The Four Color Problem

When we have a polygonal map before us, we may think of the faces as being countries or states on a map, with the ocean surrounding them in the form of the infinite face. In a good atlas the countries, together with the ocean, are colored in different colors to distinguish them from each other. This means that the coloring must be done so that countries with a common boundary have different colors. If one has a large number of colors at one's disposal, this represents no particular problem. Much more difficult is the question of finding the smallest number of colors sufficient for coloring the countries of a given map.

A famous problem is to prove that every map can be colored properly by means of four colors. The earliest trace of it we find in a letter of 23 October 1852 from the London mathematician Augustus De Morgan to Sir William Rowan Hamilton in Dublin: “A student of mine asked me today to give him a reason for a fact which I did not know was a fact and do not yet. He says that if a figure be anyhow divided and the compartments differently coloured, so that figures with any portion of common boundary line are differently coloured—four colours may be wanted but no more.”

The term “graph” in this book denotes something quite different from the graphs you may be familiar with from analytic geometry or function theory. The kind of graph you probably have dealt with consisted of the set of all points in the plane whose coordinates (x, y), in some coordinate system, satisfy an equation in x and y. The graphs we are about to study in this book are simple geometrical figures consisting of points and lines connecting some of these points; they are sometimes called “linear graphs”. It is unfortunate that two different concepts bear the same name, but this terminology is now so well established that it would be difficult to change. Similar ambiguities in the names of things appear in other mathematical fields, and unless there is danger of serious confusion, mathematicians are reluctant to alter the terminology.

The first paper on graph theory was written by the famous Swiss mathematician Euler, and appeared in 1736. From a mathematical point of view, the theory of graphs seemed rather insignificant in the beginning, since it dealt largely with entertaining puzzles. But recent developments in mathematics, and particularly in its applications, have given a strong impetus to graph theory. Already in the nineteenth century, graphs were used in such fields as electrical circuitry and molecular diagrams.

Puzzles and Directed Graphs

Previously (Section 2.6) we explained how puzzle problems can be formulated in terms of graphs. The vertices of the graph correspond to the positions in a puzzle; the edges of the graph correspond to the possible moves from one position to another. The solution of the puzzle consists in finding a path from a given initial position to one (or possibly more) terminal or winning positions.

In dealing with these puzzles, we used undirected graphs. This was based upon the tacit assumption that the moves can be made both ways from one position to another. Such a procedure is permissible for the puzzles of the ferryman, the three jealous husbands, and the moves of the knight on the chessboard.

But for many puzzles the moves can only be made in one direction, and in this case we are compelled to use directed graphs in the representation. If some moves can be made in both directions, we can include an edge for each direction, or we can use a mixed graph in which these edges are undirected. To solve the puzzle, we must find a directed path from the initial position in the graph to the desired terminal position.

We shall illustrate these remarks by considering an ancient and familiar puzzle.

Relations and Graphs

So far we have discussed a variety of uses of graphs. Applications to everyday problems and to games and puzzles were considered. Our choice of topics had the advantage that we could deal with well-known and simple concepts. In this chapter, we shall strive to make clear that graphs are closely related to (indeed, are only a different way of formulating) some of the most fundamental concepts of mathematics in general.

A mathematical system, as we usually encounter it, consists of a set of objects or elements. For instance, we deal commonly with numbers and these may belong to more or less general types; we may discuss the set of integers, the positive numbers, the rational numbers, real numbers, imaginary numbers, or complex numbers. In algebra, we are concerned with elements which can be added, subtracted, multiplied, and so on. In geometry, we ordinarily have before us a set of points or special categories of points like straight lines, circles, planes, etc. In logic, we deal with the properties of statements of various kinds.

To construct a mathematical theory we need more than these elements; we need relations between them. Let us illustrate this: in the case of numbers, we have equal numbers a and b; in formal mathematical terminology, we write a = b. We also have numbers a and b which are different, and we write a ≠ b.