Acyclic edge coloring of sparse graphs

TL;DR

This paper establishes bounds on the acyclic chromatic index for graphs with bounded maximum average degree, showing that certain sparse graphs, including triangle-free planar graphs, can be acyclically edge-colored with a limited number of colors.

Contribution

It proves new upper bounds on the acyclic chromatic index for graphs with maximum average degree less than 4 and 3, extending understanding of acyclic edge coloring in sparse graphs.

Findings

If mad(G)<4, then χ'a(G) ≤ Δ(G)+2.

If mad(G)<3, then χ'a(G) ≤ Δ(G)+1.

Every triangle-free planar graph is acyclically edge (Δ(G)+2)-colorable.

Abstract

A proper edge coloring of a graph $G$ is called acyclic if there is no bichromatic cycle in $G$. The acyclic chromatic index of $G$, denoted by $\chi'_a(G)$, is the least number of colors $k$ such that $G$ has an acyclic edge $k$-coloring. The maximum average degree of a graph $G$, denoted by $\mad(G)$, is the maximum of the average degree of all subgraphs of $G$. In this paper, it is proved that if $\mad(G)<4$, then $\chi'_a(G)\leq{\Delta(G)+2}$; if $\mad(G)<3$, then $\chi'_a(G)\leq{\Delta(G)+1}$. This implies that every triangle-free planar graph $G$ is acyclically edge $(\Delta(G)+2)$-colorable.