Graph Theory, Proceedings of the Conference on Graph Theory, Cambridge

213 Pages·1982·2.614 MB·English

by  Béla Bollobás (Eds.)

GRAPH THEORY
General Editor
Peter L. HAMMER, University of Waterloo, Ont., Canada
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, USA.
J.H. VAN LINT, California Institute of Technology, Pasadena, CA, U.S.A.
G.-C. ROTA, Massachusetts Institute of Technology, Cambridge, MA, U.S.A.
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK OXFORD
NORTH-HOLLAND
MATHEMATICS STUDIES 62
Annals of Discrete Mathematics (13)
General Editor: Peter L. Hammer
University of Waterloo, Ont., Canada
Graph Theory
Proceedings of the Conference
on Graph Theory, Cambridge
Editor:
Bela BOLLOBAS
Department of Pure Mathematics
and Mathematical Statistics
University of Cambridge
Cambridge CB2 lSB, England
1982
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK OXFORD
North-Holland Publishing Company, 1982
AII rights reserved. Nop art of this publication may be reproduced, stored in a retrievalsystem
or transmitted,i n any otherf orm or by any means, electronic, mechanical, photocopying, recording
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ISBN: 0 444 86449 0
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Library of Congress Calaloging in Publication Data
Main entry under title:
Graph theory.
(Annals of discrete mathematics ; U) (North-
Holland mathematics studies ; 62)
Papers presented at the Cambridge Graph Theory
Conference, held at Trinity College 11-l.3 Mar. p81.
1. Graph theory--Congresses. I. Bollob&, Bela.
11. Cambrid e Gra h Theory Conference (1 81 :
Trinity Colfege, eniversity of Cambridge? 111. Se-
ries. IV. Series: North-Holland mathematics
studies ; 62.
QA166.G718 1982 5ll'.5 82-8098
ISBN 0-444-86449-0 AACF2
-
PRINTED IN THE NETHERLANDS
FOREWORD
The Cambridge Graph Theory Conference, held at Trinity College from 11 to
13 March 1981, brought together top ranking workers from diverse areas of the
subject. The papers presented were by invitation only. This volume contains most
of the contniutions, suitably refereed and revised.
For many years now, graph theory has been developing at a great pace and
in many directions. In order to emphasize the variety of questions and to preserve
the freshness of research, the theme of the meeting was not restricted. Consequent-
ly, the papers in this volume deal with many aspects of graph theory, including
colouring, connectivity, cycles, Ramsey theory, random graphs, flows, simplicial
decompositions and directed graphs. A number of other papers are concerned with
related areas, including hypergraphs, designs, algorithms, games and social models.
This wealth of topics should enhance the attractiveness of the volume.
It is a pleasure to thank Mrs. J.E. Scutt and Mrs. B. Sharples for retyping
most of the papers so quickly and carefully, and to acknowledge the financial
assistance of the Department of Pure Mathematics and Mathematical Statistics
of the University of Cambridge. Above all, warm thanks are due to the particip-
ants for the exciting lectures and lively discussions at the meeting and for the many
excellent papers in this volume.
€#la Bollobis
25th February 1982
Baton Rouge
This Page Intentionally Left Blank
TABLE OF CONTENTS
Foreword V
J. AKIYAMA, K. AND0 and H. MXZUNO, Characterizationsa nd
classifications of bioonnected graphs 1
R.A. BAN, Line graphs and their chromatic polynomials 15
J.C. BERMOND, C. DELORME and G. FARHI, Large graphs with
given degree and diameter 111 23
B. BOLLOBh, Distinguishing vertices of random graphs 33
B. BOLLOBh and F. HARARY, The trail number of a graph 51
J.H. CONWAY and M.J.T. GUY, Message graphs 61
D.E. DAYKIN and P. FRANKL, Sets of graph colourings 65
P. DUCHET and H. MEYNIEL, On Hadwiger’s number and
the stability number 71
H. DE FRAYSSEIX and P. ROSENSTIEHL, A depth-first-search
characterization of planarity 75
J.L. GROSS, Graph-theoreticalm odel of social organization 81
R. I-I&XKVIST, Odd cycles of specified length in non-bipartite graphs 89
R. HALIN, Simplicial decompositions: Some new aspects
and applications 101
F. HARARY, Achievement and avoidance games for graphs 111
viii Cbnrenrs
A J.W. HILTON, Embedding incompletel atm rectangles 121
AJ.W. HILTON and C.A. RODGER, Edgecolouring regular
bipartite graphs 139
AJ. WSFIEU) and D J.A. WELSH, Some colouring problems
and their complexity 159
A. PAPAIOANNOU,A Hamilbnian game 171
J. SHEEHAN, Finite Ramsey theory and strongly regular graphs 179
R. TINDELL, The connectivities of a graph andi ts complement 191
Annalsof Discrete Mathematics 13 (1982) 1-14
0 North-Holland Publishing Company
CHARACTERIZATIONS AND CLASSIFICATIONS
OF BICONNECTED GRAPHS
J. AKIYAMA, K. AND0 and H. MIZUNO
Nippon Ika University, Kawasaki, Japan
and
University of Electrocommunications, Tokyo, Japan
Dedicated to Frank Harary for his 60th birthday
A graph G is said to be biconnected if both G and
its complement 5 are connected. In this paper
we mainly deal with biconnected graphs and classify
a set G of biconnected graphs into several classes
in terms of the number of cutvertices (or endvertices)
of G and E. Furthermore we give structural charac-
terizations of these classes, that is, characteri-
zations of graphs G such that G has m cut-
E
vertices (endvertices) and has n cutvertices
(endvertices) respectively, where m and n are
given, 1 5 m 2 2 and 1 6 n 5 2.
5 1. INTRODUCTION
A graph G is biconnected if both G and its complement are
connected. In this paper we are mainly concerned with biconnected
graphs. We denote by G the set of all biconnected graphs and by
Gp the set of all biconnected graphs of order p. We use the nota-
tions and terminology of C11 or C21. The characterization of bi-
connected graphs is well known, which is stated as follows:
Theorem A. C1, p. 261 A graph G of order p(p 2 2) is biconnected
if and only if neither G nor E contains a complete bigraph
K(m,n) as a spanning subgraph for some m and n with m + n = p.
We denote by c(G), e(G) the number of cutvertices and endvertices
of a connected graph G, respectively. We say the cutvehtex-type
of G E G is (m,n) if {c(G), c(c)}= cm,n). Similarly the
cndvektex-type of G E G is (m,n) if {e(G), e(E)j = {m,n). Note
that the cutvertex-type (the endvertex-type) of a graph G E is
the same as one of its complement E from the definition. The
1