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The bandwidth theorem for locally dense graphs

Part of: Graph theory

Published online by Cambridge University Press:  04 November 2020

Katherine Staden  and
Andrew Treglown
Katherine Staden
Affiliation:
Mathematical Institute, University of Oxford, Oxford, OX2 6GGUnited Kingdom; E-mail: staden@maths.ox.ac.uk
Andrew Treglown
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TTUnited Kingdom; E-mail: a.c.treglown@bham.ac.uk

Abstract

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The bandwidth theorem of Böttcher, Schacht, and Taraz [Proof of the bandwidth conjecture of Bollobás andKomlós, Mathematische Annalen, 2009] gives a condition on the minimum degree of an n-vertex graph G that ensures G contains every r-chromatic graph H on n vertices of bounded degree and of bandwidth $o(n)$ , thereby proving a conjecture of Bollobás and Komlós [The Blow-up Lemma, Combinatorics, Probability, and Computing, 1999]. In this paper, we prove a version of the bandwidth theorem for locally dense graphs. Indeed, we prove that every locally dense n-vertex graph G with $\delta (G)> (1/2+o(1))n$ contains as a subgraph any given (spanning) H with bounded maximum degree and sublinear bandwidth.

Keywords

bandwidthembeddingregularity method

MSC classification

Primary: 05C35: Extremal problems
Type
Discrete Mathematics
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e42
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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