Upper Bounds on the Signed $$(k,k)$$ ( k , k ) -Domatic Number of Digraphs

Abstract

Let \(D\) be a simple digraph with vertex set \(V(D)\), and let \(f:V(D)\rightarrow \{-1,1\}\) be a two-valued function. If \(k\ge 1\) is an integer and \(\sum _{x\in N^-[v]}f(x)\ge k\) for each \(v\in V(D)\), where \(N^-[v]\) consists of \(v\) and all vertices of \(D\) from which arcs go into \(v\), then \(f\) is a signed \(k\)-dominating function on \(D\). A set \(\{f_1,f_2,\ldots ,f_d\}\) of distinct signed \(k\)-dominating functions on \(D\) with the property that \(\sum _{i=1}^df_i(x)\le k\) for each \(x\in V(D)\), is called a signed \((k,k)\)-dominating family (of functions) on \(D\). The maximum number of functions in a signed \((k,k)\)-dominating family on \(D\) is the signed \((k,k)\)-domatic number of \(D\). In this article, we mainly present upper bounds on the signed \((k,k)\)-domatic number, in particular for regular digraphs.