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 ∑u∈N(v)f(u)≥k for every v∈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,…,fd} of distinct signed total Italian k-dominating functions on G with the property that ∑di=1fi(v)≤k for each v∈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 dkstI(G). In this paper we initiate the study of signed total Italian k-domatic numbers in graphs, and we present sharp bounds for dkstI(G). In addition, we determine the signed total Italian k-domatic number of some graphs.