The acyclic chromatic number of a graph is the least number of colors needed to properly color its vertices so that none of its cycles has only two colors. The acyclic chromatic index is the analogous graph parameter for edge colorings. We first show that the acyclic chromatic index is at most $2\Delta-1$, where $\Delta$ is the maximum degree of the graph. We then show that for all $\epsilon >0$ and for $\Delta$ large enough (depending on $\epsilon$), the acyclic chromatic number of the graph is at most $\lceil(2^{-1/3} +\epsilon) {\Delta}^{4/3} \rceil +\Delta+ 1$. Both results improve long chains of previous successive advances. Both are algorithmic, in the sense that the colorings are generated by randomized algorithms. However, in contrast with extant approaches, where the randomized algorithms assume the availability of enough colors to guarantee properness deterministically, and use additional colors for randomization in dealing with the bichromatic cycles, our algorithms may initially generate colorings that are not necessarily proper; they only aim at avoiding cycles where all pairs of edges, or vertices, that are one edge, or vertex, apart in a traversal of the cycle are homochromatic (of the same color). When this goal is reached, they check for properness and if necessary they repeat until properness is attained.

翻译：图的无圈色数是指对其顶点进行正常染色所需的最少颜色数，使得图中没有环仅包含两种颜色。无圈色指数是边染色的相应图参数。我们首先证明无圈色指数至多为$2\Delta-1$，其中$\Delta$是图的最大度。然后证明，对于所有$\epsilon >0$且当$\Delta$足够大（依赖于$\epsilon$）时，图的无圈色数至多为$\lceil(2^{-1/3} +\epsilon) {\Delta}^{4/3} \rceil +\Delta+ 1$。这两个结果改进了先前一系列长期积累的进展。两者都是算法性的，即染色由随机算法生成。然而，与现有方法不同——现有随机算法假设有足够多的颜色来保证确定性正常性，并利用额外颜色随机化处理双色环——我们的算法初始生成的染色未必正常；它们仅旨在避免环中所有相隔一条边或一个顶点的边对或顶点对同色（相同颜色）。当达到此目标时，算法检查正常性，并在必要时重复直至实现正常性。