, then change the color of the root of H2 to Cs and use 26 on v. Then h(B,,) = h(Bzr+3)S 26 + 5 = 2'+' + 2'-3+ ( r - 1). If r 2 5, then corrresponding vertices on level r - 3 of HI and Hz have the same special colors and are adjacent to the same special colors on level r - 2. Use 2r-4 new special colors on these 2'-4 vertices on level r - 3 of H,. Now if the roots of HI and H2 are colored with numbers we color v with yet another new special color. If the roots of H1 and H, are colored with a special color, then recolor the root of H2 with another new special color and color v with the number 2'+' - ( r - 5). Now we have used T+* numbers and 2'-4+ T4 + ( r - 2 ) + 1 = r3 + ( r - 1) special colors on &+3. We now check that the assignment of 2'+' + 2'-3 ( r - 1) colors to B2r+3is harmonious. By inductive assumption the coloring of H1 is harmonious, and it follows that the coloring of H2 is also. Now H I U H2 is harmoniously colored because the numbers used in H2 are not used in H I and the only time special colors are adjacent in H1 we have introduced new special colors in H,. Furthermore, it is clear that we have harmoniously colored the root and the endvertices of B2r+3.Thus the theorem is proved. 0

+

+

+

+

Theorem 12. For r 2 6, h(Bzr) 3(2'-')

+ 2'-' + ( r - 3).

Proof. Color BZrp1with 2'+2'-'+ ( r - 3 ) colors as in Theorem 11. Now the endvertices of B2'-' are colored with 2' colors, each occurring 2'-' times. Join two new vertices to each endvertex. For each color i on level 2r - 1 color one of its level 2r neighbors with each of the colors 2' + 1, . . . , 2' + 2'-'. Thus we have h(B,,) S 2' + 2'-s + ( r - 3) + 2'-' = 3(2'-') + T P 5+ ( r - 3). 0 22r-2

Although the determination of h ( T ) when T is a complete n-ary tree with t levels is apparently difficult we have found the exact value of h ( T ) when T has t = 3 levels.


157

Theorem 13. Let T be a complete n-ary tree on 3 levels. Then h ( T ) = [3(n + 1)/21. Proof. Color the root v with A and its neighbors vl,.. . , v, with 1 , . . . , n respectively. Let t = [ ( n - 1)/2]. For each level 2 vertex color n - t of its level 3 neighbors with color n + 1, n + 2, . . . , n + (n - t) respectively. Then color the remaining neighbors of vj with j + 1, j +2, . . . ,j + t (modn). Now color j , l 6 j s n is adjacent to colors A, n + 1 , . . . , 2n - t and colors j + 1, j 1, . . . ,j + t, j - t (mod n ) . It follows that we have a 2n + 1 - t = [3(n + 1)/21 harmonious coloring of T. We show now that there is no harmonious coloring of T with fewer colors. The color A , used on the root v, cannot be used on any other vertex of T. There are n2 edges which are not incident with v. At most C(n, 2) of these edges can join vertices colored 1, . . . , n. Thus at least n2 - C(n, 2) = n ( n + 1)/2 of the edges must have its level 3 vertex colored with a new color. Hence there must be a level 2 vertex which is adjacent to at least [ ( n + 1)21 new colors and h ( T ) n + 1 + [ ( n 1)/2] = [3(n + 1)/2]. 0

+

References [l] 0. Frank, F. Harary and M. Plantholt, The line-distinguishing chromatic number of a graph, Ars Combinatorica 14 (1982) 241-252. [2] J . Hopcroft and M.S. Krishnamoorthy, On the harmonious colorings of graphs, SIAM J . Alg. Disc. Math. 4 (1983) 306-31 1. [3] B. Jackson, Some cycle decompositions of complete graphs, preprint, 24 pages, [4] A. Kotzig, On the decomposition of the complete graph into 4k-gons, Mat. Fyz. Casopis 15 (1965) 229-233 (in Russian). [5] S . Lee and J. Mitchem, An upper bound for the harmonious chromatic number of a graph, J. Graph Theory 11 (1987) 565-567. [6] Z. Miller and D. Pritikin, The harmonious coloring number of a graph, preprint, 24 pages. [7] A. Rosa, On the cyclic decomposition of the complete graph into (4m +2)-gons, Mat. Fyz. casopis, 16 (1966) 349-353.



159

WEAK BIPOLARIZABLE GRAPHS Stephan OLARIU Department of Computer Science, Old Dominion University, Norfolk, V A 23508, U.S.A .

We characterize a new class of perfectly orderable graphs and give a polynomial-time recognition algorithm, together with linear-time optimization algorithms for this class of graphs.

1. Introduction A linear order < on the set of vertices of a graph C is perfect in the sense of Chqatal [3] if no induced path with vertices a, b, c, d and edges ab, bc, cd has a < b and d < c . Graphs which admit a perfect order are termed perfectly orderable. Recognizing perfectly orderable graphs in polynomial time seems to be a difficult problem. Quite naturally, this motivated the study of particular classes of perfectly orderable graphs. Such classes have been studied by Golumbic, Monma and Trotter [7], ChvAtal, Hoang, Mahadev, and de Werra [4],Hoang and Khouzam [9], and Preissmann, de Werra and Mahadev [12]. Recently, Hertz and de Werra [8] proposed to call a graph G bipolarizable if G admits a linear order < on the set V of its vertices such that b < a and c < d whenever { a , b, c, d } induces a path in G with edges ab, bc, cd. They characterize bipolarizable graphs by forbidden subgraphs and prove that both bipolarizable graphs and their complements are perfectly orderable. In this paper we first define and characterize the class of weak bipolarizable graphs which properly contain the class of bipolarizable graphs. This characterization can be exploited to obtain a polynomial-time recognition algorithm for weak bipolarizable graphs. Finally, given a weak bipolarizable graph G, we show how an algorithm of Rose, Tarjan and Lueker [13] can be used to obtain efficiently a linear order on the vertices of G. As soon as this is done, an algorithm of ChvAtal, Hoang, Mahadev and de Werra [4] can be used to optimize weak bipolarizable graphs in linear time. Given a graph G, we shall let G denote the complement of G; if x is a vertex in G , then N,(x) stands for the set of all the vertices in G which are adjacent to x; N&) denotes the set of all the vertices in G which are adjacent to x in G (whenever possible, we shall write simply N ( x ) and N ' ( x ) ) . We shall let GHstand for the subgraph of C induced by H ; Ck(Pk) will stand for an induced chordless cycle (path) with k vertices. 0012-365X/89/$3.50 0 1989, Elsevier Science Publishers B.V.(North-Holland)

160

S. Olariu

El

Fl

F2

Fig. 1

Fig. 2.

A graph G is called triangulated if every cycle of length greater than three in G has a chord. Dirac [5] proved that every triangulated graph contains a simplicia1 vertex: this is a vertex w such that N ( w ) is a clique. A proper subset H (IHI 2 2 ) of vertices of G will be referred to as homogeneous if every vertex outside H is either adjacent to all the vertices in H or to none of them. A graph G will be called a weak bipolarizable graph if G has no induced subgraph isomorphic to C, ( k 3 5 ) , p5 or to one of the graphs F,, 6 in Fig. 1. Since every forbidden subgraph of a weak bipolarizable graph is also a forbidden subgraph of a bipolarizable graph it follows that every bipolarizable graph is also weak bipolarizable. In addition, note that the graph in Fig. 2 is a weak bipolarizable graph but not a bipolarizable graph. Therefore, the class of weak bipolarizable graphs properly contains the class of bipolarizable graphs. As it turns out, the class of weak bipolarizable graphs also contains all triangulated graphs, all Welsh-Powell opposition graphs (see Olariu [lo]), all superbrittle graphs (see Preissmann, de Werra, and Mahadev [12]) and all superfragile graphs (see Preissmann, de Werra, and Mahadev [12]).

2. The results

The following theorem provides a characterization of the class of weak bipolarizable graphs.

Weak bipolarizable g r a p h

161

Theorem 1. For a graph G the following three statements are equivalent: (i) G is a weak bipolarizable graph (ii) Every induced subgraph H of G is triangulated, or H contains a homogeneous set which induces a connected subgraph of G (iii) Every induced subgraph H of G is triangulated or H contains a homogeneous set. Proof. To prove the implication (i)+ (ii), consider a graph G = (V, E) that satisfies (i). Assuming the implication (i)+ (ii) true for graphs with fewer vertices than G, we only need prove that G itself satisfies (ii). If G contains a homogeneous set with the property mentioned in (ii), then we are done. We shall assume, therefore, that G contains no such homogeneous set. We want to show that, with this assumption, G is triangulated. For this purpose, we only need show that G has no induced C4. Suppose not; now some vertices x , y , z , t induce a C4 with edges xy, y z , zt, tx E E. Consider the component F of the subgraph of G induced by N ( y ) n N ( t ) , containing x and z . By assumption, F is not a homogeneous set, and thus there exists a vertex u in V - F, adjacent to some but not all vertices in F. By connectedness of F in G, we find non-adjacent vertices x ’ , z‘ in F such that ux’ E E and uz‘ $ E. Trivially, u is not in N ( y ) n N ( t ) , and hence u is adjacent to at most one of y , t. If u is adjacent to precisely one of y , t then { u , x ’ , y , z‘, t } induces a p5,a contradiction. Now u is adjacent to neither y nor t. Write N ( x ’ )fl N ( z ‘ ) = U, U U, in such a way that every vertex in Ul is adjacent to u, and no vertex in Uo is adjacent to u. By the above argument, y and t belong to U, and thus 1 &,I 2 2. Observe that every vertex in Ul is adjacent to every vertex in U,, for otherwise { u , p , q, x ’ , z ’ } induces a 4, for any non-adjacent vertices p in U, and q in U , . Consider the connected component H of the subgraph of G induced by U, that contains the vertices y and t. Since H is not homogeneous, there must exist a vertex v in V - H adjacent to some but not all vertices in H. Trivially, v is not in { x ’ , z’, u } U U, U U , . By connectedness of H in G, we find non-adjacent vertices y‘, t‘ in H such that vy’ E E, vt’ $ E. Now v is adjacent to at most one of the vertices x ’ and z’. If v is adjacent to precisely one of them, then { v , x ’ , z ’ , y ’ , t ’ } induces a p’, a contradiction. Thus, v is adjacent to neither x ’ nor z’. By definition of U,, u is adjacent to neither y ’ nor t‘. However, this implies that {u, v, x ’ , y ’ , z’, t ’ } induces either an Z$ or an F , , depending on whether or not uv E E. This proves that G is triangulated, as claimed.

S. Olariu

162

The implication (ii)+ (iii) is trivial. To prove (iii)+ (i) we only need observe that if a graph G does not satisfy (i), then (iii) fails. This completes the proof of the theorem. 0 Consider a graph G1 and a graph G2 containing at least two vertices, and let v be an arbitrary vertex in G1. It is customary to say that a graph G arises from C , and G, by substitution if G is obtained as follows: (*) delete the vertex v from C , , and (**) join each vertex in G2 by an edge to every neighbour of v in GI. If G arises by substitution from graphs G1 and G,, then we shall say that G is substitution-composite. It is a simple observation that a graph G is substitutioncomposite if and only if G contains a homogeneous set. Now the equivalence (i) @ (iii) in Theorem 1 can be rephrased as follows.

Corollary la. A graph G is weak bipolarizable if and only if every induced subgraph of G is either triangulated or substitution -composite. Let Y be the class of graphs defined as follows: (v1) if G is triangulated, then C is in Y. (v2) if G' is obtained from a graph GI in Y and a triangulated graph G, by substitution, then G' is in Y.

Theorem 2. Y is precisely the class of weak bipolarizable graphs. Proof. To begin, we claim that every graph in Y is weak bipolarizable. For this purpose, let C be an arbitrary in Y. Assuming (1) to be true for all graphs with fewer vertices than G, we only need prove that G itself is weak bipolarizable. This, however, follows immediately from the observation that G is either triangulated or it contains a homogeneous set. Now Theorem 1 guarantees that G is weak bipolarizable. Conversely, we claim that every weak bipolarizable graph is in Y.

(2)

Let G be a weak bipolarizable graph. Assume that (2) holds for all graphs with fewer vertices than G. If G is triangulated, then G is in Y b y (1$1).Now we may assume that C is not triangulated. Theorem 1 guarantees that G contains a homogeneous set. Let H be a minimal homogeneous set in C (here, minimal is meant with respect to set inclusion, not cardinality). By Theorem 1, H must be triangulated. By the induction hypothesis, the graph induced by ( V - H) U { h } is in Y, for any choice of h in H. Hence, by (v2), G itself is in Y, as claimed. 0

Weak bipolarizable graphs

163

We shall refer to a graph G which contains no homogeneous set as substitution-prime. For later reference we shall make the following simple observation, whose justification is immediate.

Observation 1. If a graph G with a homogeneous set H contains an induced substitution-primesubgraph F, then either every vertex of F belongs to H or else F and H have at most one vertex in common. Let E be a class of graphs such that all forbidden graphs for E are substitution-prime.

Theorem 3. If G arises by substitution from graphs GI and G2in 2,then G is also in 2. Proof, Suppose not; now G must contain an induced subgraph F isomorphic to a forbidden graph for the class E . By assumption, F is an induced subgraph of neither Gi (i = 1, 2). By Observation 1, F has precisely one vertex in common with G2. However, this implies that GI has an induced subgraph isomorphic to F, a contradiction. 0 Theorem 1 and Theorem 3 provide the basis for a polynomial-time recognition algorithm for weak bipolarizable graphs. In addition, we shall rely on algorithms to recognize triangulated graphs (see, for example, Rose, Tarjan and Leuker [13]), as well as polynomial time algorithms to detect the presence of a homogeneous set in a graph (see Spinrad [ll]). The following two-step algorithm recognizes weak bipolarizable graphs.

Algorithm Recognize( G ) ; {Input: A graph G = (V, E ) . Output: ‘Yes’ if G is weak bipolarizable; ‘No’ otherwise.} Step 1. Call Check(G) Step 2. Return(‘Yes’); stop. Procedure Check( G ) ; begin if G is not triangulated then if G is not substitution-composite then begin return(‘N0’); stop end

164

S. Olariu

else begin {now C contains a homogeneous set H ; let H' stand for the set of all the remaining vertices in G.} Check( G H ;) pick an arbitrary vertex h in H; Check(G{h)"H,) end end; {Check} The correctness of this algorithm follows directly from Theorem 1 and Theorem 2. Furthermore, its running time is clearly bounded by O(n3):to see this, note that Check is invoked O(n) times for a graph G with n vertices. Each invocation of Check runs in O(n') time since the recognition of triangulated graphs [13] and the detection of a homogeneous set [ l l ] are both performed in 0(n2)time. Given a P4with vertices a, b, c, d and edges ab, bc, cd, the vertices a and d are called endpoints and the vertices b, c are called midpoints of the P4. We shall say that a vertex x in a graph G is semi-simplicial if x is midpoint of no P4 in G. Trivially, every simplicia1 vertex is also semi-simplicial, but not conversely. A linear order < on the vertex-set V of G is said to be a (semi-)perfect elimination if the corresponding ordering x , , x 2 , . . . ,x, of the vertices of G with xi <xi iff i <j satisfies xi is a (semi-)simplicia1vertex in G , , , , ,,..., + ,..) for every i.

(3)

It is immediate that every graph G with a semi-perfect elimination is brittle in the sense of Chv6tal [2]: every induced subgraph H of C contains a vertex which is either midpoint or endpoint of no P4 in H. Furthermore, it is an easy observation that every brittle graph is also perfectly orderable. Hertz and De Werra [8] demonstrated that bipolarizable graphs are brittle; we extend their result by showing that weak bipolarizable graphs are also brittle. Actually, we also exhibit a linear-time (and thus optimal) algorithm that finds a perfect order for any weak bipolarizable graph. The details are spelled out in Theorem 4. Rose, Tarjan and Lueker [13] proposed a linear-time search technique which is referred to as Lexicographic Breadth-First Search (LBFS, for short). They prove that a graph G is triangulated if, and only if, any ordering of the vertices of G produced by LBFS is a perfect elimination. We shall use their algorithm to obtain a perfect order on the set of vertices of a weak bipolarizable graph. To make our exposition self-contained, we give the details of LBFS.

Weak bipolarizable graphs

165

procedure LBFS(G); {Input: the adjacency list of G ; Output: an ordering u of the vertices of G} begin for every vertex w in V do label(w) -0; for i t n downto 1 do begin pick an unnumbered vertex v with the largest label; u(v) c i ; {assign to v number i } for each unnumbered w E N ( v ) do add i to label ( w ) end end; Note that we can think of the output of LBFS as a linear order < on V by setting u $ by Farr [5] and then extended by Edwards [4] who proved

260

A.D. Petford, D.J.A. Welsh

that for k a 3 the conjecture is true if a > ( k -3)/(k -2) and false for O < a ~ ( -3)/(k k -2). In addition to the “random benchmark” test graphs used in the above simulation we have tried the algorithm out on some “apparently difficult” graphs sent to us as a challenge by R . Irving and K.W. Regan. These examples consisting of two Kneser type graphs of 70 and 130 vertices respectively together with a graph of 341 vertices which was the line graph of a planar graph with no small reducible configuration were correctly coloured in a matter of seconds by the programme (written in the language C) on a Perkin Elmer mini-computer. We have made a preliminary approach to extending the above methods to the k-colouring problem for general k > 3. In principle this should not present any greater problem. In practice this does not seem to be the case, preliminary investigations for the case of k = 10 suggest that the method is not as good; one reason for this may be that the amount of experimentation needed to find a good transition function in this case takes much more space and time. Another feature of the method is that preliminary work to start off with a “good colouring” in the sense that the initial bad set Bo was small did not seem to speed up the algorithm. This can be explained by regarding such a good colouring as approaching a local optimum which is a long way in the metric of exhanges from the true global optimum. Finally we remark that since the above experiments were first carried out in 1985 Zerovnik [12] has checked the algorithm by independently verifying our results using a different language and machine at the University of Ljubljana.

Acknowledgement We would like to thank R.W. Irving and K.W. Regan for communicating their “difficult” test graphs, D.E. Blackwell for allowing the use of the computing facilities in the Department of Astrophysics and J. Zerovnik for communicating his results to us.

References [l] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan, 1976). [2] P.J. Donnelly and D.J.A. Welsh, Finite particle systems and infection models, Math. Proc. Camb. Phil. SOC.94 (1983) 167-182. [3] P.J. Donnelly and D.J.A. Welsh, The antivoter problem: random 2-colourings of graphs, Graph Theory and Combinatorics (ed. B. Bollobas) (Academic Press, 1984) 133-144. [4] K.J. Edwards, The complexity of colouring problems on dense graphs, Theoretical Computer Science, 43 (1986) 337-343. [5] G. Farr (private communication, 1985). [6] M.R. Garey and D.S. Johnson, Computers and Intractability (W.H. Freeman and Co., San Francisco, 1979).

Randomised 3-colouring algorithm

261

[7] M.R. Garey, D.S. Johnson and L. Stockmeyer, Some simplified NP-complete graph problems, Theor. Comput. Sci. 1 (1976) 237-267. [8] S. Kirkpatrick, C.D. Gelatt Jr. and M.P. Vecchi, Optimisation by simulated annealing, Science, 220, NO. 4598 (1983) 671-680. [9] L. KucCra, Expected behaviour of graph colouring algorithms, Lecture Notes in Computer Science, 56 (1977) 477-483. [lo] D.J.A. Welsh, Randomised algorithms, Discrete Applied Maths. 5 (1983) 133-145. [ 1I] D.J.A. Welsh, Correlated percolation and repulsive particle systems, Proc. Conference Heidelberg (Sept. 1984), Stochastic Spatial Processes (ed. P. Tautu) Springer Lecture Notes 1212, 300-311. [12] J. Zerovnik (private communication, 1986).

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