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A spanning bandwidth theorem in random graphs

Part of: Graph theory

Published online by Cambridge University Press:  13 December 2021

Peter Allen ,
Julia Böttcher ,
Julia Ehrenmüller ,
Jakob Schnitzer  and
Anusch Taraz
Peter Allen
Affiliation:
London School of Economics, Department of Mathematics, Houghton Street, London WC2A 2AE, UK
Julia Böttcher*
Affiliation:
London School of Economics, Department of Mathematics, Houghton Street, London WC2A 2AE, UK
Julia Ehrenmüller
Affiliation:
Technische Universität Hamburg, Institut fìr Mathematik, Am Schwarzenberg-Campus 3, 21073 Hamburg, Germany
Jakob Schnitzer
Affiliation:
Universität Hamburg, Fachbereich Mathematik, Bundesstrasse 55, 20146 HamburgGermany
Anusch Taraz
Affiliation:
Technische Universität Hamburg, Institut fìr Mathematik, Am Schwarzenberg-Campus 3, 21073 Hamburg, Germany
*
*Corresponding author. Email: j.boettcher@lse.ac.uk
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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.

Keywords

random graphsspanning subgraphssparse regularity

MSC classification

Primary: 05C80: Random graphs
Secondary: 05C35: Extremal problems
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 31 , Issue 4 , July 2022 , pp. 598 - 628
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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