### Abstract

Let (G_{t}) be the random graph process (G_{0}) is edgeless and G_{t} is obtained by adding a uniformly distributed new edge to G_{t}-1), and let tk denote the minimum time t such that the k-core of G_{t} (its unique maximal subgraph with minimum degree at least k) is nonempty. For any fixed k ≥3, the k-core is known to emerge via a discontinuous phase transition, where at time t = tk its size jumps from 0 to linear in the number of vertices with high probability (w.h.p.). It is believed that for any k ≥3, the core is Hamiltonian upon creation w.h.p., and Bollob́as, Cooper, Fenner and Frieze further conjectured that it in fact admits ≥(k - 1)/2 edgedisjoint Hamilton cycles. However, even the asymptotic threshold for Hamiltonicity of the k-core in G(n, p) was unknown for any k. We show here that for any fixed k ≥15, the k-core of G_{t} is w.h.p. Hamiltonian for all t ≥tk, that is, immediately as the k-core appears and indefinitely afterwards. Moreover, we prove that for large enough fixed k the k-core contains ≥(k - 3)/2 edge-disjoint Hamilton cycles w.h.p. for all t ≥tk.

Original language | English (US) |
---|---|

Pages (from-to) | 161-188 |

Number of pages | 28 |

Journal | Proceedings of the London Mathematical Society |

Volume | 109 |

Issue number | 1 |

DOIs | |

State | Published - 2014 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Proceedings of the London Mathematical Society*,

*109*(1), 161-188. https://doi.org/10.1112/plms/pdu003

**Cores of random graphs are born Hamiltonian.** / Krivelevich, Michael; Lubetzky, Eyal; Sudakov, Benny.

Research output: Contribution to journal › Article

*Proceedings of the London Mathematical Society*, vol. 109, no. 1, pp. 161-188. https://doi.org/10.1112/plms/pdu003

}

TY - JOUR

T1 - Cores of random graphs are born Hamiltonian

AU - Krivelevich, Michael

AU - Lubetzky, Eyal

AU - Sudakov, Benny

PY - 2014

Y1 - 2014

N2 - Let (Gt) be the random graph process (G0) is edgeless and Gt is obtained by adding a uniformly distributed new edge to Gt-1), and let tk denote the minimum time t such that the k-core of Gt (its unique maximal subgraph with minimum degree at least k) is nonempty. For any fixed k ≥3, the k-core is known to emerge via a discontinuous phase transition, where at time t = tk its size jumps from 0 to linear in the number of vertices with high probability (w.h.p.). It is believed that for any k ≥3, the core is Hamiltonian upon creation w.h.p., and Bollob́as, Cooper, Fenner and Frieze further conjectured that it in fact admits ≥(k - 1)/2 edgedisjoint Hamilton cycles. However, even the asymptotic threshold for Hamiltonicity of the k-core in G(n, p) was unknown for any k. We show here that for any fixed k ≥15, the k-core of Gt is w.h.p. Hamiltonian for all t ≥tk, that is, immediately as the k-core appears and indefinitely afterwards. Moreover, we prove that for large enough fixed k the k-core contains ≥(k - 3)/2 edge-disjoint Hamilton cycles w.h.p. for all t ≥tk.

AB - Let (Gt) be the random graph process (G0) is edgeless and Gt is obtained by adding a uniformly distributed new edge to Gt-1), and let tk denote the minimum time t such that the k-core of Gt (its unique maximal subgraph with minimum degree at least k) is nonempty. For any fixed k ≥3, the k-core is known to emerge via a discontinuous phase transition, where at time t = tk its size jumps from 0 to linear in the number of vertices with high probability (w.h.p.). It is believed that for any k ≥3, the core is Hamiltonian upon creation w.h.p., and Bollob́as, Cooper, Fenner and Frieze further conjectured that it in fact admits ≥(k - 1)/2 edgedisjoint Hamilton cycles. However, even the asymptotic threshold for Hamiltonicity of the k-core in G(n, p) was unknown for any k. We show here that for any fixed k ≥15, the k-core of Gt is w.h.p. Hamiltonian for all t ≥tk, that is, immediately as the k-core appears and indefinitely afterwards. Moreover, we prove that for large enough fixed k the k-core contains ≥(k - 3)/2 edge-disjoint Hamilton cycles w.h.p. for all t ≥tk.

UR - http://www.scopus.com/inward/record.url?scp=84904337738&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84904337738&partnerID=8YFLogxK

U2 - 10.1112/plms/pdu003

DO - 10.1112/plms/pdu003

M3 - Article

AN - SCOPUS:84904337738

VL - 109

SP - 161

EP - 188

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

SN - 0024-6115

IS - 1

ER -