Consider a balanced bipartite graph $G=(U,V,E)$, i.e., a bipartite graph with $|U|=|V|$. An independent set $I$ of $G$ is *balanced* if $|I \cap U| = |I \cap V|$.

The *bipartite independence number* of $G$, usually denote in literature by $i_{BIP}(G)$ or $\alpha_{BIP}(G)$, is the maximum size of a balanced independent set of $G$.

The bipartite independence number is known for the $n$-dimensional hypercube $Q_n$ (see Ramras 2010 and Barber 2012).

Q1) Is there any other graph for which it is known exactly? Is there any non-trivial *upper* bound on $i_{BIP}(G)$? If we add the hypothesis that $G$ is $k$-regular, do these two answers change?

Q2) In particular, is the bipartite independence number known for the $2m \times 2n$ square grid or for the $2m \times 2n$ toric square grid?