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

Abstract

An edge coloring of a graph G is to color all its edges such that adjacent edges receive different colors. It is acyclic if the subgraph induced by any two colors does not contain a cycle. Fiamcik (Math Slovaca 28:139-145, 1978) and Alon et al. (J Graph Theory 37:157-167, 2001) conjectured that every simple graph with maximum degree \(\Delta \) is acyclically edge \((\Delta + 2)\)-colorable — the well-known acyclic edge coloring conjecture. 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 contains a local structure in one of the eight characterized groups; we then deal with each local structure to color the edges in the graph acyclically using no more than \(\Delta + 5\) colors by an induction on the number of edges.