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Colouring Squares of Claw-free Graphs

  • Rémi de Joannis de Verclos (a1), Ross J. Kang (a2) and Lucas Pastor (a1)

Is there some absolute $\unicode[STIX]{x1D700}>0$ such that for any claw-free graph $G$ , the chromatic number of the square of $G$ satisfies $\unicode[STIX]{x1D712}(G^{2})\leqslant (2-\unicode[STIX]{x1D700})\unicode[STIX]{x1D714}(G)^{2}$ , where $\unicode[STIX]{x1D714}(G)$ is the clique number of $G$ ? Erdős and Nešetřil asked this question for the specific case where $G$ is 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.

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This research was supported by a Van Gogh grant, reference 35513NM and by ANR project STINT, reference ANR-13-BS02-0007. Author R. J. K. is currently supported by a NWO Vidi Grant, reference 639.032.614.

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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
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