Acyclic Edge Coloring through the Lovász Local Lemma

Abstract

We give a probabilistic analysis of a Moser-type algorithm for the Lovász 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 $\Delta$ has an acyclic proper edge coloring with at most $\lceil 3.74(\Delta -1)\rceil +1$ colors, whereas, previously, the best bound was $4(\Delta -1)$. 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.

The proof of Lemma 5 has been corrected