A scaling limit for the length of the longest cycle in a sparse random graph

Anastos, Michael; Frieze, Alan

doi:10.48550/arxiv.1907.03657

Cited by 4 publications

(21 citation statements)

References 16 publications

(28 reference statements)

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“…( Let us also note that very recently, Anastos and Frieze [3] determined L asymptotically in the range when p = c/n for a sufficiently large constant c (in particular, c is much larger than 1).…”

Section: Theorem 1 Let L Denote the Length Of The Longest Path In G(n P)mentioning

confidence: 94%

“…Thus the main difficulty is to prove Step (3). Recall that a k-set K containing J may not be queried for one of two reasons:…”

Section: Algorithm: Pathfindermentioning

confidence: 99%

“…ε 3 n → ∞ very slowly), this may not tend to zero. If we were to assume the slightly stronger condition of ε 3 n (ln n) 3 → ∞ in Theorem 4, then this would not be an issue and we would not need to handle the case 2 ≤ j = k − 1 separately.…”

mentioning

confidence: 99%

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Longest paths in random hypergraphs

Cooley¹,

Garbe²,

Hng³

et al. 2020

Preprint

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Given integers k, j with 1 ≤ j ≤ k − 1, we consider the length of the longest j-tight path in the binomial random k-uniform hypergraph H k (n, p). We show that this length undergoes a phase transition from logarithmic length to linear and determine the critical threshold, as well as proving upper and lower bounds on the length in the subcritical and supercritical ranges.In particular, for the supercritical case we introduce the Pathfinder algorithm, a depth-first search algorithm which discovers j-tight paths in a k-uniform hypergraph. We prove that, in the supercritical case, with high probability this algorithm will find a long j-tight path.

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Section: Theorem 1 Let L Denote the Length Of The Longest Path In G(n P)mentioning

confidence: 94%

“…Thus the main difficulty is to prove Step (3). Recall that a k-set K containing J may not be queried for one of two reasons:…”

Section: Algorithm: Pathfindermentioning

confidence: 99%

See 1 more Smart Citation

Longest paths in random hypergraphs

Cooley¹,

Garbe²,

Hng³

et al. 2020

Preprint

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show abstract

“…Letting L max (G) denote the length of a longest cycle in G (its circumference), L max (G)/n is expected to converge in probability when p = c/n for every fixed c > 1, yet till recently this was not known for any c > 1. Anastos and Frieze [1] then proved that this holds when c > C 0 for some absolute constant C 0 , and further identified the limit f (c). The analogous result for D(n, p) was thereafter obtained by the same authors in [2].…”

Section: Introductionmentioning

confidence: 91%

Cycle lengths in sparse random graphs

Yahav¹,

Krivelevich²,

Lubetzky³

2020

Preprint

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We study the set L (G) of lengths of all cycles that appear in a random d-regular G on n vertices for a fixed d ≥ 3, as well as in Erdős-Rényi random graphs on n vertices with a fixed average degree c > 1. Fundamental results on the distribution of cycle counts in these models were established in the 1980's and early 1990's, with a focus on the extreme lengths: cycles of fixed length, and cycles of length linear in n.Here we derive, for a random d-regular graph, the limiting probability that L (G) simultaneously contains the entire range {ℓ, . . . , n} for ℓ ≥ 3, as an explicit expression θ ℓ = θ ℓ (d) ∈ (0, 1) which goes to 1 as ℓ → ∞.For the random graph G(n, p) with p = c/n, where c ≥ C 0 for some absolute constant C 0 , we show the analogous result for the range {ℓ, . . . , (1 − o(1))Lmax(G)}, where Lmax is the length of a longest cycle in G. The limiting probability for G(n, p) coincides with θ ℓ from the d-regular case when c is the integer d − 1. In addition, for the directed random graph D(n, p) we show results analogous to those on G(n, p), and for both models we find an interval of cε 2 n consecutive cycle lengths in the slightly supercritical regime p = 1+ε n .

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“…In 1986 Frieze [18] proved that if p ≥ C n then w.h.p. in G(n, p) there exists a cycle of length at least n − (1 + ε)v 1 (n, p), where v 1 (n, p) is the number of vertices of degree at most 1 and ε := ε(C) (and it was very recently improved even more by Anastos and Frieze [3]). In 1991, Luczak showed [32] that for p = ω 1 n , w.h.p.…”

Section: Introductionmentioning

confidence: 99%

Turán-type problems for long cycles in random and pseudo-random graphs

Krivelevich,

Kronenberg,

Mond

2019

Preprint

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We study the Turán number of long cycles in random graphs and in pseudo-random graphs. Denote by ex(G(n, p), H) the random variable counting the number of edges in a largest subgraph of G(n, p) without a copy of H. We determine the asymptotic value of ex(G(n, p), C t ) where C t is a cycle of length t, for p ≥ C n and A log n ≤ t ≤ (1 − ε)n. The typical behavior of ex(G(n, p), C t ) depends substantially on the parity of t. In particular, our results match the classical result of Woodall on the Turán number of long cycles, and can be seen as its random version, showing that the transference principle holds here as well. In fact, our techniques apply in a more general sparse pseudo-random setting. We also prove a robustness-type result, showing the likely existence of cycles of prescribed lengths in a random subgraph of a graph with a nearly optimal density.

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A scaling limit for the length of the longest cycle in a sparse random graph

Cited by 4 publications

References 16 publications

Longest paths in random hypergraphs

Longest paths in random hypergraphs

Cycle lengths in sparse random graphs

Turán-type problems for long cycles in random and pseudo-random graphs

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