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arXiv · 2110.00278

Polynomial bounds for chromatic number. IV. A near-polynomial bound for excluding the five-vertex path

Alex Scott·Paul Seymour·Sophie Spirkl
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Abstract

A graph G is H-free if it has no induced subgraph isomorphic to H. We prove that a $P_5$-free graph with clique number $ω\ge 3$ has chromatic number at most $ω^{\log_2(ω)}$. The best previous result was an exponential upper bound $(5/27)3^ω$, due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdos-Hajnal conjecture holds for $P_5$, which is the smallest open case. Thus there is great interest in whether there is a polynomial bound for $P_5$-free graphs, and our result is an attempt to approach that.

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Alex Scott, Paul Seymour, Sophie Spirkl. 2022-10-02. Polynomial bounds for chromatic number. IV. A near-polynomial bound for excluding the five-vertex path. https://arxiv.org/abs/2110.00278

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