The Italian domatic number of a digraph

An Italian dominating function on a digraph D with vertex set V (D) is defined as a function f : V (D)→{0; 1; 2} such that every vertex vƐ 2 V (D) with f(v) = 0 has at least two in-neighbors assigned 1 under f or one in-neighbor w with f(w) = 2. A set {f1; f2; : : : ; fd} of distinct Italian dominating functions on D with the property that ∑^d_i=1 fi(v) ≤ 2 for each v Ɛ V (D), is called an Italian dominating family (of functions) on D. The maximum number of functions in an Italian dominating family on D is the Italian domatic number of D, denoted by d_I (D). In this paper we initiate the study of the Italian domatic number in digraphs, and we present some sharp bounds for d_I (D). In addition, we determine the Italian domatic number of some digraphs.