The bandwidth theorem in sparse graphs

Allen, Peter ORCID: 0000-0001-6555-3501, Böttcher, Julia ORCID: 0000-0002-4104-3635, 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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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:	11 Oct 2024 21:21
URI:	http://eprints.lse.ac.uk/id/eprint/106618

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