 
Summary: SIAM J. DISCRETE MATH. c 2010 Society for Industrial and Applied Mathematics
Vol. 24, No. 3, pp. 892894
THE BRUNNMINKOWSKI INEQUALITY AND NONTRIVIAL
CYCLES IN THE DISCRETE TORUS
NOGA ALON AND OHAD N. FELDHEIM
Abstract. Let (Cd
m) denote the graph whose set of vertices is Zd
m in which two distinct
vertices are adjacent iff in each coordinate either they are equal or they differ, modulo m, by at
most 1. Bollob´as, Kindler, Leader, and O'Donnell proved that the minimum possible cardinality of a
set of vertices of (Cd
m) whose deletion destroys all topologically nontrivial cycles is md  (m  1)d.
We present a short proof of this result, using the BrunnMinkowski inequality, and also show that
the bound can be achieved only by selecting a value xi in each coordinate i, 1 i d, and by
keeping only the vertices whose ith coordinate is not xi for all i.
Key words. BrunnMinkowski inequality, discrete torus, nontrivial cycles
AMS subject classifications. 05C35, 05C38, 05C40
DOI. 10.1137/100789671
1. Introduction. Let (Cd
m) denote the graph whose set of vertices is Zd
