Brooks-type theorems for choosability with separation

We consider the following type of problems. Given a graph G = (V; E) and lists L(v) of allowed colors for its vertices v 2 V such that jL(v)j = p for all v 2 V and jL(u) \ L(v)j c for all uv 2 E, is it possible to nd a \list coloring", i.e., a color f (v) 2 L(v) for each v 2 V , so that f (u) 6 = f (v) for all uv 2 E ? We prove that every graph of maximum degree admits a list coloring for every such list assignment, provided p p 5:437c. Apart from a multiplicative constant, the result is tight, as lists of length p 0:5c may be necessary. Moreover, for G = K n (the complete graph on n vertices) and c = 1 (i.e., almost disjoint lists), the smallest value of p is shown to have asymptotics (1+o(1)) p n. For planar graphs and c = 1, lists of length 4 suuce.