Geodesic
The total {k}-domatic number of digraphs
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 3, pp. 461-471

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For a positive integer k, a total k-dominating function of a digraph D is a function f from the vertex set V(D) to the set 0,1,2, ...,k such that for any vertex v ∈ V(D), the condition ∑_u ∈ N^ -(v)f(u) ≥ k is fulfilled, where N¯(v) consists of all vertices of D from which arcs go into v. A set f₁,f₂, ...,f_d of total k-dominating functions of D with the property that ∑_i = 1^d f_i(v) ≤ k for each v ∈ V(D), is called a total k-dominating family (of functions) on D. The maximum number of functions in a total k-dominating family on D is the total k-domatic number of D, denoted by dₜ^k(D). Note that dₜ^1(D) is the classic total domatic number dₜ(D). In this paper we initiate the study of the total k-domatic number in digraphs, and we present some bounds for dₜ^k(D). Some of our results are extensions of well-know properties of the total domatic number of digraphs and the total k-domatic number of graphs.
Keywords: digraph, total {k}-dominating function, total {k}-domination number, total {k}-domatic number 
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Sheikholeslami, Seyed; Volkmann, Lutz. The total {k}-domatic number of digraphs. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 3, pp. 461-471. http://geodesic.mathdoc.fr/item/DMGT_2012_32_3_a6/