Ramsey-type $\chi$-bounds for $\chi$-bounded graph classes

TL;DR

This paper establishes new linear $ ext{chi}$-bounds for classes of graphs excluding certain induced subgraphs, extending previous results and providing improved bounds and proofs for specific cases involving forests and complete multipartite graphs.

Contribution

It proves new linear $ ext{chi}$-bounds for graphs excluding certain induced subgraphs, unifying and extending prior results in $ ext{chi}$-boundedness theory.

Findings

Graphs with no induced path $P$ and no induced $C_4$ are linearly $ ext{chi}$-bounded.

The paper proves $ ext{chi}$-bounds for graphs excluding certain forests and complete multipartite graphs.

Provides a new proof with better bounds for a result on induced trees in sparse graphs.

Abstract

We prove that for every path $P$, the class of graphs with no induced $P$ and no induced four-cycle $C_4$ is linearly $\chi$-bounded. More generally, we ask for which pairs $\{T,H\}$ where $T$ is a forest and $H$ is a complete multipartite graph, every graph $G$ with no induced $T$ and no induced $H$ has chromatic number at most $C \cdot R(\alpha(H),\omega(G)+1)$ for some constant $C$ depending only on $T$ and $H$, where $R(\cdot,\cdot)$ denotes the usual Ramsey numbers. We show that this holds in the following two instances, which strengthen the case $T=P$ and $H=C_4$ mentioned above: (1) every component of $T$ is a broom and $H$ is complete multipartite; or (2) $T$ is a forest and $H$ is complete bipartite. These two unify and substantially extend a number of previous results on linear and polynomial $\chi$-boundedness for various graph classes. For case (2), we also provide a new proof (with better bounds) of a recent result of Fox, Nenadov, and Pham on the existence of an induced copy of a fixed tree in a graph satisfying certain sparsity conditions.