List strong edge-coloring of graphs with maximum degree 4

Abstract

A strong edge-coloring of a graph G is an edge-coloring such that any two edges on a path of length three receive distinct colors. We denote the strong chromatic index by χs′(G) which is the minimum number of colors that allow a strong edge-coloring of G. Erdős and Nešetřil conjectured in 1985 that the upper bound of χs′(G) is 54Δ2 when Δ is even and 14(5Δ2−2Δ+1) when Δ is odd, where Δ is the maximum degree of G. The conjecture is proved right when Δ≤3. The best known upper bound for Δ=4 is 22 (Cranston, 2006). In this paper we extend the result of Cranston to the list version, that is, we prove that when Δ=4, list strong chromatic index is at most 22.