Abstract
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 indexχs′(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. (2013) and show that the strong chromatic index of a k-degenerate graph G is at most (4k−2)⋅Δ(G)−2k2+1. As a direct consequence, the strong chromatic index of a 2-degenerate graph G is at most 6Δ(G)−7, which improves the upper bound 10Δ(G)−10 by Chang and Narayanan (2013). 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)≤4Δ(G)−3; as a corollary, every minimally 2-connected graph G has strong chromatic index at most 4Δ(G)−3. Moreover, all the results in this note are best possible in some sense.
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