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.