Consider a discrete even torus $G=(V,E)$, i.e. the graph on $\lbrace 0,1,\dots,n-1 \rbrace^2$, $n$ even, where two vertices are connected by an edge only if they differ by 1 in only one coordinate, modulo $n$.

$G$ is a bipartite graph. Call $O$ and $E$ the two sets into which the vertex set $V$ is partitioned (consisting of the odd and even vertices, respectively).

Given $A \subset V$, denote by $\partial A$ the vertex boundary of $A$, i.e. the set of all vertices in $V\setminus A$ whose graph distance from $A$ is exactly $1$.

**Question:** is there any vertex-isoperimetric inequality (VIP) of the form
$$\min_{|A|=m, A \subseteq O} |\partial A| \geq f(m),$$
i.e. where the minimum is taken only over the subsets of odd vertices of $V$?

**Bonus question:** how much can be said in general about the same question, with $G$ being a bipartite regular graph?