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.