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Examples and Counterexamples in Graph Theory

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It is a real pleasure, indeed an honor, for me to have been invited by Mike Capobianco and John Molluzzo to write an introduction to this imaginative and valuable addition to graph theory. Let me therefore present a few of my thoughts on the current status of graph theory and how their work contributes to the field.

Graphs have come a long way since 1736 when Leonhard Euler applied a graph-theoretic argument to solve the problem of the seven Konigsberg bridges. At first, interest in and results involving graphs came slowly. Two centuries passed before the first book exclusively devoted to graphs was written. Its author, Denes Konig, referred to his 1936 publication as "The Theory of Finite and Infinite Graphs" (translated from the German). The results on graphs obtained during the time between Konigberg and Konig's book were indeed developing into a theory. In the past several years a number of changes have taken place in graph theory. The applicability of graphs and graph theory to a wide range of areas both within and outside mathematics has given added stature to this youthful subject. It is clear that the full potential and usefulness of graph theory is only beginning to be realized.

The growth of graph theory during its first two hundred years could in no way foreshadow the spectacular progress which this area was to make. There is little doubt that many of the early concepts and theorems (and a few recent ones as well) were influenced by attempts to settle the Four Color Conjecture. Undoubtedly, the development of graph theory was favorably affected by the resistance to proof displayed by this now famous theorem. No longer, however, is graph theory a subject which primarily deals with the Four Color Conjecture or with games and puzzles. The dynamic expansion of graph theory has lead to the development of many significant and applicable subareas with its own concepts and theorems. As with any other area of mathematics, each major theorem in graph theory has associated with it an example or class of examples which illustrate the necessity of the hypothesis, the sharpness of the result, or the falsity of the converse. In this case, the examples are, of course, graphs. In many cases, the graphs have become as famous as the theorems themselves.
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