TY - JOUR

T1 - Resilient pancyclicity of random and pseudorandom graphs

AU - Krivelevich, Michael

AU - Lee, Choongbum

AU - Sudakov, Benny

PY - 2010

Y1 - 2010

N2 - A graph G on n vertices is pancyclic if it contains cycles of length t for all 3 ≤ t ≤ n. In this paper we prove that for any fixed ε > 0, the random graph G(n, p) with p(n) » n-1/2 (i.e., with p(n)/n-1/2 tending to infinity) asymptotically almost surely has the following resilience property. If H is a subgraph of G withmaximumdegree at most (1/2-ε)np, then G-H is pancyclic. In fact, we prove a more general result which says that if p » n-1+1/(l-1) for some integer l ≥ 3, then for any ε > 0, asymptotically almost surely every subgraph of G(n, p) with minimum degree greater than (1/2 + ε)np contains cycles of length t for all l ≤ t ≤ n. These results are tight in two ways. First, the condition on p essentially cannot be relaxed. Second, it is impossible to improve the constant 1/2 in the assumption for the minimum degree. We also prove corresponding results for pseudorandom graphs.

AB - A graph G on n vertices is pancyclic if it contains cycles of length t for all 3 ≤ t ≤ n. In this paper we prove that for any fixed ε > 0, the random graph G(n, p) with p(n) » n-1/2 (i.e., with p(n)/n-1/2 tending to infinity) asymptotically almost surely has the following resilience property. If H is a subgraph of G withmaximumdegree at most (1/2-ε)np, then G-H is pancyclic. In fact, we prove a more general result which says that if p » n-1+1/(l-1) for some integer l ≥ 3, then for any ε > 0, asymptotically almost surely every subgraph of G(n, p) with minimum degree greater than (1/2 + ε)np contains cycles of length t for all l ≤ t ≤ n. These results are tight in two ways. First, the condition on p essentially cannot be relaxed. Second, it is impossible to improve the constant 1/2 in the assumption for the minimum degree. We also prove corresponding results for pseudorandom graphs.

KW - Pancyclicity

KW - Random graphs

KW - Resilience

UR - http://www.scopus.com/inward/record.url?scp=77952486679&partnerID=8YFLogxK

U2 - 10.1137/090761148

DO - 10.1137/090761148

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AN - SCOPUS:77952486679

SN - 0895-4801

VL - 24

SP - 1

EP - 16

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 1

ER -