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 χs′(G) of a graph G is the minimum number of colors in a strong edge coloring of G. Let Δ ≥ 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 Δ and girth at least 10Δ−4 has a strong edge coloring with 2Δ−1 colors. In addition, we prove that if G is a graph with girth at least 2Δ−1 and mad(G)<2+13Δ−2, where Δ is the maximum degree and Δ ≥ 4, then χs′(G)≤2Δ−1; if G is a subcubic graph with girth at least 8 and mad(G)<2+223, then χs′(G)≤5.