Acyclic Edge Coloring through the Lov\'asz Local Lemma

TL;DR

This paper improves the upper bound on the number of colors needed for acyclic edge coloring of graphs by analyzing a Moser-type probabilistic algorithm using the Lovász Local Lemma, reducing the bound from 4(Δ-1) to approximately 3.74(Δ-1).

Contribution

It provides a novel probabilistic analysis of a Moser-type algorithm for acyclic edge coloring, achieving a tighter upper bound on the acyclic chromatic index.

Findings

Acyclic chromatic index bound improved to approximately 3.74(Δ-1).

Probabilistic analysis of a Moser-type algorithm applied to dependent events.

Enhanced understanding of algorithmic applications of the Lovász Local Lemma.

Abstract

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 $\Delta$ 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.