Acyclic Edge Coloring of Graphs with Maximum Degree 4

TL;DR

This paper proves the conjecture that the acyclic chromatic index of graphs with maximum degree 4 is at most 6, for connected graphs with certain edge constraints, confirming the bound of Δ+2.

Contribution

The paper establishes the acyclic edge coloring conjecture for connected graphs with maximum degree 4 under specific edge conditions, advancing understanding of graph coloring bounds.

Findings

Proves the conjecture for graphs with Δ ≤ 4 and m ≤ 2n-1.

Shows that for such graphs, a'(G) ≤ 7.

Confirms the bound a'(G) ≤ Δ+2 for these graphs.

Abstract

An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycle s. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic e dge coloring using k colors and is denoted by $a'(G)$. It was conjectured by Alon, Sudakov and Zaks that for any simple and finite graph $G$, $a'(G)\le \Delta+2$, where $\Delta =\Delta(G)$ denotes the maximum degree of $G$. We prove the conjecture for connected graphs with $\Delta(G) \le 4$, with the additional restriction that $m \le 2n-1$, where $n$ is the number of vertices and $m$ is the number of edges in $G $. Note that for any graph $G$, $m \le 2n$, when $\Delta(G) \le 4$. It follows that for any graph $G$ if $\Delta(G) \le 4$, then $a'(G) \le 7$.