Quo Vadis, Graph Theory?A Source Book for Challenges and Directions

407 Pages·1993·19.59 MB·English

by  John Gimbel

QUO VADIS, GRAPH THEORY?
ANNALS OF DISCRETE MATHEMATICS 55
General Editor: Peter L. HAMMER
Rutgers University, New Brunswick, NJ, USA
Advisory Editors:
C. BERGE, Universite de Paris, France
R.L. GRAHAM, AT&T Bell Laboratories, NJ, USA
M.A. HARRISON, University of California, Berkeley, CA, USA
V KLEE, University of Washington, Seattle, WA, USA
J.H. VAN LINT California Institute of Technology, Pasadena, CA, USA
G. C. ROTA, Massachusetts Institute of Technology, Cambridge, MA, USA
7: TROTER, Arizona State University, Tempe, AZ, USA
QUO VADIS, GRAPH THEORY?
A Source Book for Challenges and Directions
Edited by
John GIMBEL
University of Alaska
Fairbanks, AK, USA
John W. KENNEDY and Louis V. QUINTAS
Pace University
New York, NY USA
1993
NORTH-HOLLAND- AMSTERDAM LONDON NEW YORK TOKYO
ELSEVIER SCIENCE PUBLISHERS B.V.
Sara Burgerhartstraat 25
RO. Box 21 1, 1000 AE Amsterdam, The Netherlands
Library of Congress Cataloging-in-Publication Data
Quo vadis. graph theory? a source book for challenges and directions
/ edipt.e d byc m.J oh--n G(.A nGnianlbse lo,f Jdolhsnc rWe.t eK ennantehdeyn,a tlacnsd L.o u5i5s) V. Quintas.
Includes bibliographical references and index.
ISBN 0-444-89441-1 (alk. paper)
1. Graph theory. I. Gimbel. John Gordon. 11. Kennedy, J. W.
(John W.) 111. Quintas. Louis V. IV. Series.
QAlEE.06 1993
511'.5--dC20 93-9334
CIP
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FOREWORD
In the spectrum of mathematics, graph theory, as a recognized discipline, is a relative
newcomer. The first formal paper is found in the work of Leonhard Euler in 1736. In recent
years the subject has grown rapidly so that, in today’s literature, mathematical and scientific,
graph theory papers abound with new mathematical developments and significant applica-
tions. Three factors, perhaps, account for this explosive growth of the subject:
1) Graph theory provides the natural structures from which to construct mathematical models
that are appropriate to almost all fields of scientific (natural and social) enquiry. The underly-
ing subject of study in these fields is some set of “objects” and one or more “relations”
between the objects.
2) Graph theory has developed a rich language of terms to render concise the expression of
intricate concepts associated with object-relation structures. This facilitates, indeed encour-
ages, interdisciplinary communication of ideas and techniques to the benefit of all fields that
use graph theory.
3) Graph theory offers a huge selection of intellectual challenges that range in level from sim-
ple exercises for the novice, to deep open questions for the mathematical sophisticate. Many
fascinating and compelling questions in graph theory are easy to comprehend, but their com-
plete solutions are elusive. Nevertheless, in pursuit of these solutions, graph theorists are fre-
quently rewarded by achieving results that contribute to further development of the subject.
As with any academic field, it is beneficial periodically to step back and ask: “Where is all
this activity taking us?” “What are the outstanding fundamental problems?” “What are the
next important steps we should take?” In short, “Quo Vudis, Graph Theory?” Thanks to our
contributors, this volume offers a comprehensive reference source for future directions and
open questions in graph theory.
The idea for this volume originated together with that for an international discussion meet-
ing, also under the title “Quo Vadis, Graph Theory?” held at the University of Alaska, Fair-
banks in August of 1990. By means of discussion, rather than by formal presentation of
results, participants considered significant avenues for further exploration in graph theory.
This volume is not a proceedings of that meeting; rather, it is a collection of papers written
with the discussions of that meeting as background.
The first three papers in the volume are special in that they provide the reader with com-
plementary perspectives on the future of graph theory in general. “Whither Graph Theory?”
by William T. Tutte and “The Future of Graph Theory” by BCla Bollobas each take a philo-
sophical approach. “New Directions in Graph Theory” by Fred S. Roberts offers a compre-
hensive overview of questions and developments in the subject with an emphasis on
applications. It is with these three papers that we recommend that the reader start.
The remaining papers are arranged by topic, in the order used in the paper by Roberts.
These papers elaborate on the potential for future developments in specific topics of graph
theory. Among them the reader will find a rich source of worthwhile and challenging ques-
tions that await resolution.
v1
The editors express their thanks to the contributors to this volume, their efforts especially
have made this a worthwhile task. Our thanks are also due to the referees for their thorough
efforts and useful suggestions.
We gratefully acknowledge support for this volume and for the Quo Vadis, Graph The-
ory? meeting in Alaska provided by The Air Force Office of Scientific Research, The ARC0
Foundation, The National Security Agency, The Office of Naval Research and The University
of Alaska Fairbanks.
Our special thanks are due to Michael Kazlow, Mathematics Department, Pace University
for his expertise and dedication while worlung with us on the many technical and editorial
aspects of the preparation of this volume.
We also thank Peter L. Hammer and Elsevier Science Publishers for their encouragement
in the publication of this work.
Finally we thank the University of Alaska Fairbanks and Pace University for their general
support of this project.
John Gimbel, University of Alaska Fairbanks
John W. Kennedy, Pace University, New York
Louis V. Quintas, Pace University, New York
August, 1992
vii
CONTENTS
Foreword V
Whither graph theory? W.T. T ~ E 1
The future of graph theory, B. BOLLOBAS 5
New directions in graph theory (with an emphasis on
the role of applications), F.S. ROBERTS 13
A survey of (mk,) -colorings, M. FRICK 45
Numerical decks of trees, F. GAVRIIL. ,K RASIKOanVd J. SCHONHEIM 59
The complexity of colouring by infinite vertex transitive graphs, B. BAUSLAUGH 71
Rainbow subgraphs in edge-colorings of complete graphs, P. ERDCaJnSd Z. TUZA 81
Graphs with special distance properties, M. LEWNTER 89
Probability models for random multigraphs with applications
in cluster analysis, E.A.J. GODEHARDT 93
Solved and unsolved problems in chemical graph theory, A.T. BALABAN, 109
Detour distance in graphs, G. CHARTRANGD.L,. JOHNaSn d S. TIAN 127
Integer-distance graphs, R.P. GRIMALDI 137
Toughness and the cycle structure of graphs, D. BAUERan d E. SCHMEICHEL 145
The Birkhoff-Lewis equations for graph-colorings, W.T. TurrE 153
The complexity of knots, D.J.A. WELSH 159
The impact of F-polynomials in graph theory, E.J. FARRELL 173
A note on well-covered graphs, V. CHVATAanLd P.J. SLATER 179
Cycle covers and cycle decompositions of graphs, C.-Q. ZHANG 183
Matching extensions and products of graphs, J. LIUa nd Q. Y u 191
Prospects for graph theory algorithms, R.C. READ 20 1
The state of the three color problem, R. STEINBERG 21 1
Ranking planar embeddings using PQ-trees, A. KARABEG 249
Some problems and results in cochromatic theory, P. ERD~anSd J. GIMBEL 26 1
From random graphs to graph theory, A. RUCINSKI 265
Matching and vertex packmg: How “hard”are they? M.D. PLIJMMER 275
The competition number and its variants, S.-R. KIM 3 13
Which double starlike trees span ladders? M. LEWINTEaRnd W.F. WIDULSKI 327
The randomf-graph process, K.T. BALIKJSaKnAd L.V. QUINTAS 333
Quo vadis, random graph theory? E.M. PALMER 34 1
Exploratory statistical analysis of networks, 0. FRANaKnd K. NOWTCKI 349
The Hamiltonian decomposition of certain circulant graphs, J. LIU 367
Discovery-method teaching in graph theory, P.Z. CHINN 375
Index of Key Terms 385
...
Vlll
Quo Vadis, Graph Theory?
was also the title used for
An International Conference on the Future of Graph Theory
held at University of Alaska Fairbanks, August 1990.
Sponsors
The Air Force Office of Scientific Research
The ARC0 Foundation
The College of Liberal Arts, UAF
The Department of Mathematical Sciences, UAF
The National Security Agency
The Office of Naval Research
The Vice Chancellor for Academic Affairs, UAF
Organizing Committee
Phyllis Chinn, Humboldt State University, California
John Gimbel, University of Alaska Fairbanks, Alaska
John W. Kennedy, Pace University, New York
Louis V. Quintas, Pace University, New York
Fred S. Roberts, Rutgers University and Rutcor, New Jersey
Local Organizing Committee
Ron Gatterdam
Hannibal Grubis
Dushan Jetvic
Pete Knoke
Laura Lee Potrikus
Quo Vadis, Graph Theory? J. Girnbel, J.W. Kennedy & L.V. Quintas (eds.)
Annals of Discrete Mathematics, 55, 1 4( 1993)
0 1993 Elsevier Science Publishers B.V. All rights reserved.
WHITHER GRAPH THEORY?
William T. TU’ITE
Department of Combinatorics and Optimization
University of Waterloo, Waterloo, Ontario, CANADA
Abstract
This is the text of an oration delivered at the conference Quo Vadis, Graph Theory?, held at Fairbanks,
Alaska, on August 16,1990. It enlarges upon the image of a well introduced by R.C. Read at the same
Conference. He envisaged graph theorists as situated at the bottom of a well among the graphs of sim-
plest structure, with the more interesting graphs extending upward along the well-shaft and out to the
Stars.
Friends, Romans, fellow-citizens of the Graphic Republic, mark me well, for it is of a well
that I would speak. One of the things that impressed me in the lectures we have heard was the
metaphor that as graph theorists we live at the bottom of a well. That, I recall, was the fate of
three little girls in a work we all revere [l]. Their names, if I remember rightly, were Elsie,
Tillie and Lacie, and they lived in a well. Well in, as the narrator insisted. It was a treacle-
well, and they became very ill through consuming nothing but treacle.
We do not think our well is a treacle-well; we would rather call it a nectar-well or an
ambrosia-well. We subsist upon its product and the unenlightened remark that it makes us
mentally very ill. For it fires our imaginations and we sing right merrily of graphs and
matroids. Well has it been written: “Their young men shall see visions and their old men shall
dream dreams” (Ladies, feel free to read “women” for ‘hen”i n that quotation).
A recurring vision and dream has the well well-walled with graphs. Down at the bottom is
the null graph. Careful not to step on it! There are small graphs around us and big ones higher
up. There are mighty ones miles high. Graphs growing wider still and wider through the
leagues and the light years. For it is a deep well.
We want to explore that grand array of graphs, and reduce it to order, the order of theo-
rems and algorithms. There are ways of contacting those graphs. It can be done through the
lore of large numbers, as in so many of the theorems of Erdos. Or we can look at the graphs
nearby, note regularities, state those regularities as conjectural theorems, and then try to prove
those theorems for all graphs, even for those soaring out of sight. It works sometimes, usually
by the grace of the principle of mathematical induction. Some of the proved theorems give
algorithms, and we can carry through those algorithms step by step for graphs not far away.
But not for the graphs up there in the starry immensities. Even for them we like to assert that
the algorithms exist. Moreover, we like to affirm that some of them can be carried out in poly-
nomial time, even though we cannot imagine them being carried out at all.
We have paid special attention to algorithms of practical utility, applying to low-lying
graphs. In my graph-theoretical dreams I envision someone coming upon me and speaking
thus: “Avert thine eyes from the heavens and see the graphs that may bring thee treasures on
Earth. Be thou not like Thales of old who, gazingjixedly at the stars, fell into a well”. One can
only reply “Thou warn’st me too late. I am in a well already. Well in”. But he is a prophet of
a possible future for Graph Theory.
Mind you, in some moods I have much sympathy for him. I do find it hard to believe in all
those graphs up there getting bigger and bigger as they recede into the distance. No doubt