Planarity and the Four-Color Theorem

Abstract

A graph is said to be planar if it can be drawn in the plane with no crossing edges. A classic problem based on planarity is the “utility problem”. Suppose there are three people—Jack, Jill, and Judy—living in separate houses, and also three utilities—water, gas, and electricity—each supplied by a different plant. We wish to connect each of the three houses to each of the three plants, but we don’t want any of the nine connections to cross each other. (Perhaps all of the utilities are supplied by cables or pipes buried just beneath the surface, and if two were to cross we might damage one conduit while installing the other. Oh well, nobody ever said this problem was practical, just that it was a classic.) We can easily make eight of the connections, but we run into trouble with the ninth, as shown below (left). Struggle as we may, we will be unable to make all nine connections. (We could cheat and run a conduit underneath one of the buildings, but this is not considered valid.) This graph is non-planar. The graph is shown on the right below; it is the complete bipartite graph on two sets of three vertices, meaning that it contains two sets of three vertices along with all edges joining a vertex in one set with one in the other set. This graph is usually denoted by K_3,3.