On the Number of Hamilton Cycles in Sparse Random Graphs

Glebov, Roman; Krivelevich, Michael

doi:10.1137/120884316

Cited by 28 publications

(38 citation statements)

References 35 publications

(62 reference statements)

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“…In comparison, in the Erdős–Rényi random graph model G n,p , where each edge appears independently with probability p , the expected number of Hamiltonian paths is about n when

p \sim \frac{e}{n}

, but Hamiltonian paths don't appear until

p \sim \frac{\log n}{n}

. Furthermore, for Hamiltonian cycles in the random graph process, at the moment Hamiltonicity is achieved, the number of Hamiltonian cycles jumps from 0 to

{[(1 + o (1)) \frac{\log n}{e}]}^{n}

, as shown recently by Glebov and Krivelevich (improving an earlier result of Cooper and Frieze ). It may therefore come as a surprise that random edge orderings often have Hamiltonian paths, despite the extremely low expected value.…”

Section: Introductionsupporting

confidence: 58%

Increasing Hamiltonian paths in random edge orderings

Lavrov

Loh

2015

Random Struct Algorithms

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Let f be an edge ordering of Kn: a bijection E(Kn)→{1,2,…,(n2)}. For an edge e∈E(Kn), we call f(e) the label of e. An increasing path in Kn is a simple path (visiting each vertex at most once) such that the label on each edge is greater than the label on the previous edge. We let S(f) be the number of edges in the longest increasing path. Chvátal and Komlós raised the question of estimating m(n): the minimum value of S(f) over all orderings f of Kn. The best known bounds on m(n) are n−1≤m(n)≤(12+o(1))n, due respectively to Graham and Kleitman, and to Calderbank, Chung, and Sturtevant. Although the problem is natural, it has seen essentially no progress for three decades. In this paper, we consider the average case, when the ordering is chosen uniformly at random. We discover the surprising result that in the random setting, S(f) often takes its maximum possible value of n – 1 (visiting all of the vertices with an increasing Hamiltonian path). We prove that this occurs with probability at least about 1/ e. We also prove that with probability 1‐ o(1), there is an increasing path of length at least 0.85 n, suggesting that this Hamiltonian (or near‐Hamiltonian) phenomenon may hold asymptotically almost surely. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 588–611, 2016

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“…In comparison, in the Erdős–Rényi random graph model G n,p , where each edge appears independently with probability p , the expected number of Hamiltonian paths is about n when

p \sim \frac{e}{n}

, but Hamiltonian paths don't appear until

p \sim \frac{\log n}{n}

. Furthermore, for Hamiltonian cycles in the random graph process, at the moment Hamiltonicity is achieved, the number of Hamiltonian cycles jumps from 0 to

{[(1 + o (1)) \frac{\log n}{e}]}^{n}

Section: Introductionsupporting

confidence: 58%

Increasing Hamiltonian paths in random edge orderings

Lavrov

Loh

2015

Random Struct Algorithms

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“…This lemma conveniently summarizes the application of the Ore‐Ryser theorem and Egorychev‐Falikman theorem to give a lower bound for the number of 1‐factors in a pseudorandom almost‐regular digraph. (We remark that the statement of , Lemma 4] is for oriented graphs, but the proof applies equally well in the setting of arbitrary directed graphs.) Lemma Let D be a directed graph on

[n]

(with loops allowed), and consider some

r = r false(n false) ≫ \log \log n

.…”

Section: Proof Outline and Ingredientsmentioning

confidence: 99%

“…lower bound on the total number of 1‐factors in

D_{*}'

(in Lemma 6 in the next section). This will be accomplished with a greedy matching argument combined with the following lemma of Glebov and Krivelevich , Lemma 4]. This lemma conveniently summarizes the application of the Ore‐Ryser theorem and Egorychev‐Falikman theorem to give a lower bound for the number of 1‐factors in a pseudorandom almost‐regular digraph.…”

Section: Proof Outline and Ingredientsmentioning

confidence: 99%

“…For example, Cooper and Frieze proved that above the (hitting‐time) threshold for Hamiltonicity, random graphs have at least

{(\log n)}^{(1 - ε) n}

Hamilton cycles for any fixed

ε > 0

. More recently, Glebov and Krivelevich proved that if m is above this Hamiltonicity threshold, then a random m ‐edge undirected graph in fact has

n! {false(m / false(\begin{matrix} n \\ 2 \end{matrix} false) (1 + o (1)) false)}^{n}

Hamilton cycles. See also the work of Janson on the number of Hamilton cycles in denser random graphs.…”

Section: Introductionmentioning

confidence: 99%

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Counting Hamilton cycles in sparse random directed graphs

Ferber¹,

Kwan²,

Sudakov³

2018

Random Struct Algorithms

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Let Dfalse(n,pfalse) be the random directed graph on n vertices where each of the nfalse(n−1false) possible arcs is present independently with probability p. A celebrated result of Frieze shows that if p≥false(logn+ωfalse(1false)false)/n then Dfalse(n,pfalse) typically has a directed Hamilton cycle, and this is best possible. In this paper, we obtain a strengthening of this result, showing that under the same condition, the number of directed Hamilton cycles in Dfalse(n,pfalse) is typically n!false(pfalse(1+ofalse(1false)false)false)n. We also prove a hitting‐time version of this statement, showing that in the random directed graph process, as soon as every vertex has in‐/out‐degrees at least 1, there are typically n!false(logn/nfalse(1+ofalse(1false)false)false)n directed Hamilton cycles.

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“…In the random/pseudo-random setting, building on ideas of Krivelevich [25], in [17] Glebov and Krivelevich showed that for p ≥ ln n+ln ln n+ω (1) n and for a typical G ∼ G(n, p) we have h(G) = (1 − o(1)) n n!p n . That is, the number of Hamilton cycles is, up to a sub-exponential factor, concentrated around its mean.…”

Section: Introductionmentioning

confidence: 99%

Packing, counting and covering Hamilton cycles in random directed graphs

2017

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A Hamilton cycle in a digraph is a cycle that passes through all the vertices, where all the arcs are oriented in the same direction. The problem of finding Hamilton cycles in directed graphs is well studied and is known to be hard. One of the main reasons for this, is that there is no general tool for finding Hamilton cycles in directed graphs comparable to the so called Posá 'rotation-extension' technique for the undirected analogue. Let D(n, p) denote the random digraph on vertex set [n], obtained by adding each directed edge independently with probability p. Here we present a general and a very simple method, using known results, to attack problems of packing and counting Hamilton cycles in random directed graphs, for every edge-probability p > log C (n)/n. Our results are asymptotically optimal with respect to all parameters and apply equally well to the undirected case.

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On the Number of Hamilton Cycles in Sparse Random Graphs

Abstract: Abstract. We prove that the number of Hamilton cycles in the random graph. Furthermore, we prove the hitting-time version of this statement, showing that in the random graph process, the edge that creates a graph of minimum degree 2 creates ln n e n (1+o(1)) n Hamilton cycles a.a.s.

Cited by 28 publications

References 35 publications

Increasing Hamiltonian paths in random edge orderings

Increasing Hamiltonian paths in random edge orderings

Counting Hamilton cycles in sparse random directed graphs

Packing, counting and covering Hamilton cycles in random directed graphs

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