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 \(k \geq 1\) is an integer and \(\sum_{x\in N[v]}f(x) \geq k\) for each \(v \in V(G)\), where \(N[v]\) is the closed neighborhood of \(v$, then \(f\) is a signed \(k\)-dominating function on \(G\). A set \(\{f_1, f_2, \ldots, f_d\}\) of distinct signed \(k\)-dominating functions on \(G\) with the property that \(\sum_{i=1}{d}f_i(v) \leq j\) for each \(x \in V(G)\), is called a signed \((j, k)\)-dominating family (of functions) on \(G\), where \(j \geq 1\) 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 \(d_{jkS}(G)\).