Asymptotics of the list‐chromatic index for multigraphs

Abstract

The list-chromatic index, χl′(G) of a multigraph G is the least t such that if S(A) is a set of size t for each A∈E≔E(G), then there exists a proper coloring σ of G with σ(A)∈S(A) for each A∈E. The list-chromatic index is bounded below by the ordinary chromatic index, χ′(G), which in turn is at least the fractional chromatic index, χ′*(G). In previous work we showed that the chromatic and fractional chromatic indices are asymptotically the same; here we extend this to the list-chromatic index: χl′(G)∼χ′*(G) as χl′(G)→∞. The proof uses sampling from “hard-core” distributions on the set of matchings of a multigraph to go from fractional to list colorings. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 117–156, 2000