A new upper bound on the acyclic chromatic indices of planar graphs

TL;DR

This paper establishes a new upper bound of  on the acyclic chromatic index for planar graphs, improving previous bounds and advancing understanding of edge colorings in planar graph theory.

Contribution

It proves that the acyclic chromatic index of any planar graph is at most , significantly improving previous bounds of 2.

Findings

Proved that for planar graphs, a'(G) .

Improved the upper bound from 2 to  for acyclic chromatic index.

Advances theoretical understanding of acyclic edge coloring in planar graphs.

Abstract

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index $a'(G)$ of $G$ is the smallest integer $k$ such that $G$ has an acyclic edge coloring using $k$ colors. It was conjectured that $a'(G)\le \Delta+2$ for any simple graph $G$ with maximum degree $\Delta$. In this paper, we prove that if $G$ is a planar graph, then $a'(G)\leq\Delta +7$. This improves a result by Basavaraju et al. [{\em Acyclic edge-coloring of planar graphs}, SIAM J. Discrete Math., 25 (2011), pp. 463-478], which says that every planar graph $G$ satisfies $a'(G)\leq\Delta +12$.