Domatic Number in Cartesian Graph.

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.