Acyclic Edge coloring of Planar Graphs

TL;DR

This paper proves that for planar graphs, the acyclic chromatic index is at most the maximum degree plus 12, advancing understanding of acyclic edge coloring bounds.

Contribution

It establishes a new upper bound of  + 12 for the acyclic chromatic index of planar graphs, improving previous bounds and supporting the conjecture for specific graph classes.

Findings

Proved that acyclic chromatic index of planar graphs  + 12

Supports the conjecture that a'(G)  + 2 for all graphs

Provides new techniques for acyclic edge coloring in planar graphs

Abstract

An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by $a'(G)$. It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that $a'(G)\le \Delta+2$, where $\Delta =\Delta(G)$ denotes the maximum degree of the graph. We prove that if $G$ is a planar graph with maximum degree $\Delta$, then $a'(G)\le \Delta + 12$.