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Keywords:

  • list coloring;
  • random list

Abstract

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 inline image of size inline image. Such a list assignment is called a random inline image-list assignment. In this paper, we are interested in determining the asymptotic probability (as inline image) of the existence of a proper coloring ϕ of G, such that inline image for every vertex v of G. We show, for all fixed k and growing n, that if inline image, then the probability that G has such a proper coloring tends to 1 as inline image. A similar result for complete graphs is also obtained: if inline image and L is a random inline image-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 inline image.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 317-327, 2014