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Asymptotics of the list-chromatic index for multigraphs. (English) Zbl 0956.05038
Author’s abstract: The list-chromatic index, $$\chi'_{l}(G)$$, of a multigraph $$G$$ is the least $$t$$ such that if $$S(A)$$ is a set of size $$t$$ for each $$A\in E(G)$$, then there exists a proper coloring $$\sigma$$ of $$G$$ with $$\sigma (A)\in S(A)$$ for each $$A\in E(G)$$. The list-chromatic index is bounded below by the ordinary chromatic index, $$\chi '(G)$$, which in turn is at least the fractional chromatic index, $$\chi ^{\prime*}(G)$$. 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: $$\chi '_{l}(G)\sim \chi ^{\prime*}(G)$$ as $$\chi '_{l}(G)\rightarrow \infty$$. The proof uses sampling from “hard-core” distributions on the set of matchings of a multigraph to go from fractional to list coloring.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 60C05 Combinatorial probability 05C80 Random graphs (graph-theoretic aspects)
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