FOUR-COLOUR MAP PROBLEM

problem in topology, originally posed in the early 1850s and not solved until 1976, that required finding the minimum number of different colours required to colour a map such that no two adjacent regions (i.e., with a common boundary segment) are of the same colour. Three colours are not enough, as can be seen in a map of four regions with each region contacting three other regions. It had been proved mathematically that five colours will always suffice; and no map had ever been found empirically on which four colours would not do. As is often the case in mathematics, consideration of the problem provided the impetus for the discovery of related results in topology and combinatorics. A similar problem had been solved for the seemingly more complicated situation of a map drawn on a torus (doughnut-shaped surface), where seven colours were known to be the minimum. The four-colour problem was solved in 1976 by a group of mathematicians at the University of Illinois, directed by Kenneth Appel and Wolfgang Haken, which after four years had completed an unprecedented synthesis of computer search and theoretical reasoning. Appel and Haken created a catalog of 1,936 unavoidable configurations that must be present in any graph, no matter how large. Then they showed how each of these configurations could be reduced to a smaller one so that, if the smaller one could be coloured with four colours, so could the original catalog configuration. Thus, if there were a map that could not be coloured with four colours, they could use their catalog to find a smaller map that also could not be four-coloured, and then a smaller one still, and so on. Eventually this reduction process would lead to a map with only three or four regions that, supposedly, could not be coloured with four colours. This absurd result, which is derived from the hypothesis that a map requiring more than four colours might exist, leads to the conclusion that no such map can exist. All maps are in fact four-colourable. The strategy involved in this proof dates back to 1879 when the British mathematician A.B. Kempe produced a short list of unavoidable configurations and then showed how to reduce each to a smaller case. Appel and Haken replaced Kempe's brief list with their catalog of 1,936 cases, each involving up to 500,000 logical options for full analysis. Their complete proof, itself several hundred pages long, required more than 1,000 hours of computer calculations.