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Economical Elimination of Cycles in the Torus

Published online by Cambridge University Press:  01 September 2009

NOGA ALON
Affiliation:
Sackler School of Mathematics and Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, 69978, Israel and IAS, Princeton, NJ 08540, USA (e-mail: nogaa@tau.ac.il)
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Abstract

Let m > 2 be an integer, let C2m denote the cycle of length 2m on the set of vertices [−m, m) = {−m, −m + 1, . . ., m − 2, m − 1}, and let G = G(m, d) denote the graph on the set of vertices [−m, m)d, in which two vertices are adjacent if and only if they are adjacent in C2m in one coordinate, and equal in all others. This graph can be viewed as the graph of the d-dimensional torus. We prove that one can delete a fraction of at most of the vertices of G so that no topologically non-trivial cycles remain. This is tight up to the logd factor and improves earlier estimates by various researchers.

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Copyright © Cambridge University Press 2009

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References

[1]Alon, N. (1986) Eigenvalues and expanders. Combinatorica 6 8396.CrossRefGoogle Scholar
[2]Alon, N. and Klartag, B. Economical toric spines via Cheeger's inequality. To appear.Google Scholar
[3]Bobkov, S., Houdré, C. and Tetali, P. (2000) λ, vertex isoperimetry and concentration. Combinatorica 20 153172.CrossRefGoogle Scholar
[4]Bollobás, B., Kindler, G., Leader, I. and O'Donnell, R. (2006) Eliminating cycles in the discrete torus. In LATIN 2006: Theoretical Informatics, Vol. 3887 of Lecture Notes in Computer Science, Springer, Berlin, pp. 202210. Also: Algorithmica 50 (2008) 446–454.Google Scholar
[5]Kindler, G., O'Donnell, R., Rao, A. and Wigderson, A. (2008) Rounding schemes and cubical tilings with sphere-like surface area. In Proc. FOCS 2008, pp. 189–198.Google Scholar
[6]Raz, R. (2008) A counterexample to strong parallel repetition. In Proc. FOCS 2008, pp. 369–373.Google Scholar

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