**Authors:** Seyed Mahmoud Sheikholeslami, Lutz Volkmann

**Keywords:** Total Italian domination number, Total Italian domatic number.

### Abstract

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.

Seyed Mahmoud Sheikholeslami

ORCID: https://orcid.org/0000-0003-2298-4744

Department of Mathematics

Azarbaijan Shahid Madani University

Tabriz, I. R. Iran

E-mail:

Lutz Volkmann

ORCID: https://orcid.org/0000-0003-3496-277X

Lehrstuhl C für Mathematik

RWTH Aachen University

52062 Aachen, Germany

E-mail:

### DOI

https://doi.org/10.56415/csjm.v31.09
### Fulltext

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