The double Italian domatic number of a graph

Abstract

A double Italian dominating function on a graph \( G \) with vertex set \( V(G) \) is defined as a function \( f : V(G) \to \{0,1,2,3\} \) such that each vertex \( u \in V(G) \) with \( f(u) \in \{0,1\} \) has the property that \( \sum_{x \in N[u]} f(x) \geq 3 \), where \( N[u] \) is the closed neighborhood of \( u \). A set \( \{f_1, f_2, \dots, f_d\} \) of distinct double Italian dominating functions on \( G \) with the property that \( \sum_{i=1}^{d} f_i(v) \leq 3 \) for each \( v \in V(G) \) is called a \textit{double Italian dominating family} (of functions) on \( G \). The maximum number of functions in a double Italian dominating family on \( G \) is the double Italian domatic number of \( G \), denoted by \( dd_I(G) \). We initiate the study of the double Italian domatic number, and we present different sharp bounds on \( dd_I(G) \). In addition, we determine the double Italian domatic number of some classes of graphs.