Continued Fractions and the 4-Color Theorem

Abstract

We study the geometry of some proper 4-colorings of the vertices of sphere triangulations with degree sequence 6,...,6,2,2,2. Such triangulations are the simplest examples which have non-negative combinatorial curvature. The examples we construct, which are roughly extremal in some sense, are based on a novel geometric interpretation of continued fractions. We also present a conjectural sharp "isoperimetric inequality" for colorings of this kind of triangulation.

Lemma 3.1 in the previous version was not correct, and this left a gap in the proof of Statement 1 of Theorem 1.1. This version fixes Lemma 3.1 and thereby removes the gap