Acyclic Edge colorings of 2-degenerate graphs

TL;DR

This paper proves that 2-degenerate graphs can be acyclic edge colored with at most one more than their maximum degree, confirming a conjecture for this class of graphs.

Contribution

It establishes that the acyclic chromatic index of 2-degenerate graphs is at most Δ+1, improving previous bounds and confirming a conjecture for this graph class.

Findings

Acyclic chromatic index of 2-degenerate graphs is at most Δ+1.

Confirms the conjecture by Alon, Sudakov, and Zaks for 2-degenerate graphs.

Provides a stronger bound than the general conjecture for this class.

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 $a'(G)$. A graph is called 2-$degenerate$ if any of its induced subgraph has a vertex of degree at most 2. The class of 2-$degenerate graphs$ properly conta in $series$-$parallel graphs$, $outerplanar graphs$, \emph{non-regular subcubic graphs}, \emph{planar graphs of girth at least 6} and \emph{circle graphs of girth at least 5} as subclasses. It was conjectur ed by Alon, Sudakov and Zaks (and earlier by Fiamcik) that $a'(G)\le \Delta+2$, where $\Delta =\Delta(G)$ denotes the maximum deg ree of the graph. We prove the conjecture for 2-$degenerate$ graphs: in fact we prove a stronger bound . We prove that if $G$ is a 2-degenerate graph with maximum degree $\Delta$, then $a'(G)\le \Delta + 1$.