Strong edge-colorings of sparse graphs with 3Δ − 1 colors

Abstract

A strong edge-coloring of a graph G is a proper edge-coloring such that every path of length 3 uses three different colors. The strong chromatic index of G, denoted by χs′(G), is the least possible number of colors in a strong edge-coloring of G. Let G be a graph, mad(G) be the maximum average degree and Δ be the maximum degree of G. In this paper, we prove that if Δ≥6 and mad(G)<238, then χs′(G)≤3Δ−1; if Δ≥7 and mad(G)<269, then χs′(G)≤3Δ−1, which partially improves the result of Choi et al. (2018) who proved that if Δ≥7 and mad(G)<3, then χs′(G)≤3Δ.