The total {k}-domatic number of digraphs

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, . . . , fd} of total {k}-dominating functions of D with the property that ∑ d i=1 fi(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 d {k} t (D). Note that d {1} t (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 d {k} t (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.