A General Upper Bound On The List Chromatic Number Of Locally Sparse Graphs

Suppose that G is a graph with maximum degree d(G) and for every vertex v in G, the neighborhood of v contains at most d(G)2/f (f > 1) edges. We show that the list chromatic number of G is at most Kd(G)/ log f , for some positive constant K. This result is sharp up to the multiplicative constant K and strengthens previous results by Kim [Kim], Johansson [Joh], Alon, Krivelevich and Sudakov [AKSu], and the present author [Vu1]. This also motives several interesting questions. As an application, we derive several upper bounds for the strong (list) chromatic index of a graph, under various assumptions. These bounds extend earlier results by Faudree, Gyárfás, Schelp and Zs. Tuza [FGST] and Madhian [ Mah] and determine, up to a constant factor, the strong (list) chromatic index of a random graph. Another application is an extension of a result of Kostochka and Steibitz [KS] concerning the structure of list critical graphs.