On the global offensive alliance number of a graph

An offensive alliance in a graph $\mathrm{\Gamma }=(V,E)$ is a set of vertices $S\subset V$ where for every vertex $v$ in its boundary it holds that the majority of vertices in $v$’s closed neighborhood are in $S$. In the case of strong offensive alliance, strict majority is required. An alliance $S$ is called global if it affects every vertex in $V\mathrm{\backslash }S$, that is, $S$ is a dominating set of $\mathrm{\Gamma }$. The offensive alliance number $a_o(\mathrm{\Gamma })$ (respectively, strong offensive alliance number $a_{\widehat{o}}(\mathrm{\Gamma })$) is the minimum cardinality of an offensive (respectively, strong offensive) alliance in $\mathrm{\Gamma }$. The global offensive alliance number ${\gamma }_o(\mathrm{\Gamma })$ and the global strong offensive alliance number ${\gamma }_{\widehat{o}}(\mathrm{\Gamma })$ are defined similarly. Clearly, $a_o(\mathrm{\Gamma })\le {\gamma }_o(\mathrm{\Gamma })$ and $a_{\widehat{o}}(\mathrm{\Gamma })\le {\gamma }_{\widehat{o}}(\mathrm{\Gamma })$. It was shown in [Discuss. Math. Graph Theory, 24 (2004), no. 2, 263-275] that $ a_o(\Gamma)\le \frac{2n}{3}$and$ a_{\hat{o}}(\Gamma)\le \frac{5n}{6}$,where$n$ denotes the order of $\mathrm{\Gamma }$. In this paper we obtain several tight bounds on ${\gamma }_o(\mathrm{\Gamma })$ and ${\gamma }_{\widehat{o}}(\mathrm{\Gamma })$ in terms of several parameters of $\mathrm{\Gamma }$. For instance, we show that $\frac{2m+n}{3\Delta+1} \le \gamma_o(\Gamma)\le \frac{2n}{3}$and$\frac{2(m+n)}{3\Delta+2} \le\gamma_{\hat{o}}(\Gamma)\le \frac{5n}{6}$,where$m$denotesthesizeof$\Gamma$and$\Delta$ its maximum degree (the last upper bound holds true for all $\mathrm{\Gamma }$ with minimum degree greatest or equal to two).