Eliminating Cycles in the Discrete Torus

Abstract

In this paper we consider the following question: how many vertices of the discrete torus must be deleted so that no topologically nontrivial cycles remain?

We look at two different edge structures for the discrete torus. For (ℤ ^d _m )₁, where two vertices in ℤ _m are connected if their ℓ ₁ distance is 1, we show a nontrivial upper bound of $d^{\log_{2}(3/2)}m^{d-1}\approx d^{0.6}m^{d-1}$ on the number of vertices that must be deleted. For (ℤ ^d _m )_∞, where two vertices are connected if their ℓ _∞ distance is 1, Saks et al. (Combinatorica 24(3):525–530, 2004) already gave a nearly tight lower bound of d(m−1)^d−1 using arguments involving linear algebra. We give a more elementary proof which improves the bound to m ^d−(m−1)^d, which is precisely tight.