The Italian domatic number of a digraph

Abstract

An {\em Italian dominating function} on a digraph $D$ with vertex set $V(D)$ is defined as a function $f\colon V(D)\to \{0, 1, 2\}$ such that every vertex $v\in V(D)$ 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 $\{f_1,f_2,\ldots,f_d\}$ of distinct Italian dominating functions on $D$ with the property that $\sum_{i=1}^d f_i(v)\le 2$ for each $v\in V(D)$, is called an {\em Italian dominating family} (of functions) on $D$. The maximum number of functions in an Italian dominating family on $D$ is the {\em 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.

Keywords

• Digraphs
• Italian dominating function
• Italian domination number
• Italian domatic number.