On strong edge-coloring of graphs with maximum degree 5

Abstract

A strong edge-coloring of a graph G is a proper edge-coloring such that any two edges with distance at most 2 receive 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. Erdős and Nešetřil conjectured that every graph G with maximum degree Δ(G) has χs′(G)≤54Δ(G)2−12Δ(G)+14 if Δ(G) is odd and χs′(G)≤54Δ(G)2 if Δ(G) is even. In this paper, we prove that if G is a graph with Δ(G)≤5 and maximum average degree less than 225, then χs′(G)≤29. Our result implies that Erdős’ conjecture holds for the case Δ(G)=5, if G has no subgraph with average degree at least 225.