On strong edge-coloring of graphs with maximum degree 5

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 chi(s)'(G), is the least possible number of colors in a strong edge-coloring of G. Erdos and Nesetril conjectured that every graph G with maximum degree Delta(G) has chi(s)'(G <= 5/4 Delta(G)(2) - 1/2 increment (G) + 1/4 if increment (G) is odd and chi(s)'(G <= 5/4 Delta(G)(2) if Delta(G) is even. In this paper, we prove that if G is a graph with increment (G) <= 5 and maximum average degree less than 22/5, then chi(s)'(G <= 29. Our result implies that Erdos' conjecture holds for the case Delta(G) = 5, if G has no subgraph with average degree at least 22/5 .(c) 2023 Published by Elsevier B.V.