1.1. The scope of the work. Sometimes it seems that our society faces overwhelmingly difficult problems, problems involving energy, transportation, pollution, perturbed ecosystems, urban services, the economy, genetic changes, social inequalities, and so on. Increasingly, mathematics is being used, at least in small ways, to tackle these problems. In these lectures we shall examine the role of one branch of mathematics, graph theory, in applications to such problems of society.
We have chosen to present mathematical topics from the field of graph theory because graphs have wide-ranging applicability and because it is possible in graph theory to bring a previously unfamiliar scientist to the frontiers of research rather quickly. The choice of topics from within graph theory and even more so the order of presentation of these topics is not typical of the graph theory literature. Rather, the topics were chosen to best illustrate the applications, and to lead into them as quickly as possible. Some of the more traditional topics of graph theory, such as colorability, independence, and eulerian chains, are not covered until fairly late. Then, they are presented with an emphasis on results of applied interest.
We have tried to be self-contained in preparing these notes. However, they are written on a research level, with the goal being to present results at the frontiers of current graph-theoretical work. The reader will find some of these same topics discussed at a more leisurely pace in Roberts (1976a). These lectures are in some sense a continuation of the topics presented in Chapters 3 and 4 of that book. They go beyond the results stated there, present more recent work, and introduce a variety of additional applied and graph-theoretical topics.
The problems of society with which we are concerned are extremely complex and wide-ranging. At the outset, let us put into perspective the role of graph theory in particular and mathematics in general vis-à-vis these problems. We will not claim that graph theory alone can solve these problems. Nor will we argue that they cannot be solved without graph theory. Rather, we hope to demonstrate that the use of precise, graph-theoretical reasoning can cast light on such problems, provide tools to help in making decisions about them, and help in finding answers to a variety of specific questions which arise in the attempt to tackle the broader issues.
Graph theory is a tool for formulating problems, making them precise, and defining fundamental interrelationships. Sometimes, as we shall see, simply formulating a problem precisely helps us to understand it better. The very act of formulation is an aid to understanding.