Acyclic Edge Coloring through the Lov\'asz Local Lemma

We give a probabilistic analysis of a Moser-type algorithm for the Lov'{a}sz Local Lemma (LLL), adjusted to search for acyclic edge colorings of a graph. We thus improve the best known upper bound to acyclic chromatic index, also obtained by analyzing a similar algorithm, but through the entropic method (basically counting argument). Specifically we show that a graph with maximum degree $\mathrm{\Delta }$ has an acyclic proper edge coloring with at most $\lceil 3.74(\Delta-1)\rceil+1 $colors,whereas,previously,thebestboundwas$4(\Delta-1)$. The main contribution of this work is that it comprises a probabilistic analysis of a Moser-type algorithm applied to events pertaining to dependent variables.