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 (ℤmd)1, where two vertices in ℤm are connected if their l1 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 (ℤmd)∞, 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 md−(m−1)d, which is precisely tight.
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