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de Joannis de Verclos, Rémi; Kang, Ross J.; Pastor, Lucas
Colouring squares of claw-free graphs
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 $G$? Erdős and Nešetřil 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 Erdős and Nešetřil.

Keywords:graph colouring, Erdős--Nešetřil conjecture, claw-free graphs
Categories:05C15, 05C35, 05C70

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