Signed \(k\)-Domatic Numbers of Graphs

Abstract

Let \(k\) be a positive integer, and let \(G\) be a simple graph with vertex set \(V(G)\). A function \(f: V(G) \rightarrow \{-1, 1\}\) is called a signed \(k\)-dominating function if \(\sum_{u \in N(v)} f(u) \geq k\) for each vertex \(v \in V(G)\). A set \(\{f_1, f_2, \ldots, f_d\}\) of signed \(k\)-dominating functions on \(G\) with the property that \(\sum_{i=1}^{d} f_i(v) \leq 1\) for each \(v \in V(G)\), is called a signed \(k\)-dominating family (of functions) on \(G\). The maximum number of functions in a signed \(k\)-dominating family on \(G\) is the signed \(k\)-domatic number of \(G\), denoted by \(d_{kS}(G)\). In this paper, we initiate the study of signed \(k\)-domatic numbers in graphs and we present some sharp upper bounds for \(d_{kS}(G)\). In addition, we determine the signed \(k\)-domatic number of complete graphs.