The {\em 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 {\em 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 $\lceil(4^{-1/3} +\epsilon) {\Delta}^{4/3} \rceil +\Delta+ 1$. Both results improve long chains of previous successive advances. Both are algorithmic, in the sense that the colorings are generated by randomized algorithms. Previous randomized algorithms assume the availability of enough colors to guarantee properness deterministically and use additional colors in dealing with the bichromatic cycles in a randomized fashion. In contrast, our algorithm initially generates colorings that are not necessarily proper; it only aims 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, the algorithm checks for properness and if necessary it repeats until properness is attained. Thus savings in the number of colors is attained.

翻译：图的**无环色数**是指对顶点进行正常着色所需的最少颜色数，使得图中没有任何一个环仅使用两种颜色。**无环色指数**是边着色的相应图参数。我们首先证明无环色指数至多为$2\Delta-1$，其中$\Delta$为图的最大度。随后证明，对所有$\epsilon>0$且$\Delta$足够大（依赖于$\epsilon$），图的无环色数至多为$\lceil(4^{-1/3} +\epsilon) {\Delta}^{4/3} \rceil +\Delta+ 1$。这两个结果改进了先前一系列逐步推进的经典结论。两者均具有算法意义，即染色由随机算法生成。以往的随机算法假设有足够多的颜色以确保确定性正常性，并通过额外颜色以随机方式处理双色环。相比之下，我们的算法初始生成的染色未必正常；它仅旨在避免环中沿遍历方向相隔一条边（或一个顶点）的任意一对边（或顶点）为同色。达成此目标后，算法检验正常性，必要时重复直至达到正常性。从而节省了颜色数量。