Eliminating cycles in the discrete torus


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 md ) 1, where two vertices in ℤ m are connected if their ℓ 1 distance is 1, we show a nontrivial upper bound of d log2(3/2)m d-1 d 0.6m d-1 on the number of vertices that must be deleted. For (ℤ md ) ∞, 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. © 2007 Springer Science+Business Media, LLC.

Publication Title

Algorithmica (New York)