On b-acyclic chromatic number of a graph

Abstract

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,ldots ,k} such that c(u)ne c(v) for any uvin E(G) and the induced subgraph on vertices of any two colors i,jin {1,ldots ,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_{textrm{b}}(G) of G. We present several properties of A_{textrm{b}}(G). In particular, we derive A_{textrm{b}}(G) for several known graph families, derive some bounds for A_{textrm{b}}(G), compare A_{textrm{b}}(G) with some other parameters and generalize some influential tools from b-colorings to acyclic b-colorings.