Signed total (k, k)-domatic number 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 Sigma(x is an element of N-(v)) f(x) =>= k for each v is an element of V (G), where N- (v) consists of all vertices of D from which arcs go into v, then f is a signed total k-dominating function on D. A set {f(1), f(2), ..., f(d)} of signed total k-dominating functions on D with the property that Sigma(d)(i = 1) f(i)(x) <= k for each x is an element of V (D), is called a signed total (k, k)-dominating family (of functions) on D. The maximum number of functions in a signed total (k, k)-dominating family on D is the signed total (k, k)-domatic number on D, denoted by d(st)(k)(D). In this paper we initiate the study of the signed total (k, k)-domatic number of digraphs, and we present different bounds on d(st)(k)(D). Some of our results are extensions of known properties of the signed total domatic number d(st)(D) = d(st)(1)(D) of digraphs D as well as the signed total domatic number d(st)(G) of graphs G, given by Henning (Ars Combin. 79:277-288, 2006).