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

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

Collections: Mathematics