The Acyclic Chromatic Index is Less than the Double of the Max Degree

The acyclic chromatic index of a (simple) graph $G$ is the least number of colors needed to properly color its edges (no adjacent edges have the same color) such that none of its cycles is bichromatic. In this work, we show that $2Delta-1$ colors are sufficient to produce such a coloring. In contrast with most extant algorithmic work, where the algorithms deal with properness deterministically and use randomness only to deal with the bichromatic cycles, our randomized, Moser-like algorithm produces random colorings, in a structured way, until it reaches one that is proper and acyclic.