Four-coloring Eulerian triangulations of the torus

TL;DR

This paper establishes that all Eulerian triangulations of the torus with representativity at least 10 are 4-colorable, providing explicit bounds and demonstrating the sharpness of these bounds with a counterexample at 7.

Contribution

It provides an explicit representativity bound of 10 for 4-colorability of Eulerian torus triangulations, improving previous general results and showing the bound's optimality.

Findings

Eulerian triangulations of the torus with representativity ≥ 10 are 4-colorable

A non-4-colorable Eulerian triangulation exists with representativity 7

The bound on representativity cannot be lowered below 8

Abstract

Hutchinson, Richter and Seymour [J. Combin. Theory Ser. B 84 (2002), 225-239] showed that every Eulerian triangulation of an orientable surface that has a sufficiently high representativity is 4-colorable. We give an explicit bound on the representativity in the case of the torus by proving that every Eulerian triangulation of the torus with representativity at least 10 is 4-colorable. We also observe that the bound on the representativity cannot be decreased to less than 8 as there exists a non-4-colorable Eulerian triangulation of the torus with representativity 7.