Economical toric spines via Cheeger's inequality

Let G _∞=(C _m ^d)∞ 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 m-cycle C _m in each coordinate. Let G _∞=(C _m ^d)∞ 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 C _m and equal in all others. Both graphs can be viewed as graphs of the d-dimensional torus. We prove that one can delete O(√dm ^d-1) vertices of G ₁ so that no topologically nontrivial cycles remain. This improves an O(d log ₂ (3/2)m ^d-1) estimate of Bollobás, 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(√d/m) fraction of the edges of G _∞ so that no topologically nontrivial cycles remain in this graph. Our technique also yields a short proof of a recent result of Kindler, O'Donnell, Rao and Wigderson; there is a subset of the continuous d-dimensional torus of surface area O(√d) that intersects all nontrivial cycles. All proofs are based on the same general idea: the consideration of random shifts of a body with small boundary and no nontrivial cycles, whose existence is proved by applying the isoperimetric inequality of Cheeger or its vertex or edge discrete analogues.