Solution of the heawood map-coloring problem.

One of the most fascinating problems in mathematics is the four-color conjecture, and in spite of the fact that we have nothing new to add, a short discussion of the matter is important for our purposes. The fascination of the problem is almost certainly due to the fact that the relevant question may be stated so as to be intelligible to the general public. Are four colors always enough to obtain a coloring of the countries of any map on a sphere? It is only necessary to clarify the italicized words above for the general reader to understand and, if inquisitive, to become interested in the problem. A country must be connected; hence Pakistan, which consists of two disjoined parts, does not qualify. The reader begins to realize that we are considering an abstraction which has little resemblance to political reality. In reference to the term map on a sphere, there are no oceans; every point on the sphere is either inside exactly one country or is on the frontiers of two or more countries. Two countries are adjacent if they have a common line of frontier points. Thus France and Spain are adjacent, but the states of Colorado and Arizona are not, in spite of the fact that they have one frontier point in common. The negating factor is that there is no common line of frontier points. A coloring of a map on a sphere is an assignment of one color to each country so that no pair of adjacent countries is assigned the same color. Thus two countries having the property observed above (Colorado and Arizona) may be assigned the same color, but countries like France and Spain must be assigned different colors. The minimum number of colors which suffices to color a given map is called the chromatic number of the map. The maximum, m, of the chromatic numbers for all maps on the sphere is called the chromatic number of the sphere. Thus we can be assured that any map on the sphere can be colored by using no more than m colors. The question is: "What is m?" It is easy to see that there is a map on a sphere that consists of four countries each adjacent to the other three. Hence this map has four as its chromatic number. Consequently, m> 4. This leads to the classical question: "Does m = 4?" No one knows the answer. It can be shown, however, that m < 5. M\any attempts have been made to settle the matter. One of the most notable was made by the English barrister Kempe, who claimed the result in 1880. In 1890 Heawood1 discovered an error in Kempe's proof and went on to consider the problem for surfaces more complicated than a sphere. The simplest in the hierarchy of such surfaces is a torus, or the surface of a tire. The terms "country," "map," "adjacent," etc., have meaning on such a surface, and Heawood showed that the chromatic number of a torus is seven. The standard topological model of a surface (or orientable two-dimensional manifold) S, of genus p is a sphere with p handles attached to it. (One may also think of the surfaces of a Swiss cheese with p holes through it.) Thus a torus is