Claw-free graphs, skeletal graphs, and a stronger conjecture on $\omega$, $\mathrm{\Delta }$, and $\chi$

The second author’s $\omega$, $\mathrm{\Delta }$, $\chi$ conjecture proposes that every graph satisties $\chi \le \lceil \frac{1}{2}(\mathrm{\Delta }+1+\omega )\rceil$. In this paper we prove that the conjecture holds for all claw-free graphs. Our approach uses the structure theorem of Chudnovsky and Seymour. Along the way we discuss a stronger local conjecture, and prove that it holds for claw-free graphs with a three-colourable complement. To prove our results we introduce a very useful $\chi$-preserving reduction on homogeneous pairs of cliques, and thus restrict our view to so-called “skeletal” graphs.