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The bandwidth theorem in sparse graphs

Allen, Peter, Böttcher, Julia, Ehrenmüller, Julia and Taraz, Anusch (2020) The bandwidth theorem in sparse graphs. Advances in Combinatorics, 2020 (1). 1 - 60. ISSN 2517-5599

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Identification Number: 10.19086/aic.12849

Abstract

The bandwidth theorem [Mathematische Annalen, 343(1):175–205, 2009] states that any n-vertex graph G with minimum degree [Formula Presented] contains all n-vertex k-colourable graphs H with bounded maximum degree and bandwidth o(n). We provide sparse analogues of this statement in random graphs as well as pseudorandom graphs. More precisely, we show that for p ≫[Formula Presented] asymptotically almost surely each spanning subgraph G of G(n, p) with minimum degree [Formula Presented] pn contains all n-vertex k-colourable graphs H with maximum degree ∆, bandwidth o(n), and at least Cp−2 vertices not contained in any triangle. A similar result is shown for sufficiently bijumbled graphs, which, to the best of our knowledge, is the first resilience result in pseudorandom graphs for a rich class of spanning subgraphs. Finally, we provide improved results for H with small degeneracy, which in particular imply a resilience result in G(n, p) with respect to the containment of spanning bounded degree trees for p ≫[Formula Presented].

Item Type: Article
Official URL: https://www.advancesincombinatorics.com/
Additional Information: © 2020 The Authors
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 24 Sep 2020 12:51
Last Modified: 20 Oct 2020 06:07
URI: http://eprints.lse.ac.uk/id/eprint/106618

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