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arXiv · 2107.11780

Polynomial bounds for chromatic number. II. Excluding a star-forest

Alex Scott·Paul Seymour·Sophie Spirkl
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Abstract

The Gyarfas-Sumner conjecture says that for every forest $H$, there is a function $f$ such that if $G$ is $H$-free then $\chi(G)\le f(\omega(G))$ (where $\chi, \omega$ are the chromatic number and the clique number of $G$). Louis Esperet conjectured that, whenever such a statement holds, $f$ can be chosen to be a polynomial. The Gyarfas-Sumner conjecture is only known to be true for a modest set of forests $H$, and Esperet's conjecture is known to be true for almost no forests. For instance, it is not known when $H$ is a five-vertex path. Here we prove Esperet's conjecture when each component of $H$ is a star.

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Alex Scott, Paul Seymour, Sophie Spirkl. 2021-07-25. Polynomial bounds for chromatic number. II. Excluding a star-forest. https://arxiv.org/abs/2107.11780

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