List star edge coloring of sparse graphs

Abstract

A star-edge coloring of a graph G is a proper edge coloring such that every 2-colored connected subgraph of G is a path of length at most 3. For a graph G, let the list star chromatic index of G, chs′(G), be the minimum k such that for any k-uniform list assignment L for the set of edges, G has a star-edge coloring from L. Dvořák et al. (2013) asked whether the list star chromatic index of every subcubic graph is at most 7. In Kerdjoudj et al. (2017) we proved that it is at most 8. In this paper we give a partial answer to the question of Dvořák et al. (2013) by proving that if the maximum average degree of a subcubic graph G is less than 3011 then chs′(G)≤7.We consider also graphs with any maximum degree, we proved that if the maximum average degree of a graph G is less than 73 (resp., 52,83), then chs′(G)≤2Δ(G)−1 (resp., chs′(G)≤2Δ(G), chs′(G)≤2Δ(G)+1).