 
Summary: Economical toric spines via Cheeger's Inequality
Noga Alon
Bo'az Klartag
Abstract
Let G = (Cd
m) denote the graph whose set of vertices is {1, . . . , m}d
, where two distinct
vertices are adjacent iff they are either equal or adjacent in Cm in each coordinate. Let
G1 = (Cd
m)1 denote the graph on the same set of vertices in which two vertices are adjacent iff
they are adjacent in one coordinate in Cm and equal in all others. Both graphs can be viewed
as graphs of the ddimensional torus. We prove that one can delete O(
dmd1
) vertices of G1
so that no topologically nontrivial cycles remain. This improves an O(dlog2(3/2)
md1
) estimate
of Bollob´as, Kindler, Leader and O'Donnell. We also give a short proof of a result implicit in
a recent paper of Raz: one can delete an O(
