Eliminating cycles in the discrete torus

Béla Bollobás, Guy Kindler*, Imre Leader, Ryan O'Donnell

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


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 m d ) 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 d 0.6m d-1 on the number of vertices that must be deleted. For (ℤ m d ) , 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.

Original languageAmerican English
Pages (from-to)446-454
Number of pages9
Issue number4
StatePublished - Apr 2008
Externally publishedYes


  • Discrete torus
  • Foam
  • Multicut
  • Tiling


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