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 ${\displaystyle {\sum }_{u\in N^{-}\left(v\right)}f\left(u\right)\ge k}$ is fulfilled, where N¯(v) consists of all vertices of D from which arcs go into v. A set ${\displaystyle \{f₁,f₂,...,f_d\}}$ of total {k}-dominating functions of D with the property that ${\displaystyle {\sum }_{i=1}^d{f}_i\left(v\right)\le k}$ 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 ${\displaystyle d{ₜ}^{\left\{k\right\}}\left(D\right)}$. Note that ${\displaystyle d{ₜ}^{\left\{1\right\}}\left(D\right)}$ is the classic total domatic number ${\displaystyle dₜ\left(D\right)}$. In this paper we initiate the study of the total {k}-domatic number in digraphs, and we present some bounds for ${\displaystyle d{ₜ}^{\left\{k\right\}}\left(D\right)}$. 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.