# Acyclic edge-coloring of planar graphs: $\Delta$ colors suffice when $\Delta$ is large

## TL;DR

This paper proves that for sufficiently large maximum degree, planar graphs can be acyclically edge-colored with a number of colors equal to their maximum degree, confirming a longstanding conjecture.

## Contribution

The paper establishes that large maximum degree planar graphs have an acyclic chromatic index equal to their maximum degree, confirming a conjecture by Cohen, Havet, and Müller.

## Key findings

- Acyclic chromatic index equals maximum degree for large Δ in planar graphs.
- Confirmed the conjecture that large Δ ensures optimal acyclic edge-coloring.
- Provides a theoretical proof for a longstanding open problem.

## Abstract

An \emph{acyclic edge-coloring} of a graph $G$ is a proper edge-coloring of $G$ such that the subgraph induced by any two color classes is acyclic. The \emph{acyclic chromatic index}, $\chi'_a(G)$, is the smallest number of colors allowing an acyclic edge-coloring of $G$. Clearly $\chi'_a(G)\ge \Delta(G)$ for every graph $G$. Cohen, Havet, and M\"{u}ller conjectured that there exists a constant $M$ such that every planar graph with $\Delta(G)\ge M$ has $\chi'_a(G)=\Delta(G)$. We prove this conjecture.