Polynomial bounds for chromatic number VI. Adding a four-vertex path

TL;DR

This paper advances understanding of polynomial $hi$-boundedness in graph classes, proving that disjoint unions involving good forests and paths are also good, thus broadening the class of graphs with polynomial chromatic bounds.

Contribution

It proves that the disjoint union of a good forest and a four-vertex path is good, extending the class of graphs with polynomial chromatic bounds.

Findings

Disjoint union of a good forest and a four-vertex path is good.

If every component of a forest is good, then the union with a path is also good.

Graphs avoiding certain forests and paths are polynomially $hi$-bounded.

Abstract

A class of graphs is $\chi$-bounded if there is a function $f$ such that every graph $G$ in the class has chromatic number at most $f(\omega(G))$, where $\omega(G)$ is the clique number of $G$; the class is polynomially $\chi$-bounded if $f$ can be taken to be a polynomial. The Gy\'arf\'as-Sumner conjecture asserts that, for every forest $H$, the class of $H$-free graphs (graphs with no induced copy of $H$) is $\chi$-bounded. Let us say a forest $H$ is good if it satisfies the stronger property that the class of $H$-free graphs is polynomially $\chi$-bounded. Very few forests are known to be good: for example, it is open for the five-vertex path. Indeed, it is not even known that if every component of a forest $H$ is good then $H$ is good, and in particular, it was not known that the disjoint union of two four-vertex paths is good. Here we show the latter, and more generally, that if $H$ is good then so is the disjoint union of $H$ and a four-vertex path. We also prove a more general result: if every component of $H_1$ is good, and $H_2$ is any path (or broom) then the class of graphs that are both $H_1$-free and $H_2$-free is polynomially $\chi$-bounded.