Total Italian domatic number of graphs

Let $G$ be a graph with vertex set $V(G)$. An \textit{Italian dominating function} (IDF) on a graph $G$ is a function $f:V(G)\longrightarrow \{0,1,2\}$ such that every vertex $v$ with $f(v)=0$ is adjacent to a vertex $u$ with $f(u)=2$ or to two vertices $w$ and $z$ with $f(w)=f(z)=1$. An IDF $f$ is called a \textit{total Italian dominating function} if every vertex $v$ with $f(v)\ge 1$ is adjacent to a vertex $u$ with $f(u)\ge 1$. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct total Italian dominating functions on $G$ with the property that $\sum_{i=1}^df_i(v)\le 2$ for each vertex $v\in V(G)$, is called a \textit{total Italian dominating family} (of functions) on $G$. The maximum number of functions in a total Italian dominating family on $G$ is the \textit{total Italian domatic number} of $G$, denoted by $d_{tI}(G)$. In this paper, we initiate the study of the total Italian domatic number and present different sharp bounds on $d_{tI}(G)$. In addition, we determine this parameter for some classes of graphs.