Let G = G(n) be a graph on n vertices with girth at least g and maximum degree bounded by some absolute constant Delta. Assign to each vertex v of G a list L(v) of colors by choosing each list independently and uniformly at random from all 2-subsets of a color set e of size sigma (n). In this paper we determine, for each fixed g and growing n, the asymptotic probability of the existence of a proper coloring phi such that phi(v) is an element of L(v) for all v is an element of V(G). In particular, we show that if g is odd and sigma (n) = omega(n(1/(2g-2))), then the probability that G has a proper coloring from such a random list assignment tends to 1 as n --> infinity. Furthermore, we show that this is best possible in the sense that for each fixed odd g and each n >= g, there is a graph H = H(n, g) with bounded maximum degree and girth g, such that if sigma (n) = 0(n(1/(2g-2))), then the probability that H has a proper coloring from such a random list assignment tends to 0 as n --> infinity. A corresponding result for graphs with bounded maximum degree and even girth is also given. Finally, by contrast, we show that for a complete graph on n vertices, the property of being colorable from random lists of size 2, where the lists are chosen uniformly at random from a color set of size sigma (n), exhibits a sharp threshold at sigma (n) = 2n. (C) 2011 Elsevier Ltd. All rights reserved.
