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arXiv · 2202.05557

Polynomial bounds for chromatic number. V. Excluding a tree of radius two and a complete multipartite graph

Alex Scott·Paul Seymour
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Abstract

The Gy\'arf\'as-Sumner conjecture says that for every forest $H$ and every integer $k$, if $G$ is $H$-free and does not contain a clique on $k$ vertices then it has bounded chromatic number. (A graph is $H$-free if it does not contain an induced copy of $H$.) Kierstead and Penrice proved it for trees of radius at most two, but otherwise the conjecture is known only for a few simple types of forest. More is known if we exclude a complete bipartite subgraph instead of a clique: R\"odl showed that, for every forest $H$, if $G$ is $H$-free and does not contain $K_{t,t}$ as a subgraph then it has bounded chromatic number. In an earlier paper with Sophie Spirkl, we strengthened R\"odl's result, showing that for every forest $H$, the bound on chromatic number can be taken to be polynomial in $t$. In this paper, we prove a related strengthening of the Kierstead-Penrice theorem, showing that for every tree $H$ of radius two and every integer $d\ge 2$, if $G$ is $H$-free and does not contain as a subgraph the complete $d$-partite graph with parts of cardinality $t$, then its chromatic number is at most polynomial in $t$.

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Alex Scott, Paul Seymour. 2022-02-11. Polynomial bounds for chromatic number. V. Excluding a tree of radius two and a complete multipartite graph. https://arxiv.org/abs/2202.05557

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