ECONOMICAL TORIC SPINES VIA CHEEGER'S INEQUALITY

Let 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 Cm in each coordinate. Let 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 d-dimensional torus. We prove that one can delete vertices of G1 so that no topologically nontrivial cycles remain. This improves an O(dlog2 (3/2)md - 1) estimate of Bollobas, 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 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 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.