TITLE: Perimeter/Neighborhood of a graph on grid
QUESTION [5 upvotes]: Hello,
I have a $\sqrt{n}\times\sqrt{n}$ lattice graph $G=(V,E)$ i.e. vertices on said 2-dim integer lattice, and two vertices have an edge if their $L_1$ distance is one.
Now I want to claim something like this:
For any partition of $V$ into $V_1, V_2$ with $n/4\leq |V_1| \leq n/2$, there are at least $\sqrt{n}$ edges
of type $v~w$ where $v\in V_1$ and $w\in V_2$.
It seems quite easy, and I was wondering if there is a simple crisp known proof for it. It seems some geometry might help e.g. to claim that perimeter of subset $S$ of area $n/4$ is minimized when $S$ is just one rectangle etc.
Any reference for extensions to higher dimension would be helpful too. Thank you!

REPLY [2 votes]: The $n$-dimensional case is solved in Bollobás and Leader, Edge-isoperimetric inequalities in the grid, Combinatorica 11 (1991), no. 4, 299–314.  In two dimensions the optimal sets are small squares, then "half-space" rectangles, then complements of small squares.  The range you specify is exactly the range where rectangles are best, and your guess is correct: there are almost exactly $\sqrt n$ edges in the smallest edge boundary.
Ryan O'Donnell's suggestion to consider the vertex isoperimetry problem will give the answer up to a constant factor (because the degrees are bounded), but curiously the extremal sets are very different: always sets roughly of the form $x+y \leq r$.