Three-coloring triangle-free graphs on surfaces IV. Bounding face sizes of 4-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 1] f face of G (|f| & minus; 4) <= kappa(g +t + c & minus; 1) for a fixed constant kappa, thus generalizing and strengthening several known results. As a corollary, we prove that every triangle-free graph G embedded in a surface of genus g contains a set of O(g) vertices such that G & minus; X is 3-colorable. (c) 2020 Elsevier Inc.

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Keywords

• Graph coloring
• Graphs on surfaces
• Triangle-free