Powers of Hamilton cycles in random graphs and tight Hamilton cycles in random hypergraphs

Nenadov, Rajko; Škorić, Nemanja

doi:10.1002/rsa.20782

Cited by 32 publications

(48 citation statements)

References 19 publications

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“…The first lemma gives a bound on p for which a typical

G_{n, p}

contains a large F ‐matching for arbitrary F . Lemma (, Corollary 3.5]). Let F be a graph.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…The following lemma gives a lower bound on p for which

G_{n, p}

admits an

(F, x, Y)

‐matching. In order to state it we need the following version of m 1 ‐density: given a graph F and a subset

X \subseteq V (F)

we denote with m ( F, X ) the rooted ‐density of F , defined as

m (F, X) = \max_{\overset{F' \subseteq F}{e (F') > 0}} {\frac{e (F')}{v (F') - \max {1, | V (F') \cap X |}} : either X \subseteq V (F') or X \cap V (F') = \emptyset} .

Lemma (, lemma 3.3]). Let F be a graph and

x \subseteq V {(F)}^{r}

an r‐tuple of independent vertices in F, for some

r \leq v (F) - 2

.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

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Spanning universality in random graphs

Ferber

Nenadov

2018

Random Struct Algorithms

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A graph is said to be H(n,Δ) ‐universal if it contains every graph with n vertices and maximum degree at most Δ as a subgraph. Dellamonica, Kohayakawa, Rödl and Ruciński used a “matching‐based” embedding technique introduced by Alon and Füredi to show that the random graph Gn,p is asymptotically almost surely H(n,Δ) ‐universal for p=Ω((logn/n)1/Δ), a threshold for the property that every subset of Δ vertices has a common neighbor. This bound has become a benchmark in the field and many subsequent results on embedding spanning graphs of maximum degree Δ in random graphs are proven only up to this threshold. We take a step towards overcoming limitations of former techniques by showing that Gn,p is almost surely H(n,Δ) ‐universal for p=Ω(n−1/(Δ−0.5)log3n).

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“…The first lemma gives a bound on p for which a typical

G_{n, p}

contains a large F ‐matching for arbitrary F . Lemma (, Corollary 3.5]). Let F be a graph.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…The following lemma gives a lower bound on p for which

G_{n, p}

admits an

(F, x, Y)

‐matching. In order to state it we need the following version of m 1 ‐density: given a graph F and a subset

X \subseteq V (F)

we denote with m ( F, X ) the rooted ‐density of F , defined as

m (F, X) = \max_{\overset{F' \subseteq F}{e (F') > 0}} {\frac{e (F')}{v (F') - \max {1, | V (F') \cap X |}} : either X \subseteq V (F') or X \cap V (F') = \emptyset} .

Lemma (, lemma 3.3]). Let F be a graph and

x \subseteq V {(F)}^{r}

an r‐tuple of independent vertices in F, for some

r \leq v (F) - 2

.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

Spanning universality in random graphs

Ferber

Nenadov

2018

Random Struct Algorithms

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“…Similarly, the threshold of the square of a Hamilton cycle is conjectured to be n −1/2 , but this is still open. Currently, the best known upper bound, by Nenadov and Škorić [33], is off by a O(log 4 n)-factor from this conjectured threshold.…”

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confidence: 97%

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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“…A homomorphism from G to H is a function → ϕ V G V H : ( ) ( ) that preserves edges, that is, The kth power C l k of a cycle C l of length l has the same vertex set as C l and two different vertices are adjacent in C l k if and only if they are at distance at most k in C l . The existence problems of powers of cycles in random graphs have attracted a lot of attention, see [3,31,34] for example. Similar to the cycle decomposition of graphs [4], we put forward the following problem.…”

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confidence: 99%