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) -> {1, ... , k} such that c(u) &NOTEQUexpressionL; c(v) for any uv is an element of E(G) and the induced subgraph on vertices of any two colors i, j is an element of {1, ... , 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.