Signed k-domatic numbers of graphs

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 ∑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. 2010 Mathematics Subject Classification: 05C20, 05C69, 05C45