## Abstract

Let H be a given finite (possibly empty) family of connected graphs, each containing a cycle, and let G be an arbitrary finite H-free graph with minimum degree at least k. For p ∈ [0, 1], we form a p-random subgraph G_{p} of G by independently keeping each edge of G with probability p. Extending a classical result of Ajtai, Komlós, and Szemerédi, we prove that for every positive ε, there exists a positive δ (depending only on ε) such that the following holds: If p ≥ 1+ε/k, then with probability tending to 1 as k → ∞, the random graph G_{p} contains a cycle of length at least n_{H}(δk), where n_{H}(k) > k is the minimum number of vertices in an H-free graph of average degree at least k. Thus in particular G_{p} as above typically contains a cycle of length at least linear in k.

Original language | English |
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Journal | Electronic Journal of Combinatorics |

Volume | 21 |

Issue number | 1 |

State | Published - 13 Feb 2014 |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics