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

@article{Dvok2021ThreecoloringTG, title={Three-coloring triangle-free graphs on surfaces IV. Bounding face sizes of 4-critical graphs}, author={Zdeněk Dvoř{\'a}k and Daniel Kr{\'a}l and Robin Thomas}, journal={J. Comb. Theory, Ser. B}, year={2021}, volume={150}, pages={270-304} }

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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## 11 Citations

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