Signed total (k, k)-domatic number of digraphs

Abstract

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 \({\sum_{x \in N^-(v)}f(x) \ge k}\) for each \({v \in V(G)}\), 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 \({\sum_{i=1}^df_i(x)\le k}\) for each \({x \in V(D)}\) , 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).