Irreducible 4-critical triangle-free toroidal graphs

TL;DR

This paper characterizes the structure of 4-critical triangle-free graphs on the torus, identifying exactly four irreducible cases and bounding the types and number of faces such graphs can have.

Contribution

It provides a computer-assisted classification of irreducible 4-critical triangle-free toroidal graphs and establishes face-bound constraints for all such graphs.

Findings

Exactly four irreducible 4-critical triangle-free toroidal graphs identified.

Every such graph has at most four 5-faces or specific combinations involving 6- and 7-faces.

Provides structural bounds essential for classifying all 4-critical triangle-free toroidal graphs.

Abstract

The theory of Dvorak, Kral, and Thomas (2015) shows that a 4-critical triangle-free graph embedded in the torus has only a bounded number of faces of length greater than 4 and that the size of these faces is also bounded. We study the natural reduction in such embedded graphs - identification of opposite vertices in 4-faces. We give a computer-assisted argument showing that there are exactly four 4-critical triangle-free irreducible toroidal graphs in which this reduction cannot be applied without creating a triangle. Using this result, we show that every 4-critical triangle-free graph embedded in the torus has at most four 5-faces, or a 6-face and two 5-faces, or a 7-face and a 5-face, in addition to at least seven 4-faces. This result serves as a basis for the exact description of $4$-critical triangle-free toroidal graphs, which we present in a followup paper.