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John Gimbel, John W. Kennedy, “Quo Vadis, Graph Theory?: A Source Book for Challenges and Directions”

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 applications.

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 underlying

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 encourages,

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 simple

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 complete

solutions are elusive. Nevertheless, in pursuit of these solutions, graph theorists are frequently

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 meeting,

also under the title “Quo Vadis, Graph Theory?” held at the University of Alaska, Fairbanks

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 complementary

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 philosophical

approach. “New Directions in Graph Theory” by Fred S. Roberts offers a comprehensive

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 questions

that await resolution.

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 Theory?

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.

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