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We prove χ′ s (G) ≤ 1.93 Δ(G)2 for graphs of sufficiently large maximum degree where χ′ s (G) is the strong chromatic index of G. This improves an old bound of Molloy and Reed. As a by-product, we present a Talagrand-type inequality where we are allowed to exclude unlikely bad outcomes that would otherwise render the inequality unusable.

A Stronger Bound for the Strong Chromatic Index