Strong chromatic index of k-degenerate graphs

A strong edge coloring of a graph G 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. In this note, we improve a result by Dębski et al. [Strong chromatic index of sparse graphs, arXiv:1301.1992v1] and show that the strong chromatic index of a k-degenerate graph G is at most (4k−2)·∆(G)− 2k2 + 1. As a direct consequence, the strong chromatic index of a 2-degenerate graph G is at most 6∆(G) − 7, which improves the upper bound 10∆(G)− 10 by Chang and Narayanan [Strong chromatic index of 2-degenerate graphs, J. Graph Theory 73 (2013) (2) 119–126]. For a special subclass of 2-degenerate graphs, we obtain a better upper bound, namely if G is a graph such that all of its 3-vertices induce a forest, then χs(G) ≤ 4∆(G) − 3; as a corollary, every minimally 2-connected graph G has strong chromatic index at most 4∆(G) − 3. Moreover, all the results in this note are best possible in some sense.