Claw-free perfect graphs are chromatic-choosable.
We prove that every claw-free perfect graph $G$ satisfies $\ch(G)=\chi(G)=\omega(G)$. More strongly, if $S$ is a clique of simplicial vertices of $G$ and $k\ge\omega(G)$, every proper precoloring of $S$ extends to a proper coloring of $G$ from arbitrary lists of size $k$ on $V(G)\setminus S$. The proof has three coloring steps: a mixed-list theorem for cobipartite graphs, a precoloring-extension theorem for elementary graphs built on Galvin's kernel method and a homogeneous-pair exchange argument, and a polynomial method (Combinatorial Nullstellensatz) argument for graphs of interval type in…
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