Coloring graphs from random lists of fixed size


Correspondence to: C. J. Casselgren e-mail:


Let G = G(n) be a graph on n vertices with maximum degree bounded by some absolute constant Δ. Assign to each vertex v of G a list L(v) of colors by choosing each list uniformly at random from all k-subsets of a color set math formula of size math formula. Such a list assignment is called a random math formula-list assignment. In this paper, we are interested in determining the asymptotic probability (as math formula) of the existence of a proper coloring ϕ of G, such that math formula for every vertex v of G. We show, for all fixed k and growing n, that if math formula, then the probability that G has such a proper coloring tends to 1 as math formula. A similar result for complete graphs is also obtained: if math formula and L is a random math formula-list assignment for the complete graph Kn on n vertices, then the probability that Kn has a proper coloring with colors from the random lists tends to 1 as math formula.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 317-327, 2014