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 (Zd m)1, where two vertices in Zm are connected if their L1 distance is 1, we show a nontrivial upper bound of dlog2(3/2)md-1 d.6md-1 on the number of vertices that must be deleted. For (Zd m), 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)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.

• Type: Journal
• Year: 2008
• Where: ALGORITHMICA
• Authors: Béla Bollobás, Guy Kindler, Imre Leader, Ryan O'Donnell
• Keywords: ALGORITHMICA 2008, Discrete Torus, Nontrivial Cycles, Nontrivial Upper Bound