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Colouring Squares of Claw-free Graphs

Part of: Graph theory

Published online by Cambridge University Press:  09 January 2019

Rémi de Joannis de Verclos ,
Ross J. Kang  and
Lucas Pastor
Rémi de Joannis de Verclos
Affiliation:
Université Grenoble Alpes, CNRS, Grenoble INP, Laboratoire G-SCOP, 46 avenue Félix Viallet, 38031 Grenoble, France Email: remi.deverclos@g-scop.grenoble-inp.frlucas.pastor@g-scop.grenoble-inp.fr
Ross J. Kang
Affiliation:
Department of Mathematics, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, Netherlands Email: ross.kang@gmail.com
Lucas Pastor
Affiliation:
Université Grenoble Alpes, CNRS, Grenoble INP, Laboratoire G-SCOP, 46 avenue Félix Viallet, 38031 Grenoble, France Email: remi.deverclos@g-scop.grenoble-inp.frlucas.pastor@g-scop.grenoble-inp.fr
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Abstract

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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.

Keywords

graph colouringErdős–Nešetřil conjectureclaw-free graphs

MSC classification

Primary: 05C15: Coloring of graphs and hypergraphs
Secondary: 05C35: Extremal problems 05C70: Factorization, matching, partitioning, covering and packing
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 1 , February 2019 , pp. 113 - 129
Copyright
© Canadian Mathematical Society 2017 

Footnotes

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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