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