Signed k-domatic numbers of digraphs

Let D be a finite and simple digraph with vertex set V(D), and let f:V(D)?{?1,1} be a two-valued function. If k?1 is an integer and ?x?N?[v]f(x)?k for each v?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 {f1,f2,…,fd} of distinct signed k-dominating functions of D with the property that ?di=1fi(v)?1 for each v?V(D), is called a {\em signed k-dominating family} (of functions) of D. The maximum number of functions in a signed k-dominating family of D is the {\em signed k-domatic number} of D, denoted by dkS(D). In this note we initiate the study of the signed k-domatic numbers of digraphs and present some sharp upper bounds for this parameter.