The graphical balls-into-bins process is a generalization of the classical 2-choice balls-into-bins process, where the bins correspond to vertices of an arbitrary underlying graph G. At each time step an edge of G is chosen uniformly at random, and a ball must be assigned to either of the two endpoints of this edge. The standard 2-choice process corresponds to the case of G=Kn.

Let $(C_m^d)_{\infty}$ denote the graph whose set of vertices is $Z_m^d$ in which two distinct vertices are adjacent iff in each coordinate either they are equal or they differ, modulo m, by at most 1. Bollobás, Kindler, Leader, and O'Donnell proved that the minimum possible cardinality of a set of vertices of $(C_m^d)_{\infty}$ whose deletion destroys all topologically nontrivial cycles is $m^d-(m-1)^d$. We present a short proof of this result, using the Brunn–Minkowski inequality, and also show that the bound can be achieved only by selecting a value $x_i$ in each coordinate i, $1\leq i\leq d$, and by keeping only the vertices whose ith coordinate is not $x_i$ for all i.