On Partitioning Grids into Equal Parts

Let an edge cut partition the vertex set of an n-dimensional quadratic grid with the side length a into k subsets A1, ..., Ak with ||Ai| − |Aj || ≤ 1. We consider the problem of determining the minimal size c(n, k, a) of such a cut and present its asymptotic c(n, k, a) ∼ nan−1 n √ k as a, k →∞ and k/an → 0. The same asymptotic holds for partitioning of the n-dimensional torus. We present also some heuristics, which provide better partitioning for n = 2 and small k.