Tilings in randomly perturbed graphs: Bridging the gap between Hajnal‐Szemerédi and Johansson‐Kahn‐Vu

Han, Jie; Morris, Patrick; Treglown, Andrew

doi:10.1002/rsa.20981

Cited by 33 publications

(37 citation statements)

References 48 publications

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“…In most of these cases a G α , that is responsible for the lower bound, is the complete imbalanced bipartite graph K αn,(1−α)n . In this model there are also results with lower bounds on α [6,16,23,38] and for Ramsey-type problems [13,14].…”

Section: Introduction and Resultsmentioning

confidence: 98%

Random Perturbation of Sparse Graphs

Hahn‐Klimroth

Maesaka

Mogge

et al. 2021

Electron. J. Combin.

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In the model of randomly perturbed graphs we consider the union of a deterministic graph $\mathcal{G}_\alpha$ with minimum degree $\alpha n$ and the binomial random graph $\mathbb{G}(n,p)$. 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 $\mathbb{G}(n,p)$. In this note we extend this result in $\mathcal{G}_\alpha\cup\mathbb{G}(n,p)$ to sparser graphs with $\alpha=o(1)$. More precisely, for any $\varepsilon>0$ and $\alpha \colon \mathbb{N} \mapsto (0,1)$ we show that a.a.s. $\mathcal{G}_\alpha\cup \mathbb{G}(n,\beta /n)$ is Hamiltonian, where $\beta = -(6 + \varepsilon) \log(\alpha)$. If $\alpha>0$ is a fixed constant this gives the aforementioned result by Bohman, Frieze, and Martin and if $\alpha=O(1/n)$ the random part $\mathbb{G}(n,p)$ 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 $\mathbb{G}(n,p)$.

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Section: Introduction and Resultsmentioning

confidence: 98%

Random Perturbation of Sparse Graphs

Hahn‐Klimroth

Maesaka

Mogge

et al. 2021

Electron. J. Combin.

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“…In comparison, adding the sublinear independence number condition does not reduce the requirement for the Hamiltonicity (e.g., consider two disjoint cliques of equal sizes). Further comparison on clique-factor problems can be done to the work of [18]. On the other hand, as observed by Reiher and Schacht [39], the locally dense condition is quite handy in the graph setting, which, e.g., allows one to draw an arbitrarily large constant-sized clique in a linear-sized set of vertices.…”

Section: Comparison With Related Resultsmentioning

confidence: 95%

Embedding clique-factors in graphs with low $\ell$-independence number

Chang

Han

Kim³

et al. 2021

Preprint

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The following question was proposed by Nenadov and Pehova and reiterated by Knierim and Su: Given integers , r and n with n ∈ rN, is it true that every n-vertex graph G with δ(G) ≥ max{ 1 2 , r− r }n + o(n) and α (G) = o(n) contains a K r -factor? We give a negative answer for the case when ≥ 3r 4 by giving a family of constructions using the so-called cover thresholds and show that the minimum degree condition given by our construction is asymptotically best possible. That is, for all integers r, with r > ≥ 3 4 r and µ > 0, there exist α > 0 and N such that for every n ∈ rN with n > N , every n-vertex graph G with δ(G) ≥ 1 2− (r−1) + µ n and α (G) ≤ αn contains a K r -factor. Here (r − 1) is the Ramsey-Turán density for K r−1 under the -independence number condition.

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“…In other words, as long as α ∈ ( 1 3 , 2 3 ) it suffices to add a linear number of random edges to the deterministic graph G α for enforcing the square of a Hamilton cycle, and for α ≤ 1 problems: for K r -factors in [17] and for C -factors in [8,9]. However, Theorem 1.1 is to the best of our knowledge the first result exhibiting a countably infinite number of 'jumps'.…”

Section: Introductionmentioning

confidence: 99%

The square of a Hamilton cycle in randomly perturbed graphs

Böttcher¹,

Parczyk²,

Sgueglia³

et al. 2022

Preprint

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We investigate the appearance of the square of a Hamilton cycle in the model of randomly perturbed graphs, which is, for a given α ∈ (0, 1), the union of any n-vertex graph with minimum degree αn and the binomial random graph G(n, p). This is known when α > 1 2, and we determine the exact perturbed threshold probability in all the remaining cases, i.e., for each α ≤ 1 2. We demonstrate that, as α ranges over the interval (0, 1), the threshold performs a countably infinite number of 'jumps'. Our result has implications on the perturbed threshold for 2-universality, where we also fully address all open cases.

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Tilings in randomly perturbed graphs: Bridging the gap between Hajnal‐Szemerédi and Johansson‐Kahn‐Vu

Cited by 33 publications

References 48 publications

Random Perturbation of Sparse Graphs

Random Perturbation of Sparse Graphs

Embedding clique-factors in graphs with low $\ell$-independence number

The square of a Hamilton cycle in randomly perturbed graphs

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