Total k-rainbow Domatic Number

The Total k-rainbow domination number is defined by considering k types of guards or set of k colors. A location which does not have any type of guard (colour) assigned, should have all type of guards (colours) in its immediate surrounding location to protect it. For a positive integer k, a function f:V(G)?P({1,2,?,k}) is said to be a total k-rainbow dominating function if ? v ?V(G),?_(u?N[v])??f(u?)={1,2,?,k} where f(u) is nonempty subset of {1,2,?,k} and N[v] is a closed neighborhood of v. A set {f_1,f_2,?,f_d} of total k-rainbow dominating function of a graph G with the property that ?_(i=1)^d?|f_i (v)| ?k for each v?V(G), is called a total k-rainbow dominating family (functions) on G. The maximum number of functions in a total k-rainbow dominating family (TkRD family) on G is called the total k-rainbow domatic number of G, is denoted by d_trk (G). In this paper we initiate the study of total k-rainbow domatic number in graphs and we obtain d_trk (K_n )=min?{n,k},d_trk (C_n)?3 . We also proved some bounds for d_trk (G).