Signed total \(k\)-domatic numbers of digraphs (Q2853238)

Let \(D\) be a finite and simple digraph with vertex set \(V(D)\), and let \(f:V(D)\to\{-1,1\}\) be a two-valued function. This function \(f\) is a signed total \(k\)-dominating function on \(D\) if \(\sum_{x\in N^-(v)}f(x)\geq k\) for each \(v\in V(D)\), where the integer \(k\geq1\) and \(N^-(v)\) consists of all vertices of \(D\) from which arcs go into \(v\).NEWLINENEWLINE A set \(\{f_1,f_2,\dots,f_d\}\) of distinct signed total \(k\)-dominating functions of \(D\) with the property that \(\sum_{i=1}^df_i(v)\leq1\), for each \(v\in V(D)\), is called a signed total \(k\)-dominating family of functions of \(D\). The maximum number of functions in a signed total \(k\)-dominating family of \(D\), denoted by \(d^t_{kS}(D)\), is the signed total \(k\)-domatic number of \(D\). The authors initiate the study of the signed total \(k\)-domatic numbers of digraphs and present some sharp upper bounds for this parameter.