 
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 {0, . . . , m  1}d
, where two
distinct vertices are adjacent if and only if they are either equal or adjacent in the mcycle Cm
in each coordinate. Let G1 = (Cd
m)1 denote the graph on the same set of vertices in which two
vertices are adjacent if and only if 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(
