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The Brunn-Minkowski Inequality and nontrivial cycles in the discrete torus
 

Summary: The Brunn-Minkowski Inequality and nontrivial cycles in the
discrete torus
Noga Alon
Ohad N. Feldheim
March 21, 2010
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 they are either equal or 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 Brunn-Minkowski Inequality, and also show that the bound
can only be achieved by selecting a value xi in each coordinate i, 1 i d, and by keeping only
the vertices whose i-th coordinate is not xi, for all i.
Keywords: Brunn-Minkowski, Discrete torus, Nontrivial cycles.
AMS classification: 05C35,05C38,05C40.

  

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

 

Collections: Mathematics