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Squared Chromatic Number Without Claws or Large Cliques

  • Wouter Cames van Batenburg (a1) and Ross J. Kang (a2)

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.

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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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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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