Squared Chromatic Number Without Claws or Large Cliques

Abstract

AbstractLet $G$ 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.