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 $\sum _{x\in N[u]}f(x)\ge 3$, 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 $\sum _{i=1}^d{f}_i(v)\le 3$ 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 $d{d}_I(G)$. We initiate the study of the double Italian domatic number, and we present different sharp bounds on $d{d}_I(G)$. In addition, we determine the double Italian domatic number of some classes of graphs.