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!