Characterization of 4-critical triangle-free toroidal graphs

TL;DR

This paper provides a precise characterization of 3-colorability for triangle-free graphs on the torus using 186 templates, leading to an efficient algorithm for determining colorability.

Contribution

It introduces an exact template-based characterization of 3-colorability for triangle-free toroidal graphs, a novel approach in topological graph coloring.

Findings

Graphs with edge-width at least six are 3-colorable.

The template characterization enables efficient 3-colorability algorithms.

Identifies specific subgraph structures that determine non-3-colorability.

Abstract

We give an exact characterization of 3-colorability of triangle-free graphs drawn in the torus, in the form of 186 "templates" (graphs with certain faces filled by arbitrary quadrangulations) such that a graph from this class is not 3-colorable if and only if it contains a subgraph matching one of the templates. As a consequence, we show every triangle-free graph drawn in the torus with edge-width at least six is 3-colorable, a key property used in an efficient 3-colorability algorithm for triangle-free toroidal graphs.