The list chromatic number of graphs with small clique number.

Abstract We prove that every triangle-free graph with maximum degree Δ has list chromatic number at most (1 + o (1)) Δ ln ⁡ Δ. This matches the best-known upper bound for graphs of girth at least 5. We also provide a new proof that for any r ≥ 4 every K r -free graph has list-chromatic number at most 200 r Δ ln ⁡ ln ⁡ Δ ln ⁡ Δ. [ABSTRACT FROM AUTHOR]