SIGNED STAR (j, k)-DOMATIC NUMBER OF A GRAPH

Let G be a simple graph without isolated vertices with edge set E(G), and let j and k be two positive integers. A function f : E(G) -> {-1, 1} is said to be a signed star j-dominating function on G if Sigma(e is an element of E(v)) f (e) >= j for every vertex v of G, where E(v) = {uv is an element of E(G) vertical bar u is an element of N(v)}. A set {f(1), f(2), ..., f(d)} of distinct signed star j-dominating functions on G with the property that Sigma(d)(i=1) f(i)(e) <= k for each e is an element of E(G), is called a signed star (j, k)-dominating family (of functions) on G. The maximum number of functions in a signed star (j, k)-dominating family on G is the signed star (j, k)-domatic number of G denoted by d(SS)((j, k)) (G). In this paper we study properties of the signed star (j, k)-domatic number of a graph G. In particular, we determine bounds on d(SS)((j, k))(G). Some of our results extend those ones given by Atapour, Sheikholeslami, Ghameslou and Volkmann [1] for the signed star domatic number, Sheikholeslami and Volkmann [5] for the signed star (k, k)-domatic number and Sheikholeslami and Volkmann [4] for the signed star k-domatic number.