Claw-free graphs, skeletal graphs, and a stronger conjecture on $ω$, $Δ$, and $χ$

The second author's $ω$, $Δ$, $χ$ conjecture proposes that every graph satisties $χ\leq \lceil \frac 12 (Δ+1+ω)\rceil$. 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 $χ$-preserving reduction on homogeneous pairs of cliques, and thus restrict our view to so-called "skeletal" graphs.

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