Choosability with separation of complete multipartite graphs and hypergraphs

TL;DR

This paper investigates the list coloring properties of complete multipartite graphs and hypergraphs with separation constraints, establishing asymptotic bounds using probabilistic methods.

Contribution

It introduces new asymptotic results for the choosability parameter hi_{ll}(G,s) for specific classes of graphs and hypergraphs, extending prior work.

Findings

Asymptotic bounds for hi_{ll}(G,s) in balanced complete multipartite graphs

Asymptotic bounds for hi_{ll}(G,s) in complete k-partite k-uniform hypergraphs

Use of randomized constructions to derive these bounds

Abstract

For a hypergraph G and a positive integer s, let \chi_{\ell} (G,s) be the minimum value of l such that G is L-colorable from every list L with |L(v)|=l for each v\in V(G) and |L(u)\cap L(v)|\leq s for all u, v\in e\in E(G). This parameter was studied by Kratochv\'{i}l, Tuza and Voigt for various kinds of graphs. Using randomized constructions we find the asymptotics of \chi_{\ell} (G,s) for balanced complete multipartite graphs and for complete k-partite k-uniform hypergraphs.