Note on $4$-coloring $6$-regular triangulations on the torus

TL;DR

This paper addresses and corrects a gap in the classification of 4-colorable 6-regular triangulations on the torus, completing the characterization of their colorability.

Contribution

It identifies a gap in previous classification work and provides a corrected, complete characterization of 4-colorability for all such triangulations.

Findings

Identified a gap in Collins and Hutchinson's classification

Fixed the classification to complete the characterization

Unified the classification with Yeh and Zhu's results

Abstract

In 1973, Altshuler characterized the $6$-regular triangulations on the torus to be precisely those that are obtained from a regular triangulation of the $r \times s$ toroidal grid where the vertices in the first and last column are connected by a shift of $t$ vertices. Such a graph is denoted $T(r, s, t)$. In 1999, Collins and Hutchinson classified the $4$-colorable graphs $T(r, s, t)$ with $r, s \geq 3$. In this paper, we point out a gap in their classification and show how it can be fixed. Combined with the classification of the $4$-colorable graphs $T(1, s, t)$ by Yeh and Zhu in 2003, this completes the characterization of the colorability of all the $6$-regular triangulations on the torus.