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

The second author’s ω, ∆, χ conjecture proposes that every graph satisties χ ≤ d 1 2 (∆ + 1 + ω)e. 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.