Coloring graphs of various maximum degree from random lists

Let G = G(n) be a graph on n vertices with maximum degree Δ =Δ (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 Cσ=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 Δ, 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(n1/k2Δ1/k) and Δ=O(nk−1k(k3+2k2−k+1)), then the probability that G has an L-coloring tends to 1 as n→∞. If k≥2 and Δ=Ω(n1/2), then the same conclusion holds provided that σ=ω(Δ). We also give related results for other bounds on Δ, when k is constant or a strictly increasing function of n.