On chromatic transversal domination in graphs

Abstract

A proper $k$ - coloring of a graph $G$ is a function $f: V(G) \rightarrow\{1,2, \ldots, k\}$ such that $f(u) \neq f(v)$ for all $u v \in E(G)$. The color class $S_i$ is the subset of vertices of $G$ that is assigned to color $i$. The chromatic number $\chi(G)$ 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 $D \subseteq V(G)$ of vertices in a graph $G$ is called dominating set if every vertex $v \in V(G)$ is either an element of $D$ or is adjacent to an element of $D$. If $\mathscr{C}=\left\{S_1, S_2, \ldots, S_k\right\}$ is a $k$ - coloring of a graph $G$ then a subset $D$ of $V(G)$ is called a transversal of $\mathscr{C}$ if $D \cap S_i \neq \phi$ for all $i \in\{1,2, \ldots, k\}$. 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.