Colouring squares of claw-free graphs

Rémi de Joannis de Verclos, Ross J. Kang, Lucas Pastor

Is there some absolute $\epsilon >0$ such that for any claw-free graph $G$, the chromatic number of the square of $G$ satisfies $\chi (G^2)\le (2-\epsilon )\omega (G{)}^2$, where $ω(G)…