Signed total Italian k-domatic number of a graph

Let k >= 1 be an integer, and let G be a finite and simple graph with vertex set V (G). A signed total Italian k-dominating function on a graph G is a function f : V (G) -> {-1, 1, 2} such that Sigma(u epsilon N(v)) f (u) >= k for every v epsilon V (G), where N (v) is the neighborhood of v, and each vertex u with f (u) = -1 is adjacent to a vertex v with f (v) = 2 or to two vertices w and z with f (w) = f (z) = 1. A set {f1, f2,..., f(d)} of distinct signed total Italian k-dominating functions on G with the property that Sigma(d)(i)=1 fi (v) <= k for each v epsilon V (G), is called a signed total Italian k-dominating family (of functions) on G. The maximum number of functions in a signed total Italian k-dominating family on G is the signed total Italian k-domatic number of G, denoted by d(stI)(k) (G). In this paper we initiate the study of signed total Italian k-domatic numbers in graphs, and we present sharp bounds for d(stI)(k) (G). In addition, we determine the signed total Italian k-domatic number of some graphs.