Bounds on the global offensive k-alliance number in graphs

Let G = (V (G), E(G)) be a graph, and let k ≥ 1 be an integer. A set S ⊆ V (G) is called a global offensive k-alliance if |N(v) ∩ S| ≥ |N(v)−S|+k for every v ∈ V (G)−S, where N(v) is the neighborhood of v. The global offensive k-alliance number γ o (G) is the minimum cardinality of a global offensive k-alliance in G. We present different bounds on γ o (G) in terms of order, maximum degree, independence number, chromatic number and minimum degree.