Bounds on Domination Number of Complete Grid Graphs

In a paper of Cockayne et al., the authors establish an upper and a lower bound for the dominating number of the complete grid graph \(G_{n,n}\), of order \(n^2\). Namely, they proved a “formula”, and cited two questions of Paul Erdős. One of these questions was “Can we improve the order of the difference between lower and upper bounds from \(\frac{n}{5}\) to \(\frac{n}{2}\)?”. Our aim here is to give a positive answer to this question.