The acyclic chromatic index (or acyclic edge-chromatic number) of a graph is the least number of colors needed to properly color its edges so that none of its cycles has only two colors. We show that for a graph of max degree $Δ$, the acyclic chromatic index is at most $3.142(Δ-1)+1$, improving on the (best to date) bound of Fialho et al. (2020). Our improvement is made possible by considering unordered (non-plane) trees, instead of ordered (plane) ones, as witness structures for the Lovász Local Lemma, a key combinatorial tool often used in related works. The counting of these witness structures entails methods of Analytic Combinatorics.

翻译：图的无圈边色数（或称无圈边染色数）是指对其边进行正常染色所需的最少颜色数，使得图中任意圈均不包含仅由两种颜色构成的边。本文证明，对于最大度为$Δ$的图，其无圈边色数至多为$3.142(Δ-1)+1$，改进了Fialho等人（2020年）迄今最佳的上界结果。这一改进的关键在于，我们采用无序（非平面）树而非有序（平面）树作为Lovász局部引理的见证结构——该组合工具在相关研究中被广泛使用。对这些见证结构的计数涉及解析组合学的方法。