Relation Between Dominating and Total Dominating Color Transversal number of Graphs

Abstract

Total Dominating Color Transversal Set is the combination of three very well known concepts of graph theory, viz., Total Dominating Set, Transversal and Proper Coloring of vertices of a graph. It is defined as a Total Dominating Set which is also Transversal of Some $\chi$-Partition of vertices of G. Here $\chi$ is the Chromatic number of the graph G. Total Dominating Color Transversal number of a graph is the cardinality of a Total Dominating Color Transversal Set which has minimum cardinality among all such sets that the graph admits. Analogously, R. L. J. Manoharan defined Dominating Color Transversal number of a graph in [2]. In this paper we investigate some relations between these two numbers. We give sufficient examples to justify our results.