Continued Fractions and the 4-Color Theorem

TL;DR

This paper explores the geometric properties of specific 4-colorings of sphere triangulations with non-negative curvature, introducing a novel interpretation of continued fractions and proposing a conjectural isoperimetric inequality.

Contribution

It introduces a new geometric interpretation of continued fractions applied to sphere triangulations and formulates a conjecture on an optimal isoperimetric inequality for such colorings.

Findings

Constructed extremal examples of 4-colorings with non-negative curvature

Developed a novel geometric interpretation of continued fractions

Proposed a conjectural sharp isoperimetric inequality

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.