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# Colouring squares of claw-free graphs

Published:2017-12-13

• Rémi de Joannis de Verclos,
Université Grenoble Alpes, CNRS, Grenoble INP, Laboratoire G-SCOP, 46 avenue Félix Viallet, 38031 Grenoble, France
• Ross J. Kang,
Department of Mathematics, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, Netherlands
• Lucas Pastor,
Université Grenoble Alpes, CNRS, Grenoble INP, Laboratoire G-SCOP, 46 avenue Félix Viallet, 38031 Grenoble, France
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## 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 $G$? 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
 MSC Classifications: 05C15 - Coloring of graphs and hypergraphs 05C35 - Extremal problems [See also 90C35] 05C70 - Factorization, matching, partitioning, covering and packing

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