On b-acyclic chromatic number of a graph

Abstract

AbstractLet 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 $$c:V(G)\rightarrow \{1,\ldots ,k\}$$ c : V ( G ) → { 1 , … , k } such that $$c(u)\ne c(v)$$ c ( u ) ≠ c ( v ) for any $$uv\in E(G)$$ u v ∈ E ( G ) and the induced subgraph on vertices of any two colors $$i,j\in \{1,\ldots ,k\}$$ i , j ∈ { 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 $$A_{\textrm{b}}(G)$$ A b ( G ) of G. We present several properties of $$A_{\textrm{b}}(G).$$ A b ( G ) . In particular, we derive $$A_{\textrm{b}}(G)$$ A b ( G ) for several known graph families, derive some bounds for $$A_{\textrm{b}}(G),$$ A b ( G ) , compare $$A_{\textrm{b}}(G)$$ A b ( G ) with some other parameters and generalize some influential tools from b-colorings to acyclic b-colorings.