Packing, counting and covering Hamilton cycles in random directed graphs

Ferber, Asaf; Kronenberg, Gal; Long, Eoin

doi:10.1007/s11856-017-1518-7

Cited by 17 publications

(21 citation statements)

References 40 publications

(74 reference statements)

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“…Let  ba denote the set of Q ∈  which are not contained in a copy of C in Q ∪ D 2 . From Equation 8 we have E(| ba |) = o(||).…”

Section: Stagementioning

confidence: 99%

“…Here we deal with the problems of counting and packing arbitrary oriented Hamilton cycles in

D \sim scriptD false(n, p false)

, for edge‐densities

p \geq \log^{C} false(n false) false/ n

. The analogous problems regarding the "consistently oriented" Hamilton cycles has been recently treated by Kronenberg and the authors in. However, the proof method there is inapplicable to the arbitrary oriented case.…”

Section: Introductionmentioning

confidence: 99%

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Packing and counting arbitrary Hamilton cycles in random digraphs

Ferber

Long

2018

Random Struct Algorithms

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We prove packing and counting theorems for arbitrarily oriented Hamilton cycles in scriptD(n, p) for nearly optimal p (up to a logcn factor). In particular, we show that given t = (1 − o(1))np Hamilton cycles C1,…,Ct, each of which is oriented arbitrarily, a digraph scriptD∼scriptD(n, p) w.h.p. contains edge disjoint copies of C1,…,Ct, provided p=ωfalse(log3nfalse/nfalse). We also show that given an arbitrarily oriented n‐vertex cycle C, a random digraph scriptD∼scriptD(n, p) w.h.p. contains (1 ± o(1))n!pn copies of C, provided p≥log1+ofalse(1false)nfalse/n.

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“…Let  ba denote the set of Q ∈  which are not contained in a copy of C in Q ∪ D 2 . From Equation 8 we have E(| ba |) = o(||).…”

Section: Stagementioning

confidence: 99%

“…Here we deal with the problems of counting and packing arbitrary oriented Hamilton cycles in

D \sim scriptD false(n, p false)

, for edge‐densities

p \geq \log^{C} false(n false) false/ n

Section: Introductionmentioning

confidence: 99%

Packing and counting arbitrary Hamilton cycles in random digraphs

Ferber

Long

2018

Random Struct Algorithms

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show abstract

“…Conditioning on these specific

v_{1}, v_{R} \in U

, we show that with high probability there is a Hamiltonian path from

v_{1} to v_{R}

in the subgraph of

H_{p_{0}, e}

induced by

U

. By a result of [, Theorem 1.3] if

D

is a

p_{0}

‐edge percolation of the complete directed graph with

R

vertices with

p_{0} = \frac{p}{3 \log (R)} = \frac{10 \log^{4} (R)}{R}

, then with high probability every edge of

D

is contained in some Hamiltonian cycle in

D

. Note that the probability that

H_{p_{0}, e}

contains a Hamiltonian path from

v_{1} to v_{R}

is equal to the probability that

H_{p_{0}, e}

contains a Hamiltonian cycle that goes through the edge

(v_{1} \to v_{R})

, conditioned on the event that

(v_{1} \to v_{R})

is an edge in

H_{p_{0}, e}

.…”

Section: Hamiltonicity and Percolationmentioning

confidence: 90%

On percolation and ‐hardness

Bennett

Reichman

Shinkar

2018

Random Struct Algorithms

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The edge-percolation and vertex-percolation random graph models start with an arbitrary graph G, and randomly delete edges or vertices of G with some fixed probability. We study the computational complexity of problems whose inputs are obtained by applying percolation to worst-case instances. Specifically, we show that a number of classical N P-hard problems on graphs remain essentially as hard on percolated instances as they are in the worst-case (assuming N P BPP). We also prove hardness results for other N P-hard problems such as Constraint Satisfaction Problems and Subset-Sum, with suitable definitions of random deletions. Along the way, we establish that for any given graph G the independence number α(G) and the chromatic number χ(G) are robust to percolation in the following sense. Given a graph G, let G � be the graph obtained by randomly deleting edges of G with some probability p ∈ (0, 1). We show that if α(G) is small, then α(G � ) remains small with probability at least 0.99. Similarly, we show that if χ(G) is large, then χ(G � ) remains large with probability at least 0.99. We believe these results are of independent interest.chromatic number, hardness of approximation, independence number, percolation, random subgraphs Random Struct Alg. 2018;00:1-30.wileyonlinelibrary.com/journal/rsa

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“…The main step in the proof of this theorem is to construct many almost Hamilton decompositions, each of which can be further completed to a full decomposition. This is done by extending some ideas from [6] and differs from the approach used in [15]. In particular, we obtain a new and much simpler proof for the approximate version of Kelly's conjecture, originally established by Kühn, Osthus and Treglown in [16].…”

Section: Introductionmentioning

confidence: 99%

Counting Hamilton Decompositions of Oriented Graphs

Ferber

Long

Sudakov

2017

International Mathematics Research Notices

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A Hamilton cycle in a directed graph G is a cycle that passes through every vertex of G. A Hamiltonian decomposition of G is a partition of its edge set into disjoint Hamilton cycles. In the late 60s Kelly conjectured that every regular tournament has a Hamilton decomposition. This conjecture was recently settled by Kühn and Osthus [15], who proved more generally that every r-regular n-vertex oriented graph G (without antiparallel edges) with r = cn for some fixed c > 3/8 has a Hamiltonian decomposition, provided n = n(c) is sufficiently large. In this paper we address the natural question of estimating the number of such decompositions of G and show that this number is n (1−o(1))cn 2 . In addition, we also obtain a new and much simpler proof for the approximate version of Kelly's conjecture.

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Packing, counting and covering Hamilton cycles in random directed graphs

Cited by 17 publications

References 40 publications

Packing and counting arbitrary Hamilton cycles in random digraphs

Packing and counting arbitrary Hamilton cycles in random digraphs

On percolation and ‐hardness

Counting Hamilton Decompositions of Oriented Graphs

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