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∑ d i= 1 fi (v)≤ 1 for each v∈ V (D), is called a signed k-dominating family (of functions) of D. The maximum number of functions in a signed k-dominating family of D is the 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.
