The conflict-free chromatic index is at most (1+o(1)) log₂ Δ for any graph and this bound is tight.
Łuczak, Size and connectivity of the k-core of a random graph, Discrete Mathe- matics 91(1) (1991) 61–68
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Asymptotically Tight Bound for the Conflict-Free Chromatic Index
The conflict-free chromatic index is at most (1+o(1)) log₂ Δ for any graph and this bound is tight.