An improvement on the bound for the acyclic chromatic index

TL;DR

This paper improves the upper bound on the acyclic chromatic index of graphs with maximum degree Δ by using unordered trees and analytic combinatorics, reducing the previous bound.

Contribution

It introduces a novel approach using unordered trees as witness structures in the Lovász Local Lemma to tighten the bound.

Findings

A new upper bound of 3.142(Δ-1)+1 for the acyclic chromatic index.

Use of unordered trees instead of ordered trees in the proof.

Application of analytic combinatorics to count witness structures.

Abstract

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 $\Delta$, the acyclic chromatic index is at most $3.142(\Delta-1)+1$, 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 Lov\'{a}sz Local Lemma, a key combinatorial tool often used in related works. The counting of these witness structures entails methods of Analytic Combinatorics.