On the Total {k}-Domination and Total {k}-Domatic Number of Graphs.

For a positive integer k, a total {k}-dominating function of a graph G without isolated vertices is a function f from the vertex set V(G) to the set {0,1,2, . . . ,k} such that for any vertex v ∊ V(G), the condition Σ_u∊N(v) f (u) ≥ k is fulfilled, where N(v) is the open neighborhood of v. The weight of a total {k}-dominating function f is the value ω(f) = Σ_v∊V f (v). The total {k}-domination number, denoted by γ_t^{k} (G), is the minimum weight of a total {k}-dominating function on G. A set {f₁, f₂, . . . , f_d} of total {k}-dominating functions on G with the property that Σ^d_i =1 fi(v) ≤ k for each v ∊ V(G), is called a total {k}-dominating family (of functions) on G. The maximum number of functions in a total {k}-dominating family on G is the total {k}-domatic number of G, denoted by d_t^{k} (G). Note that d_t^{1} (G) is the classic total domatic number d_t(G). In this paper, we present bounds for the total {k}-domination number and total {k}-domatic number. In addition, we determine the total {k}-domatic number of cylinders and we give a Nordhaus-Gaddum type result. [ABSTRACT FROM AUTHOR]