Universality for bounded degree spanning trees in randomly perturbed graphs

Böttcher, Julia; Han, Jie; Kohayakawa, Yoshiharu; Montgomery, Richard; Parczyk, Olaf; Person, Yury

doi:10.1002/rsa.20850

Cited by 49 publications

(44 citation statements)

References 29 publications

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“…Krivelevich, Kwan, and Sudakov [25] asked if extending on their result also T pn, ∆quniversality holds in G α Y Gpn, pq for p " ωp1{nq and α ą 0. In [9] we proved this together with Böttcher, Han, Kohayakawa, Montgomery, and Person building on the method from [10].…”

Section: Introductionmentioning

confidence: 74%

2-universality in randomly perturbed graphs

Parczyk

2020

European Journal of Combinatorics

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A graph G is called universal for a family of graphs F if it contains every element F P F as a subgraph. Let Fpn, 2q be the family of all graphs with maximum degree 2. Ferber, Kronenberg, and Luh [Optimal Threshold for a Random Graph to be 2-Universal, to appear in Transactions of the American Mathematical Society] proved that there exists a C such that for p ě Cpn´2 {3 log 1{3 nq the random graph Gpn, pq a.a.s is Fpn, 2q-universal, which is asymptotically optimal. For any n-vertex graph Gα with minimum degree δpGαq ě αn Aigner and Brandt [Embedding arbitrary graphs of maximum degree two, Journal of the London Mathematical Society 48 (1993), 39-51] proved that Gα is Fpn, 2q-universal for an optimal α ě 2{3.In this note, we consider the model of randomly perturbed graphs, which is the union GαYGpn, pq. We prove that a.a.s. Gα Y Gpn, pq is Fpn, 2q-universal provided that α ą 0 and p " ωpn´2 {3 q. This is asymptotically optimal and improves on both results from above in the respective parameter. Furthermore, this extends a result of Böttcher, Montgomery, Parczyk, and Person [Embedding spanning bounded degree subgraphs in randomly perturbed graphs, arXiv:1802.04603 (2018)], who embed a given F P Fpn, 2q at these values. We also prove variants with universality for the family F ℓ pn, 2q, all graphs from Fpn, 2q with girth at least ℓ. For example, there exists an ℓ0 depending only on α such that for all ℓ ě ℓ0 already p " ωp1{nq is sufficient for F ℓ pn, 2q-universality.but the extra log n-term ist needed to guarantee that the graph has a.a.s. minimum degree 2. In the following discussion we will work with p " ωppq even though for many results a stronger variant is proved, where p ě Cp for some absolute constant C depending only on pF n q ně1 .1.1. Randomly perturbed graphs. Combining the two models from random and extremal graph theory, Bohman, Frieze, and Martin [7] introduced the model of randomly perturbed graphs G α Y Gpn, pq for any α ą 0, where, as above, G α is any n-vertex graph with minimum degree δpG α q ě αn. They show that p " ωp1{nq is sufficient to a.a.s. guarantee a Hamilton cycle in G α Y Gpn, pq for any G α . When G α is the complete unbalanced bipartite graph K αn,p1´αqn then at least a linear number of egdes is needed. Using both graphs G α and Gpn, pq this result dramatically improves on the α ě 1{2 needed in G α alone, even though adding Gpn, pq is a relatively small random perturbation. On the other hand it is also a log n-term better than the threshold in Gpn, pq alone, which is plausible as G α guarantees a large minimum degree.In recent years this model attracted a lot of attention. For a bounded degree spanning tree Krivelevich, Kwan, and Sudakov [25] proved that p " ωp1{nq also is sufficient in G α Y Gpn, pq. In G α alone α ą 1{2 is needed [22] and in Gpn, pq only recently Montgomery [28] was able to show that again log n{n gives the threshold. Similar results were proved for factors in [3] and for powers of Hamilton cycles and general bounded degree graphs in [10]. Together with Böttcher,...

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Section: Introductionmentioning

confidence: 74%

2-universality in randomly perturbed graphs

Parczyk

2020

European Journal of Combinatorics

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“…In the case of spanning bounded degree trees, in joint work with Han and Kohayakawa, we establish the following universality result in [11]. We show that G α ∪ G(n, c(α, )/n) simultaneously contains all spanning trees of maximum degree at most .…”

Section: 2mentioning

confidence: 75%

Embedding Spanning Bounded Degree Graphs in Randomly Perturbed Graphs

et al. 2020

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We study the model G α ∪ G(n, p) of randomly perturbed dense graphs, where G α is any n-vertex graph with minimum degree at least αn and G(n, p) is the binomial random graph. We introduce a general approach for studying the appearance of spanning subgraphs in this model using absorption. This approach yields simpler proofs of several known results. We also use it to derive the following two new results.For every α > 0 and 5, and every n-vertex graph F with maximum degree at most , we show that if p = ω(n −2/( +1) ), then G α ∪ G(n, p) with high probability contains a copy of F . The bound used for p here is lower by a log-factor in comparison to the conjectured threshold for the general appearance of such subgraphs in G(n, p) alone, a typical feature of previous results concerning randomly perturbed dense graphs.We also give the first example of graphs where the appearance threshold in G α ∪ G(n, p) is lower than the appearance threshold in G(n, p) by substantially more than a log-factor. We prove that, for every k 2 and α > 0, there is some η > 0 for which the kth power of a Hamilton cycle with high probability appears in G α ∪ G(n, p) when p = ω(n −1/k−η ). The appearance threshold of the kth power of a Hamilton cycle in G(n, p) alone is known to be n −1/k , up to a log-term when k = 2, and exactly for k > 2.

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“…For technical reasons, we assume that maximum over the emptyset equals 0. For several applications of concentration inequalities, it will be convenient to define the following for each j ∈ [6],…”

Section: Construction Of the Embeddingmentioning

confidence: 99%

“…Recall the definition in (3.1). For all i ∈ [r]× [2], ∈ {0, … , k +1}, ′ ∈ [k + 1], j ∈ [6], and j ′ ∈ [2], we define…”

Section: Construction Of the Embeddingmentioning

confidence: 99%

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Spanning trees in randomly perturbed graphs

Joos

Kim

2019

Random Struct Algorithms

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A classical result of Komlós, Sárközy, and Szemerédi states that every n‐vertex graph with minimum degree at least (1/2 + o(1))n contains every n‐vertex tree with maximum degree Ofalse(nfalse/normallognfalse). Krivelevich, Kwan, and Sudakov proved that for every n‐vertex graph Gα with minimum degree at least αn for any fixed α > 0 and every n‐vertex tree T with bounded maximum degree, one can still find a copy of T in Gα with high probability after adding O(n) randomly chosen edges to Gα. We extend the latter results to trees with (essentially) unbounded maximum degree; for a given nofalse(1false)≤normalΔ≤cnfalse/normallogn and α > 0, we determine up to a constant factor the number of random edges that we need to add to an arbitrary n‐vertex graph with minimum degree αn in order to guarantee with high probability a copy of any fixed n‐vertex tree with maximum degree at most Δ.

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Universality for bounded degree spanning trees in randomly perturbed graphs

Cited by 49 publications

References 29 publications

2-universality in randomly perturbed graphs

2-universality in randomly perturbed graphs

Embedding Spanning Bounded Degree Graphs in Randomly Perturbed Graphs

Spanning trees in randomly perturbed graphs

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