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تسجيل مستخدم جديدClaw-free graphs, skeletal graphs, and a stronger conjecture on $omega$, $Delta$, and $chi$
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The second authors $omega$, $Delta$, $chi$ conjecture proposes that every graph satisties $chi leq lceil frac 12 (Delta+1+omega)rceil$. 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.
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