The Brunn–Minkowski Inequality and Nontrivial Cycles in the Discrete Torus
Noga Alon, Ohad N. Feldheim · SIAM Journal on Discrete Mathematics · 2010
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.