Gr\"unbaum Colorings of Toroidal Triangulations

TL;DR

This paper proves that for any triangulation of the torus with chromatic number not equal to 5, a 3-edge-coloring exists where each face's boundary edges are all differently colored.

Contribution

It establishes a new coloring property for toroidal triangulations, extending understanding of face-edge colorings in topological graph theory.

Findings

Triangulations of the torus with chromatic number not 5 admit a 3-edge-coloring.

The result applies to all such triangulations, regardless of their specific structure.

Provides a new characterization of face-edge colorings in toroidal graphs.

Abstract

We prove that if G is a triangulation of the torus and \chi(G) \neq 5, then there is a 3-coloring of the edges of G so that the edges bounding every face are assigned three different colors.