Network-Colourings

Abstract

I wonder why problems about map-colourings are so fascinating? I know several people who have made more or less serious attempts to prove the Four-Colour Theorem, and I suppose many more have made collections of maps in the hope of hitting upon a counter-example. I like P. G. Tait’s approach myself; he removed the problem from the plane so that it could be discussed in terms of more general figures. He showed that the Four-Colour Theorem is equivalent to the proposition that if N is a connected cubical network, without an isthmus, in the plane, then the edges of N can be coloured in three colours so that the colours of the three meeting at any vertex are all different. (A cubical network is a set of points called vertices joined in pairs by simple arcs called edges, no two of which intersect except at a common vertex, in such a way that just three edges meet at each vertex. An isthmus is an edge whose removal destroys the connection of the network.) It was at first conjectured that every cubical network having no isthmus could be “three-coloured” in this way, but this was disproved by the example of Fig. 1, for which it may readily be verified that no three-colouring exists.