How false is Kempe’s proof of the Four Color Theorem? Part II

Abstract

We continue the investigation of A. B. Kempe’s flawed proof of the Four Color Theorem from a computational and historical point of view. Kempe’s “proof” gives rise to an algorithmic method of coloring plane graphs that sometimes yields a proper vertex coloring requiring four or fewer colors. We investigate a recursive version of Kempe’s method and a modified version based on the work of I. Kittell. Then we empirically analyze the performance of the implementations on a variety of historically motivated benchmark graphs and explore the usefulness of simple randomization in four-coloring small plane graphs. We end with a list of open questions and future work.