Powers of Hamiltonian cycles in randomly augmented graphs

Dudek, Andrzej; Reiher, Christian; Ruciński, Andrzej; Schacht, Mathias

doi:10.1002/rsa.20870

Cited by 36 publications

(48 citation statements)

References 23 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: 97%

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: 97%

Random Perturbation of Sparse Graphs

Hahn‐Klimroth

Maesaka

Mogge

et al. 2021

Electron. J. Combin.

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“…In this range, one thinks of the deterministic graph as “helping” G ( n , p ) to get a certain spanning structure and the observed phenomenon is usually a decrease in the probability threshold of a logarithmic factor, as is the case for Hamiltonicity as above. Recently, there has been interest in the other extreme, where one starts with a minimum degree slightly less than the extremal minimum degree threshold for a certain spanning structure and requires a small “sprinkling” of random edges to guarantee the existence of the spanning structure in the resulting graph, see for example, [14,45].…”

Section: Introductionmentioning

confidence: 99%

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

Han

Morris

Treglown

2020

Random Struct Algorithms

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A perfect Kr‐tiling in a graph G is a collection of vertex‐disjoint copies of Kr that together cover all the vertices in G. In this paper we consider perfect Kr‐tilings in the setting of randomly perturbed graphs; a model introduced by Bohman, Frieze, and Martin [7] where one starts with a dense graph and then adds m random edges to it. Specifically, given any fixed 0<α<1−1/r we determine how many random edges one must add to an n‐vertex graph G of minimum degree δ(G)≥αn to ensure that, asymptotically almost surely, the resulting graph contains a perfect Kr‐tiling. As one increases α we demonstrate that the number of random edges required “jumps” at regular intervals, and within these intervals our result is best‐possible. This work therefore closes the gap between the seminal work of Johansson, Kahn and Vu [25] (which resolves the purely random case, that is, α=0) and that of Hajnal and Szemerédi [18] (which demonstrates that for α≥1−1/r the initial graph already houses the desired perfect Kr‐tiling).

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“…In recent years, a whole host of results have been obtained concerning embedding spanning subgraphs into a randomly perturbed graph, as well as other properties of the model; see e.g. [4,6,7,9,10,15,24,29,30,36]. The model has also been investigated in the setting of directed graphs and hypergraphs (see e.g.…”

Section: 2mentioning

confidence: 99%

Ramsey properties of randomly perturbed graphs: cliques and cycles

Das

Treglown

2020

Combinator. Probab. Comp.

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Given graphs H1, H2, a graph G is (H1, H2) -Ramsey if, for every colouring of the edges of G with red and blue, there is a red copy of H1 or a blue copy of H2. In this paper we investigate Ramsey questions in the setting of randomly perturbed graphs. This is a random graph model introduced by Bohman, Frieze and Martin [8] in which one starts with a dense graph and then adds a given number of random edges to it. The study of Ramsey properties of randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali [30] in 2006; they determined how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (K3, K t ) -Ramsey (for t ≽ 3). They also raised the question of generalizing this result to pairs of graphs other than (K3, K t ). We make significant progress on this question, giving a precise solution in the case when H1 = K s and H2 = K t where s, t ≽ 5. Although we again show that one requires polynomially fewer edges than in the purely random graph, our result shows that the problem in this case is quite different to the (K3, K t ) -Ramsey question. Moreover, we give bounds for the corresponding (K4, K t ) -Ramsey question; together with a construction of Powierski [37] this resolves the (K4, K4) -Ramsey problem. We also give a precise solution to the analogous question in the case when both H1 = C s and H2 = C t are cycles. Additionally we consider the corresponding multicolour problem. Our final result gives another generalization of the Krivelevich, Sudakov and Tetali [30] result. Specifically, we determine how many random edges must be added to a dense graph to ensure the resulting graph is with high probability (C s , K t ) -Ramsey (for odd s ≽ 5 and t ≽ 4). To prove our results we combine a mixture of approaches, employing the container method, the regularity method as well as dependent random choice, and apply robust extensions of recent asymmetric random Ramsey results.

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Powers of Hamiltonian cycles in randomly augmented graphs

Cited by 36 publications

References 23 publications

Random Perturbation of Sparse Graphs

Random Perturbation of Sparse Graphs

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

Ramsey properties of randomly perturbed graphs: cliques and cycles

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