ECONOMICAL TORIC SPINES VIA CHEEGER'S INEQUALITY

Let [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] 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.