Coloring graphs of various maximum degree from random lists

Let G = G(n) be a graph on n vertices with maximum degree ∆ = ∆(n). Assign to each vertex v of G a list L(v) of colors by choosing each list independently and uniformly at random from all k-subsets of a color set C of size σ = σ(n). Such a list assignment is called a random (k, C)-list assignment. In this paper, we are interested in determining the asymptotic probability (as n → ∞) of the existence of a proper coloring φ of G, such that φ(v) ∈ L(v) for every vertex v of G, a so-called L-coloring. We give various lower bounds on σ, in terms of n, k and ∆, which ensures that with probability tending to 1 as n → ∞ there is an L-coloring of G. In particular, we show, for all fixed k and growing n, that if σ(n) = ω(n 2 ∆) and ∆ = O ( n k−1 k(k3+2k2−k+1) ) , then the probability that G has an L-coloring tends to 1 as n → ∞. If k ≥ 2 and ∆ = Ω(n), then the same conclusion holds provided that σ = ω(∆). We also give related results for other bounds on ∆, when k is constant or a strictly increasing function of n.