On the offensive alliance number for the zero divisor graph of Z_n

A nonempty subset $ D $ of vertices in a graph $ \Gamma = (V, E) $ is said is an offensive alliance, if every vertex $ v \in \partial(D) $ satisfies $ \delta_D(v) \geq \delta_{\overline{D}}(v) + 1 $; the cardinality of a minimum offensive alliance of $ \Gamma $ is called the offensive alliance number $ \alpha ^o(\Gamma) $ of $ \Gamma $. An offensive alliance $ D $ is called global, if every $ v \in V - D $ satisfies $ \delta_D(v) \geq \delta_{\overline{D}}(v) + 1 $; the cardinality of a minimum global offensive alliance of $ \Gamma $ is called the global offensive alliance number $ \gamma^o(\Gamma) $ of $ \Gamma $. For a finite commutative ring with identity $ R $, $ \Gamma(R) $ denotes the zero divisor graph of $ R $. In this paper, we compute the offensive alliance (global, independent, and independent global) numbers of $ \Gamma(\mathbb{Z}_n) $, for some cases of $ n $.