Colouring squares of claw-free graphs

TL;DR

This paper proves that for claw-free graphs, the chromatic number of their square graphs is bounded above by a quadratic function of their clique number, extending a question originally posed for line graphs.

Contribution

It establishes an affirmative answer to a generalization of a classic question about coloring squares of claw-free graphs, linking it to the original Erdős–Nešetřil problem.

Findings

Confirmed the existence of a universal epsilon for claw-free graphs.

Reduced the general problem to the original Erdős–Nešetřil question.

Extended results from line graphs to all claw-free graphs.

Abstract

Is there some absolute $\varepsilon > 0$ such that for any claw-free graph $G$, the chromatic number of the square of $G$ satisfies $\chi(G^2) \le (2-\varepsilon) \omega(G)^2$, where $\omega(G)$ is the clique number of $G$? Erd\H{o}s and Ne\v{s}et\v{r}il asked this question for the specific case of $G$ the line graph of a simple graph and this was answered in the affirmative by Molloy and Reed. We show that the answer to the more general question is also yes, and moreover that it essentially reduces to the original question of Erd\H{o}s and Ne\v{s}et\v{r}il.