# 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}
}```
• Published 25 April 2014
• Computer Science, Mathematics
• J. Comb. Theory, Ser. B
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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