A crown graph is a complete bipartite graph from which a perfect matching has been removed.
The bipartite dimension of a graph is the minimum number of complete bipartite subgraphs needed to cover all the edges of the graph.
The bipartite dimension of a crown graph with $2n$ vertices is
$\sigma(n)=\min\left\{k \hspace{0.2cm} |\hspace{0.2cm} n\leq\begin{pmatrix}k\\\lfloor k/2\rfloor\end{pmatrix}\right\}$.
This formula is described here.
My question is: how can we derive such a formula?
My actual problem is to compute the bipartite dimension of the following graph where the blue edges are removed from the crown graph. It means two perfect matchings have been removed from a complete bipartite graph.
Thanks for any advice!