Signed $(j,k)$-Domatic Numbers of Graphs

Abstract

Let $G$ be a finite and simple graph with vertex set $V(G)$, and let $f:V(G)\to \{-1,1\}$ be a two-valued function. If $k\ge 1$ is an integer and $\sum _{x\in N[v]}f(x)\ge k$ for each $v\in V(G)$, where $N[v]$ is the closed neighborhood of $v\$,then\text{(}f$ is a signed $k$-dominating function on $G$. A set $\{f_1,f_2,\dots ,f_d\}$ of distinct signed $k$-dominating functions on $G$ with the property that $\sum _{i=1}d{f}_i(v)\le j$ for each $x\in V(G)$, is called a signed $(j,k)$-dominating family (of functions) on $G$, where $j\ge 1$ is an integer. The maximum number of functions in a signed $(j,k)$-dominating family on $G$ is the signed $(j,k)$-domatic number on $G$, denoted by $d_{jkS}(G)$.