Signed star (j, k)-domatic numbers of digraphs

Abstract

Let D be a simple digraph with arc set A(D), and let j and k be two positive integers. A function f : A(D) → {-1, 1} is said to be a signed star j-dominating function (SSjDF) on D if ∑a ∈ A(v) f(a) ≥ j for every vertex v of D, where A(v) is the set of arcs with head v. A set {f1, f2, …, fd} of distinct SSjDFs on D with the property that [Formula: see text] for each a ∈ A(D), is called a signed star (j, k)-dominating (SS(j, k)D) family (of functions) on D. The maximum number of functions in a SS(j, k)D family on D is the signed star (j, k)-domatic number of D, denoted by [Formula: see text]. In this paper, we study properties of the signed star (j, k)-domatic number of a digraph D. In particular, we determine bounds on [Formula: see text]. Some of our results extend these ones given by Sheikholeslami and Volkmann [Signed star k-domination and k-domatic number of digraphs, submitted] for the signed (j, 1)-domatic number.