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Publisher DOI: 10.37236/9510
Title: Random perturbation of sparse graphs
Language: English
Authors: Hahn-Klimroth, Maximilian Grischa 
Maesaka, Giulia Satiko 
Mogge, Yannick 
Mohr, Samuel 
Parczyk, Olaf 
Issue Date: 21-May-2021
Publisher: EMIS ELibEMS
Source: Electronic Journal of Combinatorics 28 (2): #P2.26 (2021)
Abstract (english): 
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). α α α
DOI: 10.15480/882.3578
ISSN: 1077-8926
Journal: The electronic journal of combinatorics 
Institute: Mathematik E-10 
Document Type: Article
License: CC BY-ND 4.0 (Attribution-NoDerivatives) CC BY-ND 4.0 (Attribution-NoDerivatives)
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