On chromatic transversal domination in graphs
Authors :
S. K. Vaidya 1 * and A. D. Parmar 2
Author Address :
1 Department of Mathematics, Saurashtra University, Rajkot - 360 005, Gujarat, India
2 Atmiya Institute of Technology and Science for Diploma Studies, Rajkot - 360 005, Gujarat, India
*Corresponding author.
Abstract :
A proper $k$ - coloring of a graph $G$ is a function $f: V(G) o {1,2,...,k}$ such that $f(u) eq f(v)$ for all $uv in E(G)$. 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 $mathscr{C} = {S_1, S_2, ..., S_k }$ is a $k$ - coloring of a graph $G$ then a subset $D$ of $V(G)$ is called a transversal of $mathscr{C}$ if $D cap S_i eq phi$ for all $i in {1,2,...,k}$. 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.
Keywords :
Coloring, Domination, Chromatic Transversal Dominating Set..
DOI :
Article Info :
Received : February 11, 2019; Accepted : May 27, 2019.
