Planar graphs are acyclically edge $(\Delta + 5)$-colorable

TL;DR

This paper proves that all planar graphs can be acyclically edge colored with at most 1+5 colors, advancing understanding of the acyclic edge coloring conjecture for this class.

Contribution

It establishes a new upper bound of 1+5 for acyclic edge coloring of planar graphs, using discharging and induction techniques.

Findings

Every non-trivial planar graph has one of eight local structures.

Planar graphs are acyclically edge (1+5)-colorable.

Proof combines discharging method with inductive coloring.

Abstract

An edge coloring of a graph $G$ is to color all the edges in the graph such that adjacent edges receive different colors. It is acyclic if each cycle in the graph receives at least three colors. Fiam{\v{c}}ik (1978) and Alon, Sudakov and Zaks (2001) conjectured that every simple graph with maximum degree $\Delta$ is acyclically edge $(\Delta + 2)$-colorable -- the well-known acyclic edge coloring conjecture (AECC). Despite many major breakthroughs and minor improvements, the conjecture remains open even for planar graphs. In this paper, we prove that planar graphs are acyclically edge $(\Delta + 5)$-colorable. Our proof has two main steps: Using discharging methods, we first show that every non-trivial planar graph must have one of the eight groups of well characterized local structures; and then acyclically edge color the graph using no more than $\Delta + 5$ colors by an induction on the number of edges.