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arXiv · 2409.19165

Four-coloring Eulerian triangulations of the torus

Marcin Brianski·Daniel Kral·Ander Lamaison·Xichao Shu
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Abstract

Hutchinson, Richter and Seymour [J. Combin. Theory Ser. B 84 (2002), 225-239] showed that every Eulerian triangulation of an orientable surface that has a sufficiently high representativity is 4-colorable. We give an explicit bound on the representativity in the case of the torus by proving that every Eulerian triangulation of the torus with representativity at least 10 is 4-colorable. We also observe that the bound on the representativity cannot be decreased to less than 8 as there exists a non-4-colorable Eulerian triangulation of the torus with representativity 7.

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Marcin Brianski, Daniel Kral, Ander Lamaison, Xichao Shu. 2024-09-27. Four-coloring Eulerian triangulations of the torus. https://arxiv.org/abs/2409.19165

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