The total {k}‐domatic number of a graph

For a positive integer k, a total {k}‐dominating function of a graph G 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. A set {f 1,f 2,…,f d } of total {k}‐dominating functions on G with the property that ${\sum }_{i=1}^d{f}_i(v)\le k$ 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 initiate the study of the total {k}‐domatic number in graphs and we present some bounds for $d_t^{\{k\}}(G)$ . Many of the known bounds of d t (G) are immediate consequences of our results.