A large hole in pseudo-random graphs

Abstract

We show that there exist constants δ1 ,δ2 >0 such that if G is an (n,d,λ)-graph with λ/d ≤δ1 , then G contains an induced cycle of length at least δ2 n/d. We further demonstrate that, up to a constant factor, this is best possible. Utilising our techniques, we derive that the number of non-isomorphic induced subgraphs of such G is at least exponential in nlogd/d, and further demonstrate that this is tight up to a constant factor in the exponent.

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Item Type:	Article
Authors/Creators:	Diskin, S. Krivelevich, M. https://orcid.org/0000-0003-2357-4982 Markbreit, I. Zhukovskii, M. https://orcid.org/0000-0001-8763-9533
Copyright, Publisher and Additional Information:	© The Author(s), 2026. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
Keywords:	pseudo-random graphs; induced cycles; graph isomorphisms
Dates:	Accepted: 18 May 2026 Published (online): 25 June 2026 Published: 25 June 2026
Institution:	The University of Sheffield
Academic Units:	The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Computer Science (Sheffield)
Date Deposited:	30 Jun 2026 11:29
Last Modified:	30 Jun 2026 11:29
Status:	Published online
Publisher:	Cambridge University Press (CUP)
Refereed:	Yes
Identification Number:	10.1017/s0963548326100479
Open Archives Initiative ID (OAI ID):	oai:eprints.whiterose.ac.uk:242665

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A large hole in pseudo-random graphs

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