An improvement on the bound for the acyclic chromatic index

The acyclic chromatic index (or acyclic edge-chromatic number) of a graph is the least number of colors needed to properly color its edges so that none of its cycles has only two colors. We show that for a graph of max degree $Δ$, the acyclic chromatic index is at most $3.142(Δ-1)+1$, improving on the (best to date) bound of Fialho et al. (2020). Our improvement is made possible by considering unordered (non-plane) trees, instead of ordered (plane) ones, as witness structures for the Lovász Local Lemma, a key combinatorial tool often used in related works. The counting of these witness structures entails methods of Analytic Combinatorics.