On the total {k}-domination and total {k}-domatic number of graphs

For a positive integer k, a {\em 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 ?u?N(v)f(u)?k is fulfilled, where N(v) is the open neighborhood of v. The {\em weight} of a total {k}-dominating function f is the value ?(f)=?v?Vf(v). The {\em total {k}-domination number}, denoted by ?{k}t(G), is the minimum weight of a total {k}-dominating function on G. A set {f1,f2,…,fd} of total {k}-dominating functions on G with the property that ?di=1fi(v)?k for each v?V(G), is called a {\em total {k}-dominating family} (of functions) on G. The maximum number of functions in a total {k}-dominating family on G is the {\em total {k}-domatic number} of G, denoted by d{k}t(G). Note that d{1}t(G) is the classic total domatic number dt(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.