A Simple Proof of the Karakhanyan–Riordan Theorem on the Even Discrete Torus
Sergei L. Bezrukov, Uwe Leck · SIAM Journal on Discrete Mathematics · 2009
We present a simple proof of the result published by Karakhanyan [Doklady AN Arm. SSR, LXXIV (1982), pp. 61–65] and Riordan [SIAM J. Discrete Math., 11 (1998), pp. 110–127] concerning the vertex-isoperimetric problem on the n-dimensional torus. The proof method is a reduction of this problem to a similar problem on some $(2n)$-dimensional grid, for which a solution is known [B. Bollobás and I. Leader, J. Combin. Theory, A-56 (1991), pp. 47–62]. We also simplify and clarify the structure of the involved isoperimetric order.