Table Of ContentCycles and Rays
NATO ASI Series
Advanced Science Institutes Series
A Series presenting the results of activities sponsored by the NA TO Science Committee,
which aims at the dissemination of advanced scientific and technological knowledge,
with a view to strengthening links between scientific communities.
The Series is published by an international board of publishers in conjunction with the
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and Physical Sciences Dordrecht, Boston and London
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G Ecological Sciences Berlin, Heidelberg, New York, London,
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Series C: Mathematical and Physical Sciences· Vol. 301
Cycles and Rays
edited by
Gena Hahn
Departement d'informatique et de Recherche Operationnelle,
Universite de Montreal,
Montreal, Quebec, Canada
Gert Sabidussi
Departement de Mathematiques et de Statistique,
Universite de Montreal,
Montreal, Quebec, Canada
and
Robert E. Woodrow
Department of Mathematics and Statistics,
University of Calgary,
Calgary, Alberta, Canada
Kluwer Academic Publishers
Dordrecht / Boston / London
Published in cooperation with NATO Scientific Affairs Division
Proceedings of the NATO Advanced Research Workshop on
Cycles and Rays: Basic Structures in Finite and Infinite Graphs
Montreal, Canada
May 3-9,1987
Library of Congress Cataloging In Publication Data
C¥cll. and ra¥s I Idltld b~ Gina Hahn, Girt S.blduSS1, Roblrt E.
Woodrow.
p. el. -- (NATO ASI urI ... Slrl .. C, fIlathellfle.l .nd
physlnl sCI'nces , vol. 301)
'PubllShld In coopiriflon wtth NATO ScI'ntHlc Affllrs DivISIon.'
lSBN·13,978-94-01!M71~S (alk, p,plr)
I. Piths Ind cycl .. (Grlph thloryJ--Congrlssn. 2. Rays (Graph
thloryJ--ConllrISSlS. I. Hlhn. GI;;a, 1949- II. S.bldussi.
Girt. III. WOOdrow, Roblrt E., 1948- IV. North Atlantic Tr .. ty
Ot-g.nlutl0n. SC1tnt1flc Affllrs DivISIon. V. Sir liS: NATO ASt
nrlls. Slrln C, MatlllnlC:ll Ind ph¥S1cal scllnclS , no. 301.
0A166.22.C9 1990
511'.S--dc~ 89-26724
ISBN·13: 978·94.() I 0-6719·5 o-ISBN-1J: 9J8.94.()()9.Q517.7
001: 10.I007f9J8.94.()()9.Q517.7
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TABLE OF CONTENTS
PREFACE vii
PARTICIPANTS ix
CONTRIBUTORS xiii
R. AHARONI, Linkability in Countable-Like Webs 1
B. ALSPACH, J .-C. BERMOND, D. SOTTEAU Decomposition into
Cycles I: Hamilton Decompositions 9
G.D. BADENHORST, T. STURM, An Order- and Graph- Theoretical
Characterisation of Weakly Compact Cardinals 19
J.A BONDY, Small Cycle Double Covers of Graphs 21
A. BOUCHET, Ie-Transformations, Local Complementations
and Switching 41
J.M. BROCHET, M. POUZET, Two Extremal Problems in Infinite
Ordered Sets and Graphs: Infinite Versions of Menger and
Gallai-Milgram Theorems for Ordered Sets and Graphs 51
N. CHAKROUN, D. SOTTEAU, Chvatal-Erdos Theorem for Digraphs 75
G. FAN, Long Cycles and the Co diameter of a Graph II 87
H. FLEISCHNER, B. JACKSON, Compatible Euler Tours in Eulerian
~~~ ~
A.J.W. HILTON, C.A. RODGER, Edge-Colouring Graphs and
Embedding Partial Triple Systems of Even Index 101
vi TABLE OF CONTENTS
W. IMRICH, S. KRSTIC, E.C. TURNER, On the Rank of Fixed Point
Sets of Automorphisms of Free Groups 113
F. JAEGER, On Transition Polynomials of 4-Regular Graphs 123
W. MADER, On Infinite n-Connected Graphs 151
E. C. MILNER, Ordered Graphs Without Infinite Paths 161
M.A. PICARDELLO, W. WOESS, Ends of Infinite Graphs, Potential
Theory and Electrical Networks 181
N. POLAT, Topological Aspects ofInfinite Graphs 197
G. SABIDUSSI, Dendroids, End-Separators, and Almost
Circuit-Connected Trees 221
N. SAUER, M. EL-ZAHAR, Partition Theorems for Graphs Respecting
the Chromatic Number 237
M.E. WATKINS, Vertex-Transitive Graphs That Are Not
Cayley Graphs 243
INDEX 257
PREFACE
What is the "archetypal" image that comes to mind when one thinks of an infinite
graph? What with a finite graph - when it is thought of as opposed to an infinite one?
What structural elements are typical for either -by their presence or absence -yet provide
a common ground for both?
In planning the workshop on "Cycles and Rays" it had been intended from the outset
to bring infinite graphs to the fore as much as possible. There never had been a graph
theoretical meeting in which infinite graphs were more than "also rans", let alone one in
which they were a central theme. In part, this is a matter of fashion, inasmuch as they
are perceived as not readily lending themselves to applications, in part it is a matter of
psychology stemming from the insecurity that many graph theorists feel in the face of
set theory - on which infinite graph theory relies to a considerable extent. The result is
that by and large, infinite graph theorists know what is happening in finite graphs but not
conversely. Lack of knowledge about infinite graph theory can also be found in authoritative
sources. For example, a recent edition (1987) of a major mathematical encyclopaedial
proposes to "... restrict [itself] to finite graphs, since only they give a typical theory". If
anything, the reverse is true, and needless to say, the graph theoretical world knows better.
One may wonder, however, by how much.
This volume shows that in spite of the relatively small number of practitioners, infinite
graph theory is an active and varied field. It shows that the reputation of being tainted
with set theory may well be unfounded: only two of the seven major papers on infinite
graphs use anything beyond the basic set theoretic baggage of any mathematician. It also
testifies to an inner coherence that is much greater than in finite graphs. There is quite
literally a common thread running through most of infinite graph theory: only rarely does
one encounter a situation where rays (infinite paths) do not playa significant part. It is
difficult to find a single structure in finite graph theory which enjoys the same degree of
fundamentality -in the sense of being ubiquitous. Cycles are perhaps the best candidates.
There is one context where rays and cycles meet on an equal footing - as manifestations
of the same principle, if such an expresssion be permitted - thereby providing at least an
aesthetic justification for mentioning them in the same breath: namely, as building blocks
in problems of decomposition and multiple covering. Several papers in the present volume
deal with problems of this kind.
The thirteen one-hour survey lectures given at the Workshop were evenly divided be
tween the two main topics. On the side of the Cycles, the core was provided by four lectures
on transition systems, compatibility and decomposition problems in eulerian graphs and
associated isotropic systems (Bouchet, Jackson, Jaeger), as well as hamiltonian cycle de
compositions (Bermond). Two other main lectures dealt with variations on the double
lEncyclopedic Dictionary of Mathematics, 2nd ed., MIT Press, C&J\lbridge, Mass., and London (1987),
vol. 2, p. 693.
vii
PREFACE viii
cover conjecture (Bondy), and connections between cyclicity and perfect graphs (Berge) .
On the infinite side there was a presentation of compactness arguments for the extension of
finitary properties to infinite graphs (Thomassen), an account of infinite graphs with prim
itive automorphism group (Watkins), and of simplicial decompositions of infinite graphs
(Halin). One lecture each was devoted to connections between infinite graphs and topology
(Polat) and analysis (Picardello), dealing respectively with the space of ends of a graph
and with harmonic functions on a graph. One lecture (Hajnal), involving coloured graphs,
illustrated the set theoretic side.
In addition to the 13 invited speakers there were 30 participants coming from a dozen
different countries. The meeting was intentionally kept small in order to achieve maxi
mum thematic coherence. All participants, with the exception of a few graduate students,
presented papers. It is worth mentioning that they carefully respected the theme of the
conference. To all, a most heartfelt Thank you! for their enthusiasm and the quality of
their contributions which made this Workshop a success.
Most of the funding for the Workshop came from NATO through its ARW-Programme.
An additional grant was obtained from the Natural Sciences and Engineering Research
Council of Canada. The meeting took place at the Universite de Montreal with the help of
the Departement de mathematiques et de statistique and the Centre de recherches mathe
matiques. It is with great pleasure that we express our gratitude to all these institutions.
These Proceedings have been a long time in the making, and we apologize to all con
cerned. Most of the delay was due to our scanty experience with TEX. It would have been
greater still but for the expert help of Michel Toulouse whom we owe sincere thanks.
Finally, we wish to express our gratitude to the members of the Committee of the
NATO ARW-Programme as well as -and especially so -to its Director, Dr. Craig Sinclair.
His advice and help in the months preceding the Workshop have been invaluable, and his
patience thereafter most reassuring.
August 1989 The Editors
PARTICIPANTS
Ron AHARONI Jean-Michel BROCHET
Department of Mathematics Institut de Mathematiques-Informatique
Technion Lab. Algebre Ordinale
32000 Haifa Universite Claude Bernard (Lyon I)
Israel 69622 Villeurbanne Cedex
France
Thomas ANDREAE
Institut fiir Mathematik II (WE 2) Christine CHARRETTON
FU Berlin Institut de Mathematiques-Informatique
Arnimallee 3 Lab. Algebre Ordinale
D-I000 Berlin 33 Universite Claude Bernard (Lyon I)
FRG 69622 Villeurbanne Cedex
France
Hans-Jiirgen BANDELT/KE
Genghua FAN
Faculty of Economic and Business Administration
Combinatorics & Optimization
Main Bldg.
University of Waterloo
Rijksuniversitet Limburg
Waterloo, Ontario, N2L 3Gl
Postbus 616
Canada
6200 MD Maastricht
The Netherlands
Joan FEIGENBAUM
AT&T Bell Laboratories
Claude BERGE
600 Montain Avenue
10, rue Galvani
Murray Hill, NJ 07974
75017 Paris
USA
France
Herbert FLEISCHNER
Jean-Claude BERMOND
Institut fiir Informationsverarbeitung
Informatique CRNS
Akademie der Wissenschaften
3, rue Albert Einstein AI0I0 Wien
Sophia Antipolis
Sonnenfelsgasse 19
06560 Valbonne
Austria
France
J. Adrian BONDY Stephane FOLDES
Combinatorics & Optimization GERAD
University of Waterloo Ecole des Hautes Etudes Commerciales
Waterloo, Ontario, N2L 3Gl 5255 ave Decelles
Canada Montreal, Que., H3T IV6
Canada
Andre BOUCHET
Dep. de Mathematiques et d'Informatique Roland FRAisSE
Universite du Maine Universite de Provence I
Route de Laval U.E.R. de Mathematiques
B.P.535 3, Place Victor Hugo
72017 Le Mans Cedex 13331 Marseille
France France
ix
x PARTICIPANTS
Luis A. GODDYN Bill JACKSON
Combinatorics &: Optimization Mathematical Sciences Department
University of Waterloo Goldsmiths' College
Waterloo, Ontario, N2L 3G1 London SE14 6NW
Canada United Kingdom
Lucien HADDAD Franc;ois JAEGER
Physical Science Division LSD, Institut IMAG
Scarborough College BP 68
University of Toronto 38042 St-Martin-d'Heres Cedex
Scarborough, Ontario, M1C 1A4 France
Canada
Tommy JENSEN
Gena HAHN
Combinatorics &: Optimization
Dep. d'Informatique et de Recherche Operationnelle
University of Waterloo
U niversite de Montreal
Waterloo, Ontario, N2L 3G1
C.P. 6128, Succ. A
Canada
Montreal, Que., H3C 3J7
Canada
Heinz Adolf JUNG
Andras HAJNAL Fachbereich 3- Mathematik
Mathematical institute TU Berlin
Hungarian Academy of Sciences Strasse des 17. Juni 135
1053 Budapest V. 1000 Berlin 12
ReaItanoda u. 13-15 FRG
Hungary
Franc;ois LAVIOLETTE
Rudolf HALIN Dep. de Mathematiques et de Statistique
Mathematisches Seminar Un iversite de Montreal
Universitat Hamburg C.P. 6128, Succ. A
Bundesstrasse 55 Montreal, Que., H3C 3J7
2000 Hamburg 13 Canada
FRG
Charles H.C. LITTLE
Simone HAZAN Department of Mathematics and Statistics
Department of Mathematics Massey University
University of California at Berkeley Palmerston North
Berkeley, CA 94720 New Zealand
U.S.A.
Wolfgang MADER
Pavol HELL
Un iversitat Hannover
School of Computing Science
Institut fiir Mathematik
Simon Fraser University
Welfengarten 1
Burnaby, B.C., V5A 1S6
D-3000 Hannover 1
Canada
FRG
Anthony HILTON
Department of Mathematics Eric MILNER
University of Reading Department of Mathematics and Statistics
P.O. Box 220 University of Calgary
Whiteknights, Reading RG6 2AX Calgary, Alberta, T2N 1N4
United Kingdom Canada
Petr HoRAK Massimo PICARDELLO
Katedra Matematiky Dipartimento di Matematica PUla e Applicata
SvF SVST Universita. dell'Aquila
Radlinskeho 11 67100 L'Aquila
813 68 Bratislava Italy
Czechoslovakia
