Random perturbation of sparse graphs

In the model of randomly perturbed graphs we consider the union of a deterministic graph G with minimum degree αn and the binomial random graph G(n, p). This model was introduced by Bohman, Frieze, and Martin and for Hamilton cycles their result bridges the gap between Dirac’s theorem and the results by Pósa and Korshunov on the threshold in G(n, p). In this note we extend this result in G ∪G(n, p) to sparser graphs with α = o(1). More precisely, for any ε > 0 and α: N ↦→ (0, 1) we show that a.a.s. G ∪ G(n, β/n) is Hamiltonian, where β = −(6 + ε) log(α). If α > 0 is a fixed constant this gives the aforementioned result by Bohman, Frieze, and Martin and if α = O(1/n) the random part G(n, p) is sufficient for a Hamilton cycle. We also discuss embeddings of bounded degree trees and other spanning structures in this model, which lead to interesting questions on almost spanning embeddings into G(n, p).

Metadaten
Author:	Maximilian Grischa Hahn-Klimroth ORCiD GND, Giulia Satiko Maesaka ORCiD, Yannick Mogge ORCiD GND, Samuel Mohr ORCiD GND, Olaf Parczyk ORCiD GND
URN:	urn:nbn:de:hebis:30:3-576841
DOI:	https://doi.org/10.37236/9510
ISSN:	1077-8926
Parent Title (English):	The electronic journal of combinatorics
Publisher:	EMIS ELibEMS
Place of publication:	[Madralin]
Document Type:	Article
Language:	English
Year of Completion:	2021
Year of first Publication:	2021
Publishing Institution:	Universitätsbibliothek Johann Christian Senckenberg
Release Date:	2022/05/11
Volume:	28
Issue:	issue 2, art. P2.26
Page Number:	12
First Page:	1
Last Page:	12
Note:	(c) The authors. Released under the CC BY-ND license (International 4.0).
HeBIS-PPN:	496023322
Institutes:	Informatik und Mathematik / Mathematik
Dewey Decimal Classification:	5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
MSC-Classification:	05-XX COMBINATORICS (For finite fields, see 11Txx) / 05Cxx Graph theory (For applications of graphs, see 68R10, 81Q30, 81T15, 82B20, 82C20, 90C35, 92E10, 94C15) / 05C35 Extremal problems [See also 90C35]
	05-XX COMBINATORICS (For finite fields, see 11Txx) / 05Cxx Graph theory (For applications of graphs, see 68R10, 81Q30, 81T15, 82B20, 82C20, 90C35, 92E10, 94C15) / 05C80 Random graphs [See also 60B20]
Sammlungen:	Universitätspublikationen
Licence (German):	Creative Commons - Namensnennung-Keine Bearbeitung

Open Access