Polynomial bounds for chromatic number. II. Excluding a star-forest

TL;DR

This paper proves Esperet's conjecture that for forests composed of star components, the chromatic number of H-free graphs can be bounded by a polynomial function of their clique number.

Contribution

It establishes the polynomial bound for the chromatic number in graphs excluding forests made of star components, confirming Esperet's conjecture in this case.

Findings

Polynomial bounds for chromatic number when excluding star-forest subgraphs

Confirmation of Esperet's conjecture for forests with star components

Advancement in understanding the chromatic properties of H-free graphs

Abstract

The Gyarfas-Sumner conjecture says that for every forest $H$, there is a function $f$ such that if $G$ is $H$-free then $\chi(G)\le f(\omega(G))$ (where $\chi, \omega$ are the chromatic number and the clique number of $G$). Louis Esperet conjectured that, whenever such a statement holds, $f$ can be chosen to be a polynomial. The Gyarfas-Sumner conjecture is only known to be true for a modest set of forests $H$, and Esperet's conjecture is known to be true for almost no forests. For instance, it is not known when $H$ is a five-vertex path. Here we prove Esperet's conjecture when each component of $H$ is a star.