[Sasireka et al., 3(4): April, 2014] ISSN: 2277-9655 Impact Factor: 1.852 http: // www.ijesrt.com (C) International Journal of Engineering Sciences & Research Technology [4050-4053] IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY Domatic Number in Cartesian Graph A. Sasireka *1 , P. Vijayalakshmi 2 , J. Femila Mercy Rani 3 *1,2,3 Assistant Professor, Department of Mathematics, PSNA College of Engineering and Technology, Kothandaraman nagar, Dindigul, Tamil Nadu, India sasireka.anand@gmail.com Abstract A domatic partition of a graph GH = (V, E) is a partition of V into disjoint sets V1,V2, …Vk such that each Vj is a dominating set for GH. The maximum number of dominating sets, which the vertex set of a Cartesian graph GH can be partitioned is called the domatic number of a graph GH. It is denoted by dom(GH) or d(GH). In this paper, we discuss the sharp bounds for dom(GH) and all Cartesian graphs attaining these bounds are characterized. We also describe the Cartesian product on complete graph G and H of order m and n and derive some properties and bounds on it. Keywords: Dominating set, Domatic set, Domatic number, Cartesian graph, R-graph. I. Introduction A graph in this paper shall mean a simple finite, connected and undirected graph without isolated vertices. For a graph G=(V,E), V denotes its vertex set while E denotes its edge set, unless otherwise stated, the graph G=(p,q) has p vertices and q edges. Degree of a vertex v is denoted by deg(v). The complete graph on m vertices is denoted by Km. The complement G of G is the graph with vertex set V in which two vertices are adjacent if and only if they are adjacent in G. If S is a subset of V, then S denotes the vertex induced subgraph of G induced by S. The cardinality of a set S denoted by |S|, is the number of elements that S possesses. For all terminologies and notations in Graph theory we follow [1] and in particular all terminologies regarding trees we follow [2]. The concept of domination was first introduced by Ore[7] and C.Berge[6]. Definition 1.1 Let G = (V, X) be a graph. A subset S  V(G) is said to be a dominating set [5] [11], if every vertex in V \ S is adjacent to atleast one vertex in S. The minimum cardinality taken over all minimal dominating sets is called the domination number of G and is denoted by (G). Many authors have introduced different types of domination parameters by imposing conditions on the dominating set and on the domatic number [9], [10] Definition 1.2 The graphs G & H are complete graphs with the order of m and n, size of eG and eH respectively. The Cartesian product G×H of graphs G and H is a graph such that  The vertex set of G×H is the Cartesian product V(G)×V(H) and  Any two Vertices (u,u1) and (v,v1) are adjacent in G×H if and only if either u = v and u1 is adjacent with v1 or u1 = v1 and u is adjacent with v. Every Cartesian graph is connected. Many variants of the domination number has been studied. Definition 1.3 A graph G is said to be R-graph if V(G) = V(Kmn) and the adjacency of vertices should satisfy the following conditions, i) The vertices in the same group should not be adjacent to each other.