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 $({\mathbb Z}^{d}_{m})_{1}$, where two vertices in ${\mathbb Z}_{\it m}$ are connected if their ℓ₁ distance is 1, we show a nontrivial upper bound of d $^{log_{2}{(3/2)}}{\it m}^{{\it d}-1}$ ≈ d $^{0.6} {\it m}^{{\it d}-1}$ on the number of vertices that must be deleted. For $({\mathbb Z}^{d}_{m})_{\infty}$, where two vertices are connected if their ℓ_∞ distance is 1, Saks, Samorodnitsky and Zosin [8] already gave a nearly tight lower bound of d (m-1)$^{{\it d}-1}$ using arguments involving linear algebra. We give a more elementary proof which improves the bound to ${\it m}^{d}-({\it m}-1)^{d}$, which is precisely tight.