On chromatic transversal domination in graphs

Abstract

A proper k - coloring of a graph G is a function f : V(G) → {1,2,..., k} such that f(u) 6= f(v) for all uv ∈ E(G). The color class Si is the subset of vertices of G that is assigned to color i. The chromatic number χ(G) is the minimum number k for which G admits proper k - coloring. A color class in a vertex coloring of a graph G is a subset of V(G) containing all the vertices of the same color. The set D ⊆ V(G) of vertices in a graph G is called dominating set if every vertex v ∈ V(G) is either an element of D or is adjacent to an element of D. If C = {S1,S2,...,Sk} is a k - coloring of a graph G then a subset D of V(G) is called a transversal of C if D∩Si 6= φ for all i ∈ {1,2,..., k}. A dominating set D of a graph G is called a chromatic transversal dominating set (cdt - set) of G if D is transversal of every chromatic partition of G. Here we prove some characterizations and also investigate chromatic transversal domination number of some graphs.