# A spanning bandwidth theorem in random graphs

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

Identification Number: 10.1017/S0963548321000481

## 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 https://www.cambridge.org/core/journals/combinator... © 2021 The Authors Mathematics Q Science > QA MathematicsQ Science > QA Mathematics > QA75 Electronic computers. Computer science 16 Mar 2022 12:12 13 Jun 2022 23:00 http://eprints.lse.ac.uk/id/eprint/114370