The total {k}-domatic number of a graph

For a positive integer k, a ltotal {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 is an element of V (G), the condition Sigma(u is an element of 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 Sigma(d)(i=1) f(i)(v) = k for each v is an element of 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.