Signed Distance $k$-domatic Numbers of Graphs

Abstract

Let $k$ be a positive integer and let $G$ be a simple graph with vertex set $V(G)$. If $v$ is a vertex of $G$, then the open $k$-neighborhood of $v$, denoted by $N_{k,G}(v)$, is the set $N_{k,G}(v)=\{u\mid u\ne v\text{ and }d(u,v)\le k\}$. The closed $k$-neighborhood of $v$, denoted by $N_{k,G}[v]$, is $N_{k,G}[v]=N_{k,G}(v)\cup \{v\}$. A function $f:V(G)\to \{-1,1\}$ is called a $signeddistancek-dominatingfunction$ if $\sum _{u\in N_{k,G}(v)}f(u)\ge 1$ for each vertex $v\in V(G)$. A set $\{f_1,f_2,\dots ,f_d\}$ of signed distance $k$-dominating functions on $G$ with the property that $\sum _{i=1}^d{f}_i(v)\le 1$ for each $v\in V(G)$ is called a $signeddistancek-dominatingfamily$ (of functions) on $G$. The maximum number of functions in a signed distance $k$-dominating family on $G$ is the $signeddistancek-domaticnumber$ of $G$, denoted by $d_{k,s}(G)$. Note that $d_{1,s}(G)$ is the classical signed domatic number $d_s(D)$. In this paper, we initiate the study of signed distance $k$-domatic numbers in graphs and we present some sharp upper bounds for $d_{k,s}(G)$.