Abstract
A strong edge-coloring of a graph G is a coloring of the edges of G such that each color class is an induced matching. The strong chromatic index of G is the minimum number of colors in a strong edge-coloring of G.We show that the strong chromatic index of a claw-free graph with maximum degree Δ is at most 1.125Δ2+Δ, which confirms the conjecture of Erdős and Nešetřil from 1985 for this class of graphs for Δ≥12.We also prove an upper bound of 2−1t−2Δ2 on strong chromatic index of K1,t-free graphs with maximum degree Δ for all t≥4 and give an improved result 1.625Δ2 for unit disk graphs.
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