TY - JOUR

T1 - Economical elimination of cycles in the torus

AU - Alon, Noga

PY - 2009/9

Y1 - 2009/9

N2 - 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 füraction of at most O (log d/m)$ 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.

AB - 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 füraction of at most O (log d/m)$ 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.

UR - http://www.scopus.com/inward/record.url?scp=74149087127&partnerID=8YFLogxK

U2 - 10.1017/S0963548309009997

DO - 10.1017/S0963548309009997

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:74149087127

SN - 0963-5483

VL - 18

SP - 619

EP - 627

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

IS - 5

ER -