Abstract. Assign to each vertex v of the complete graph K n on n vertices a list L(v) of colors by choosing each list independently and uniformly at random from all f (n)-subsets of a color set [n] = {1, . . . , n}, where f (n) is some integer-valued function of n. Such a list assignment L is called a random (f (n), [n])-list assignment. In this paper, we determine the asymptotic probability (as n → ∞) of the existence of a proper coloring ϕ of K n , such that ϕ(v) ∈ L(v) for every vertex v of K n . We show that this property exhibits a sharp threshold at f (n) = log n. Additionally, we consider the corresponding problem for the line graph of a complete bipartite graph K m,n with parts of size m and n, respectively. We show that if m = o( √ n), f (n) ≥ 2 log n, and L is a random (f (n), [n])-list assignment for the line graph of K m,n , then with probability tending to 1, as n → ∞, there is a proper coloring of the line graph of K m,n with colors from the lists.