Abstract
Chudnovsky and Seymour proved that every connected claw-free graph that contains a stable set of size 3 has chromatic number at most twice its clique num- ber. We improve this for small clique size, showing that every claw-free graph with clique number at most 3 is 4-choosable and every claw-free graph with clique number at most 4 is 7-choosable. These bounds are tight.
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