An algorithm for finding hamilton paths and cycles in random graphs

Bollobás, Béla; Frieze, Alan; Fenner, Trevor

doi:10.1007/bf02579321

Cited by 94 publications

(74 citation statements)

References 8 publications

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“…The proof of the main theorem is based on the ingenious rotation-extension technique, developed by Pósa [21], and applied later in a multitude of papers on Hamiltonicity, mostly of random or pseudorandom graphs (see for example [6], [9], [16], [19]). …”

Section: Proofmentioning

confidence: 99%

On covering expander graphs by hamilton cycles

Glebov

Krivelevich

Szabó

2012

Random Struct Algorithms

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Abstract. The problem of packing Hamilton cycles in random and pseudorandom graphs has been studied extensively. In this paper, we look at the dual question of covering all edges of a graph by Hamilton cycles and prove that if a graph with maximum degree ∆ satisfies some basic expansion properties and contains a family of (1 − o(1))∆/2 edge disjoint Hamilton cycles, then there also exists a covering of its edges by (1 + o(1))∆/2 Hamilton cycles. This implies that for every α > 0 and every p ≥ n α−1 there exists a covering of all edges of G(n, p) by (1 + o(1))np/2 Hamilton cycles asymptotically almost surely, which is nearly optimal.

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Section: Proofmentioning

confidence: 99%

On covering expander graphs by hamilton cycles

Glebov

Krivelevich

Szabó

2012

Random Struct Algorithms

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“…We use rotations and extensions together with property P1 to find a path of maximum length with large rotation endpoint sets (see for example [4], [10], [14], [15]). …”

Section: Constructing An Initial Long Pathmentioning

confidence: 99%

“…Since |S ′ t | ≫ log 11 n, property (4) implies that almost every vertex of S ′ t has degree at least 12. By (3), no two small vertices have a common neighbor, so…”

mentioning

confidence: 99%

Hamilton cycles in highly connected and expanding graphs

2009

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In this paper we prove a sufficient condition for the existence of a Hamilton cycle, which is applicable to a wide variety of graphs, including relatively sparse graphs. In contrast to previous criteria, ours is based on only two properties: one requiring expansion of "small" sets, the other ensuring the existence of an edge between any two disjoint "large" sets. We also discuss applications in positional games, random graphs and extremal graph theory.

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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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An algorithm for finding hamilton paths and cycles in random graphs

Cited by 94 publications

References 8 publications

On covering expander graphs by hamilton cycles

On covering expander graphs by hamilton cycles

Hamilton cycles in highly connected and expanding graphs

Hamiltonicity thresholds in Achlioptas processes

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