## Abstract

Kim and Vu made the following conjecture (Advances in Mathematics, 2004): if d ≫ log n, then the random dregular graph G(n, d) can asymptotically almost surely be “sandwiched” between G(n, p_{1}) and G(n, p_{2}) where p_{1} and p_{2} are both (1 + o(1))d/n. They proved this conjecture for log n ≪ d 6 n^{1}/^{3−}o^{(1)}, with a defect in the sandwiching: G(n, d) contains G(n, p_{1}) perfectly, but is not completely contained in G(n, p_{2}). Recently, the embedding G(n, p_{1}) ⊆ G(n, d) was improved by Dudek, Frieze, Ruci´ nski and Šileikis to d = o(n). In this paper, we prove Kim-Vu's sandwich conjecture, with perfect containment on both sides, for all d ≫ n/√log n. For d = O(n/√log n), we prove a weaker version of the sandwich conjecture with p_{2} approximately equal to (d/n) log n, without any defect. In addition to sandwiching regular graphs, our results cover graphs whose degrees are asymptotically equal. The proofs rely on estimates for the probability that a random factor of a pseudorandom graph contains a given edge, which is of independent interest. As applications, we obtain new results on the properties of random graphs with given near-regular degree sequences, including Hamiltonicity and universality in subgraph containment. We also determine several graph parameters in these random graphs, such as the chromatic number, small subgraph counts, the diameter, and the independence number. We are also able to characterise many phase transitions in edge percolation on these random graphs, such as the threshold for the appearance of a giant component.

Original language | English |
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Title of host publication | Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms |

Editors | Shuchi Chawla |

Publisher | Society for Industrial & Applied Mathematics (SIAM) |

Pages | 690-701 |

Number of pages | 12 |

ISBN (Electronic) | 9781611975994 |

DOIs | |

Publication status | Published - 2020 |

Event | ACM/SIAM Symposium on Discrete Algorithms 2020 - Salt Lake City, United States of America Duration: 5 Jan 2020 → 8 Jan 2020 Conference number: 31st https://www.siam.org/conferences/cm/conference/soda20 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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Volume | 2020-January |

### Conference

Conference | ACM/SIAM Symposium on Discrete Algorithms 2020 |
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Abbreviated title | SODA 2020 |

Country/Territory | United States of America |

City | Salt Lake City |

Period | 5/01/20 → 8/01/20 |

Internet address |