Fine structure of 4-critical triangle-free graphs III. General surfaces

TL;DR

This paper refines the structural understanding of 4-critical triangle-free graphs on surfaces, specifically addressing unresolved coloring properties related to certain substructures, thereby advancing graph coloring theory on surfaces.

Contribution

It completes the description of the structure of 4-critical triangle-free graphs on surfaces by resolving previously undetermined coloring properties of specific substructures.

Findings

Resolved coloring properties of two substructures in the graph structure

Enhanced the structural description of 4-critical triangle-free graphs

Contributed to the theory of graph coloring on surfaces

Abstract

Dvo\v{r}\'ak, Kr\'al' and Thomas gave a description of the structure of triangle-free graphs on surfaces with respect to 3-coloring. Their description however contains two substructures (both related to graphs embedded in plane with two precolored cycles) whose coloring properties are not entirely determined. In this paper, we fill these gaps.