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Dirac’s theorem for random regular graphs

Part of: Graph theory

Published online by Cambridge University Press:  28 August 2020

Padraig Condon ,
Alberto Espuny Díaz ,
António Girão ,
Daniela Kühn  and
Deryk Osthus
Padraig Condon
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK
Alberto Espuny Díaz
Affiliation:
Institut für Mathematik, Technische Universität Ilmenau, 98693 Ilmenau, Germany
António Girão*
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK
Daniela Kühn
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK
Deryk Osthus
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK
*
*Corresponding author. Email: giraoa@bham.ac.uk
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Abstract

We prove a ‘resilience’ version of Dirac’s theorem in the setting of random regular graphs. More precisely, we show that whenever d is sufficiently large compared to $\epsilon > 0$ , a.a.s. the following holds. Let $G'$ be any subgraph of the random n-vertex d-regular graph $G_{n,d}$ with minimum degree at least $$(1/2 + \epsilon )d$$ . Then $G'$ is Hamiltonian.

This proves a conjecture of Ben-Shimon, Krivelevich and Sudakov. Our result is best possible: firstly the condition that d is large cannot be omitted, and secondly the minimum degree bound cannot be improved.

MSC classification

Primary: 05C80: Random graphs
Secondary: 05C35: Extremal problems 05C45: Eulerian and Hamiltonian graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 1 , January 2021 , pp. 17 - 36
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

†

This project has received partial funding from the European Research Council (ERC) under the EuropeanUnion’sHorizon 2020 research and innovation programme (grant agreement 786198, D. Kühn and D. Osthus). The research leading to these results was also partially supported by the EPSRC, grants EP/N019504/1 (A. Girão and D. Kühn) and EP/S00100X/1 (D. Osthus), as well as the Royal Society and theWolfson Foundation (D. Kühn).

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