Chapter 9 - Map Coloring

Summary

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.”