Triangles in randomly perturbed graphs

Böttcher, Julia ORCID: 0000-0002-4104-3635, Parczyk, Olaf, Sgueglia, Amedeo and Skokan, Jozef ORCID: 0000-0003-3996-7676 (2023) Triangles in randomly perturbed graphs. Combinatorics, Probability and Computing, 32 (1). 91 - 121. ISSN 0963-5483

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Abstract

We study the problem of finding pairwise vertex-disjoint triangles in the randomly perturbed graph model, which is the union of any n-vertex graph G satisfying a given minimum degree condition and the binomial random graph G(n, p). We prove that asymptotically almost surely G∪G(n, p) contains at least min{δ(G),⌊n/3⌋} pairwise vertex-disjoint triangles, provided p ≥ C log n/n, where C is a large enough constant. This is a perturbed version of an old result of Dirac. Our result is asymptotically optimal and answers a question of Han, Morris, and Treglown [RSA, 2021, no. 3, 480–516] in a strong form. We also prove a stability version of our result, which in the case of pairwise vertex-disjoint triangles extends a result of Han, Morris, and Treglown [RSA, 2021, no. 3, 480–516]. Together with a result of Balogh, Treglown, and Wagner [CPC, 2019, no. 2, 159–176] this fully resolves the existence of triangle factors in randomly perturbed graphs. We believe that the methods introduced in this paper are useful for a variety of related problems: we discuss possible generalisations to clique factors, cycle factors, and 2-universality.

Item Type:	Article
Official URL:	https://www.cambridge.org/core/journals/combinator...
Additional Information:	© 2022 The Authors
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	31 May 2022 10:51
Last Modified:	21 Sep 2023 16:24
URI:	http://eprints.lse.ac.uk/id/eprint/115257

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