Signed total Italian k-domatic number of a graph

Document Type : Original paper

Author

RWTH Aachen University

Abstract

Let $k\ge 1$ be an integer, and let $G$ be a finite and simple graph with vertex set $V(G)$. A  signed total Italian $k$-dominating function on a graph $G$ is a function $f:V(G)\longrightarrow \{-1, 1, 2\}$ such that $\sum_{u\in N(v)}f(u)\ge k$ for every $v\in V(G)$, where $N(v)$ is the neighborhood of $v$, and each vertex $u$ with $f(u)=-1$ is adjacent to a vertex $v$ with $f(v)=2$ or to two vertices $w$ and $z$ with $f(w)=f(z)=1$. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct signed total Italian $k$-dominating functions on $G$ with the property that $\sum_{i=1}^df_i(v)\le k$ for each $v\in V(G)$, is called a signed total Italian $k$-dominating family (of functions) on $G$. The maximum number of functions in a signed total Italian $k$-dominating family on $G$ is the  signed total Italian k-domatic number of $G$, denoted by $d_{stI}^k(G)$. In this paper we initiate the study of signed total Italian k-domatic numbers in graphs, and we present sharp bounds for $d_{stI}^k(G)$. In addition, we determine the signed total Italian k-domatic number of some graphs.

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