The strong chromatic index of sparse graphs

A coloring of the edges of a graph G is strong if each color class is an induced matching of G. The strong chromatic index of G, denoted by χs′(G), is the least possible number of colors in a strong edge coloring of G. In this note, we prove that χs′(G)≤(4k−1)Δ(G)−k(2k+1)+1 for every k-degenerate graph G. This confirms the strong version of a conjecture stated recently by Chang and Narayanan [4]. Our approach also allows to improve the upper bound from [4] for chordless graphs. We get that χs′(G)≤4Δ−3 for any chordless graph G. Both bounds remain valid for the list version of the strong edge coloring of these graphs.