Odd graph and its applications to the strong edge coloring

A strong edge coloring of a graph is a proper edge coloring in which every color class is an induced matching. The strong chromatic index chi '(s)(G) of a graph G is the minimum number of colors in a strong edge coloring of G. Let Delta >= 4 be an integer. In this note, we study the odd graphs and show the existence of some special walks. By using these results and Chang's et al. (2014) ideas, we show that every planar graph with maximum degree at most Delta and girth at least 10 Delta -4 has a strong edge coloring with 2 Delta - 1 colors. In addition, we prove that if G is a graph with girth at least 2 Delta - 1 and mad (G) < 2 + 1/3 Delta-2, where Delta is the maximum degree and Delta >= 4, then chi '(s) (G) <= 2 Delta - 1; if G is a subcubic graph with girth at least 8 and mad (G) < 2 + 2 23, then chi ' s(G) <= 5. (c) 2017 Elsevier Inc. All rights reserved.