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Squared Chromatic Number Without Claws or Large Cliques

Part of: Graph theory

Published online by Cambridge University Press:  09 January 2019

Wouter Cames van Batenburg  and
Ross J. Kang
Wouter Cames van Batenburg
Affiliation:
Computer Science Department, Faculté des Sciences, Université Libre de Bruxelles, Campus de la Plaine, CP212, B-1050 Brussels, Belgium Email: [email protected]
Ross J. Kang
Affiliation:
Department of Mathematics, Radboud University Nijmegen, Postbus 9010, 6500 GL Nijmegen, The Netherlands Email: [email protected]
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Abstract

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Let $G$ be a claw-free graph on $n$ vertices with clique number $\unicode[STIX]{x1D714}$, and consider the chromatic number $\unicode[STIX]{x1D712}(G^{2})$ of the square $G^{2}$ of $G$. Writing $\unicode[STIX]{x1D712}_{s}^{\prime }(d)$ for the supremum of $\unicode[STIX]{x1D712}(L^{2})$ over the line graphs $L$ of simple graphs of maximum degree at most $d$, we prove that $\unicode[STIX]{x1D712}(G^{2})\leqslant \unicode[STIX]{x1D712}_{s}^{\prime }(\unicode[STIX]{x1D714})$ for $\unicode[STIX]{x1D714}\in \{3,4\}$. For $\unicode[STIX]{x1D714}=3$, this implies the sharp bound $\unicode[STIX]{x1D712}(G^{2})\leqslant 10$. For $\unicode[STIX]{x1D714}=4$, this implies $\unicode[STIX]{x1D712}(G^{2})\leqslant 22$, which is within 2 of the conjectured best bound. This work is motivated by a strengthened form of a conjecture of Erdős and Nešetřil.

Keywords

graph colouringErdős–Nešetřil conjectureclaw-free graph

MSC classification

Primary: 05C15: Coloring of graphs and hypergraphs
Secondary: 05C35: Extremal problems 05C70: Factorization, matching, partitioning, covering and packing
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 1 , March 2019 , pp. 23 - 35
Copyright
© Canadian Mathematical Society 2018 

Footnotes

Author W. C. v. B. was supported by NWO grant 613.001.217. Author R. J. K. is supported by a NWO Vidi Grant, reference 639.032.614.

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