TY - JOUR

T1 - Regular graphs whose subgraphs tend to be acyclic

AU - Alon, Noga

AU - Bachmat, Eitan

PY - 2006/10/1

Y1 - 2006/10/1

N2 - Motivated by a problem that arises in the study of mirrored storage systems, we describe, for any fixed ε, δ > 0, and any integer d ≥ 2, explicit or randomized constructions of d-regular graphs on n > n0(ε,δ) vertices in which a random subgraph obtained by retaining each edge, randomly and independently, with probability ρ = 1-ε/d-1, is acyclic with probability at least 1 - δ. On the other hand we show that for any d-regular graph G on n > n1(ε, δ) vertices, a random subgraph of G obtained by retaining each edge, randomly and independently, with probability ρ = 1+ε/d-1, does contain a cycle with probability at least 1 - δ. The proofs combine probabilistic and combinatorial arguments, with number theoretic techniques.

AB - Motivated by a problem that arises in the study of mirrored storage systems, we describe, for any fixed ε, δ > 0, and any integer d ≥ 2, explicit or randomized constructions of d-regular graphs on n > n0(ε,δ) vertices in which a random subgraph obtained by retaining each edge, randomly and independently, with probability ρ = 1-ε/d-1, is acyclic with probability at least 1 - δ. On the other hand we show that for any d-regular graph G on n > n1(ε, δ) vertices, a random subgraph of G obtained by retaining each edge, randomly and independently, with probability ρ = 1+ε/d-1, does contain a cycle with probability at least 1 - δ. The proofs combine probabilistic and combinatorial arguments, with number theoretic techniques.

UR - http://www.scopus.com/inward/record.url?scp=33749362325&partnerID=8YFLogxK

U2 - 10.1002/rsa.20107

DO - 10.1002/rsa.20107

M3 - Article

AN - SCOPUS:33749362325

VL - 29

SP - 324

EP - 337

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 3

ER -