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

TL;DR

This paper proves that graphs excluding a certain tree of radius two and a complete multipartite subgraph have a chromatic number bounded by a polynomial in the size of the excluded subgraph, advancing the understanding of the Gyárfás-Sumner conjecture.

Contribution

It extends polynomial bounds on chromatic number to graphs excluding a radius-two tree and a complete multipartite subgraph, generalizing previous results.

Findings

Chromatic number is polynomially bounded under the given exclusions.

Generalizes previous bounds for trees of radius two.

Strengthens the connection between forbidden subgraphs and chromatic bounds.

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 $t$.