Let $G$ be a graph. An acyclic $k$-coloring of $G$ is a map $c:V(G)\rightarrow \{1,\dots,k\}$ such that $c(u)\neq c(v)$ for any $uv\in E(G)$ and the subgraph induced by the vertices of any two colors $i,j\in \{1,\dots,k\}$ is a forest. If every vertex $v$ of a color class $V_i$ misses a color $\ell_v\in\{1,\dots,k\}$ in its closed neighborhood, then every $v\in V_i$ can be recolored with $\ell_v$ and we obtain a $(k-1)$-coloring of $G$. If a new coloring $c'$ is also acyclic, then such a recoloring is an acyclic recoloring step and $c'$ is in relation $\triangleleft_a$ with $c$. The acyclic b-chromatic number $A_b(G)$ of $G$ is the maximum number of colors in an acyclic coloring where no acyclic recoloring step is possible. Equivalently, it is the maximum number of colors in a minimum element of the transitive closure of $\triangleleft_a$. In this paper, we consider $A_b(G)$ of cubic graphs.

翻译：设$G$为一个图。$G$的一个无环$k$着色是一个映射$c:V(G)\\rightarrow \\{1,\\dots,k\\}$，使得对于任意$uv\\in E(G)$有$c(u)\\neq c(v)$，且任意两种颜色$i,j\\in \\{1,\\dots,k\\}$所对应的顶点诱导子图构成一个森林。若颜色类$V_i$中的每个顶点$v$在其闭邻域中缺失某种颜色$\\ell_v\\in\\{1,\\dots,k\\}$，则每个$v\\in V_i$可被重新着色为$\\ell_v$，从而得到$G$的一个$(k-1)$着色。若新着色$c'$也是无环的，则此类重新着色称为无环重着色步骤，且$c'$与$c$满足关系$\\triangleleft_a$。$G$的无环b色数$A_b(G)$是指不存在无环重着色步骤的无环着色中的最大颜色数。等价地，它是$\\triangleleft_a$传递闭包中最小元素的最大颜色数。本文研究立方图的$A_b(G)$。