Hamiltonian circuits in random graphs

Pósa, L.

doi:10.1016/0012-365x(76)90068-6

Cited by 406 publications

(300 citation statements)

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“…The technique involves the rotation of paths attributed to Posa [10]. The underlying idea is that if there is a path of maximal length then edges from the endpoints of the path must connect to the center of the path and can be used to rotate the path to create a new maximal path.…”

Section: Introductionmentioning

confidence: 99%

Small Spectral Gap in the Combinatorial Laplacian Implies Hamiltonian

Butler

Chung

2010

Ann. Comb.

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Abstract. We consider the spectral and algorithmic aspects of the problem of finding a Hamiltonian cycle in a graph. We show that a sufficient condition for a graph being Hamiltonian is that the nontrivial eigenvalues of the combinatorial Laplacian are sufficiently close to the average degree of the graph. An algorithm is given for the problem of finding a Hamiltonian cycle in graphs with bounded spectral gaps which has complexity of order n c ln n .

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Section: Introductionmentioning

confidence: 99%

Small Spectral Gap in the Combinatorial Laplacian Implies Hamiltonian

Butler

Chung

2010

Ann. Comb.

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“…As we already mentioned in the introduction, a key tool of our proof is the celebrated rotationextension technique, developed by Pósa [24] and applied in several subsequent papers on Hamiltonicity of random and pseudo-random graphs (cf., e.g., [11], [17], [20], [25]). Below we will cover this approach, including a key lemma and its proof.…”

Section: Pósa's Rotation-extension Techniquementioning

confidence: 99%

Hamiltonicity thresholds in Achlioptas processes

Krivelevich

Lubetzky

Sudakov

2010

Random Struct. Alg.

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In this paper we analyze the appearance of a Hamilton cycle in the following random process. The process starts with an empty graph on n labeled vertices. At each round we are presented with K = K(n) edges, chosen uniformly at random from the missing ones, and are asked to add one of them to the current graph. The goal is to create a Hamilton cycle as soon as possible.We show that this problem has three regimes, depending on the value of K. For K = o(log n), the threshold for Hamiltonicity is 1+o(1) 2K n log n, i.e., typically we can construct a Hamilton cycle K times faster that in the usual random graph process. When K = ω(log n) we can essentially waste almost no edges, and create a Hamilton cycle in n + o(n) rounds with high probability. Finally, in the intermediate regime where K = Θ(log n), the threshold has order n and we obtain upper and lower bounds that differ by a multiplicative factor of 3.

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“…In our proofs we shall use the following tool due to Pósa [20]. Let P be a path in a graph G, say from u and v. Given a vertex x ∈ P , we write x − for the vertex preceding x on P , and x + for the vertex following x on P (whenever these exist).…”

Section: Forcing Long Cyclesmentioning

confidence: 99%

“…As all paths derived from P have the same vertex set as P , we have S(P ) ⊆ V (P ). The following lemma is a one-sided variant of Pósa's Lemma [20]; see also [7,Lemma 6.3]. Lemma 2.6 Let G be a graph, let P = u .…”

Section: Forcing Long Cyclesmentioning

confidence: 99%

Global Connectivity And Expansion: Long Cycles and Factors In f-Connected Graphs

et al. 2006

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Given a function f : N → R, call an n-vertex graph f -connected if separating off k vertices requires the deletion of at least f (k) vertices whenever k ≤ (n − f (k))/2. This is a common generalization of vertex connectivity (when f is constant) and expansion (when f is linear). We show that an f -connected graph contains a cycle of length linear in n if f is any linear function, contains a 1-factor and a 2-factor if f (k) ≥ 2k + 1, and contains a Hamilton cycle if f (k) ≥ 2(k + 1) 2 . We conjecture that linear growth of f suffices to imply hamiltonicity.

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Hamiltonian circuits in random graphs

Cited by 406 publications

References 1 publication

Small Spectral Gap in the Combinatorial Laplacian Implies Hamiltonian

Small Spectral Gap in the Combinatorial Laplacian Implies Hamiltonian

Hamiltonicity thresholds in Achlioptas processes

Global Connectivity And Expansion: Long Cycles and Factors In f-Connected Graphs

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