On the global offensive alliance number of a graph
  An offensive alliance in a graph $\Gamma=(V,E)$ 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 $V\backslash S$, that is, $S$ is a dominating set of $\Gamma$. The offensive alliance number $a_o(\Gamma)$ (respectively, strong offensive alliance number $a_{\hat{o}}(\Gamma)$) is the minimum cardinality of an offensive (respectively, strong offensive) alliance in $\Gamma$. The global offensive alliance number $\gamma_o(\Gamma)$ and the global strong offensive alliance number $\gamma_{\hat{o}}(\Gamma)$ are defined similarly. Clearly, $a_o(\Gamma)\le \gamma_o(\Gamma)$ and $a_{\hat{o}}(\Gamma)\le \gamma_{\hat{o}}(\Gamma)$. 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 $\Gamma$. In this paper we obtain several tight bounds on $\gamma_o(\Gamma)$ and $\gamma_{\hat{o}}(\Gamma)$ in terms of several parameters of $\Gamma$. 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 $m$ denotes the size of $\Gamma$ and $\Delta$ its maximum degree (the last upper bound holds true for all $\Gamma$ with minimum degree greatest or equal to two). 
J. M. Sigarreta, J. A. Rodriguez