A Simple Proof of the Karakhanyan--Riordan Theorem on the Even Discrete Torus

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. Bollobas and I. Leader, J. Combin. Theory, A-56 (1991), pp. 47-62]. We also simplify and clarify the structure of the involved isoperimetric order.