# Weak signed Roman k-domatic number of a graph

Document Type : Original paper

Author

RWTH Aachen University

Abstract

Let $k\ge 1$ be an integer. A { weak signed Roman $k$-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 closed neighborhood of $v$.
A set $\{f_1,f_2,\ldots,f_d\}$ of distinct weak signed Roman $k$-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
{ weak signed Roman $k$-dominating family} (of functions) on $G$. The maximum number of functions
in a  weak signed Roman $k$-dominating family on $G$ is the { weak signed Roman $k$-domatic number} of $G$,
denoted by $d_{wsR}^k(G)$. In this paper we initiate the study of the weak signed Roman $k$-domatic number
in graphs, and we present sharp bounds for $d_{wsR}^k(G)$. In addition, we determine the weak signed Roman
$k$-domatic number of some graphs.

Keywords

Main Subjects

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