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.

图G的边着色是将其所有边着色，使得相邻边接收不同的颜色。如果任意两种颜色导出的子图不包含环，则它是无环的。 Fiamcik (Math Slovaca 28:139-145, 1978) 和 Alon 等人。 (J Graph Theory 37:157-167, 2001) 推测每个具有最大度\(\Delta \) 的简单图都是非循环边\((\Delta + 2)\) -可着色 — 著名的非循环边着色猜想。尽管有许多重大突破和微小改进，即使对于平面图，这个猜想仍然是开放的。在本文中，我们证明平面图是非循环边缘\((\Delta + 5)\)可着色的。我们的证明有两个主要步骤：使用放电方法，我们首先证明每个非平凡平面图都包含八个特征组之一的局部结构；然后，我们通过对边数的归纳，使用不超过\(\Delta + 5\)种颜色来处理每个局部结构，以非循环方式对图中的边进行着色。