Polynomial bounds for chromatic number. III. Excluding a double star

TL;DR

This paper proves that for graphs excluding a double star as an induced subgraph, the chromatic number can be bounded by a polynomial function of the clique number, advancing the understanding of the Gyárfás-Sumner conjecture.

Contribution

It establishes that the function bounding the chromatic number for graphs excluding a double star can be chosen to be polynomial, confirming a polynomial bound in this case.

Findings

Chromatic number is polynomially bounded for graphs excluding a double star.

Confirms the Gyárfás-Sumner conjecture for double stars with polynomial bounds.

Advances understanding of graph coloring in relation to forbidden induced subgraphs.

Abstract

A double star is a tree with two internal vertices. It is known that the Gy\'arf\'as-Sumner conjecture holds for double stars, that is, for every double star $H$, there is a function $f$ such that if $G$ does not contain $H$ as an induced subgraph then $\chi(G)\le f(\omega(G))$ (where $\chi, \omega$ are the chromatic number and the clique number of $G$). Here we prove that $f$ can be chosen to be a polynomial.