Three-coloring triangle-free graphs on surfaces IV. Bounding face sizes of 4-critical graphs

TL;DR

This paper establishes a linear bound on the sum of face sizes in 4-critical triangle-free graphs embedded on surfaces, relating face sizes to genus, triangles, and specific 4-cycles.

Contribution

It provides a new linear bound on face sizes in 4-critical graphs on surfaces, linking topological and combinatorial properties.

Findings

Sum of face lengths (>=5) is linearly bounded by g+t+c-1

Introduces bounds relating surface genus, triangles, and 4-cycles

Advances understanding of face structure in critical graphs

Abstract

Let G be a 4-critical graph with t triangles, embedded in a surface of genus g. Let c be the number of 4-cycles in G that do not bound a 2-cell face. We prove that the sum of lengths of (>=5)-faces of G is at most linear in g+t+c-1.