On b-acyclic chromatic number of a graph

Let $G$ be a graph. We introduce the acyclic b-chromatic number of $G$ as an analogue to the b-chromatic number of $G$. An acyclic coloring of a graph $G$ 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 induced subgraph on vertices of any two colors $i,j\in \{1,\dots,k\}$ induces a forest. On the set of all acyclic colorings of $G$ we define a relation whose transitive closure is a strict partial order. The minimum cardinality of its minimal element is then the acyclic chromatic number $A(G)$ of $G$ and the maximum cardinality of its minimal element is the acyclic b-chromatic number $A_b(G)$ of $G$. We present several properties of $A_b(G)$. In particular, we derive $A_b(G)$ for several known graph families, derive some bounds for $A_b(G)$, compare $A_b(G)$ with some other parameters and generalize some influential tools from b-colorings to acyclic b-colorings.