Three-color graph as the 1-skeleton of the 2-sphere triangulation

TL;DR

This paper investigates three-color edge colorings of 2-sphere triangulations' 1-skeletons, enumerates small cases, and demonstrates unique colorings for triangulations with fewer than eight vertices.

Contribution

It introduces a method for enumerating and analyzing three-colorings of 2-sphere triangulations with up to 8 vertices, including specific findings for small triangulations.

Findings

All triangulations with up to 8 vertices are colorable with the described method.

Identified a triangulation with 6 and 7 vertices each having two different colorings.

Triangulations with fewer than 8 vertices have exactly one coloring each.

Abstract

The paper is devoted to finding the colorings of the edges of the 1-skeleton of triangulations of the 2-sphere in three colors so that for each face all three of its sides have different colors. First, by the method of adding one vertex inside the triangle or on its side, we enumerate all tiangulations with no more than 8 vertices. Next, one triangulation with 6 and 7 vertices, each with two different colors, was found. And finally, it is shown that other triangulations, which have less than 8 vertices, have one coloring each.