Allen, Peter ORCID: 0000-0001-6555-3501, Böttcher, Julia ORCID: 0000-0002-4104-3635, Ehrenmüller, Julia, Schnitzer, Jakob and Taraz, Anusch
(2021)
*A spanning bandwidth theorem in random graphs.*
Combinatorics Probability and Computing.
ISSN 0963-5483

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## Abstract

The bandwidth theorem of Böttcher, Schacht and Taraz states that any n-vertex graph G with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)n$ contains all n-vertex k-colourable graphs H with bounded maximum degree and bandwidth o(n). Recently, a subset of the authors proved a random graph analogue of this statement: for $p\gg \big(\tfrac{\log n}{n}\big)^{1/\Delta}$ a.a.s. each spanning subgraph G of G(n,p) with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)pn$ contains all n-vertex k-colourable graphs H with maximum degree $\Delta$ , bandwidth o(n), and at least $C p^{-2}$ vertices not contained in any triangle. This restriction on vertices in triangles is necessary, but limiting. In this paper, we consider how it can be avoided. A special case of our main result is that, under the same conditions, if additionally all vertex neighbourhoods in G contain many copies of $K_\Delta$ then we can drop the restriction on H that $Cp^{-2}$ vertices should not be in triangles.

Item Type: | Article |
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Official URL: | https://www.cambridge.org/core/journals/combinator... |

Additional Information: | © 2021 The Authors |

Divisions: | Mathematics |

Subjects: | Q Science > QA Mathematics Q Science > QA Mathematics > QA75 Electronic computers. Computer science |

Date Deposited: | 16 Mar 2022 12:12 |

Last Modified: | 13 Jun 2022 23:00 |

URI: | http://eprints.lse.ac.uk/id/eprint/114370 |

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