Three-coloring triangle-free graphs on surfaces IV. Bounding face sizes of 4-critical graphs
Zdenek Dvorak, Daniel Kral, Robin Thomas

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.
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