New Bounds for the Acyclic Chromatic Index

TL;DR

This paper establishes new upper bounds for the acyclic chromatic index of graphs, especially those excluding certain bipartite subgraphs or with large girth, using advanced probabilistic methods.

Contribution

It introduces bounds on the acyclic chromatic index for graphs with forbidden bipartite subgraphs and large girth, extending previous results with novel probabilistic techniques.

Findings

If G contains no copy of a bipartite graph H, then a'(G) ≤ 3Δ(G) + o(Δ(G)).

For graphs with large girth, a'(G) ≤ (2+ε)Δ(G) + o(Δ(G)).

Improves bounds on acyclic edge coloring using the Lovász Local Lemma.

Abstract

An edge coloring of a graph $G$ is called an acyclic edge coloring if it is proper and every cycle in $G$ contains edges of at least three different colors. The least number of colors needed for an acyclic edge coloring of $G$ is called the acyclic chromatic index of $G$ and is denoted by $a'(G)$. Fiam\v{c}ik and independently Alon, Sudakov, and Zaks conjectured that $a'(G) \leq \Delta(G)+2$, where $\Delta(G)$ denotes the maximum degree of $G$. The best known general bound is $a'(G)\leq 4(\Delta(G)-1)$ due to Esperet and Parreau. We apply a generalization of the Lov\'{a}sz Local Lemma to show that if $G$ contains no copy of a given bipartite graph $H$, then $a'(G) \leq 3\Delta(G)+o(\Delta(G))$. Moreover, for every $\varepsilon>0$, there exists a constant $c$ such that if $g(G)\geq c$, then $a'(G)\leq(2+\varepsilon)\Delta(G)+o(\Delta(G))$, where $g(G)$ denotes the girth of $G$.