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 ${\displaystyle \gamma {ₒ}^k\left(G\right)}$ is the minimum cardinality of a global offensive k-alliance in G. We present different bounds on ${\displaystyle \gamma {ₒ}^k\left(G\right)}$ in terms of order, maximum degree, independence number, chromatic number and minimum degree.