Loose Hamilton cycles in 3-uniform hypergraphs of high minimum degree

Kühn, Daniela; Osthus, Deryk

doi:10.1016/j.jctb.2006.02.004

Cited by 115 publications

(118 citation statements)

References 19 publications

(51 reference statements)

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“…For k-graphs with k ≥ 3, a cycle may be defined in many ways (see, e.g., [3], [13] and [14]). Here by a cycle of length l ≥ k + 1 we mean a k-graph whose vertices can be ordered cyclically v 1 , .…”

Section: Introductionmentioning

confidence: 99%

Computational Complexity of the Hamiltonian Cycle Problem in Dense Hypergraphs

Karpiński

Ruciński

Szymańska

2010

LATIN 2010: Theoretical Informatics

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Abstract.We study the computational complexity of deciding the existence of a Hamiltonian Cycle in some dense classes of k-uniform hypergraphs. Those problems turned out to be, along with the hypergraph Perfect Matching problems, exceedingly hard, and there is a renewed algorithmic interest in them. In this paper we design a polynomial time algorithm for the Hamiltonian Cycle problem for k-uniform hypergraphs with density at least 1 2 + , > 0. In doing so, we depend on a new method of constructing Hamiltonian cycles from (purely) existential statements which could be of independent interest. On the other hand, we establish NP-completeness of that problem for density at least 1 k − . Our results seem to be the first complexity theoretic results for the Dirac-type dense hypergraph classes.

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Section: Introductionmentioning

confidence: 99%

Computational Complexity of the Hamiltonian Cycle Problem in Dense Hypergraphs

Karpiński

Ruciński

Szymańska

2010

LATIN 2010: Theoretical Informatics

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“…Let K 3 4 − 2e, K 3 4 − e and K 3 4 denote the 3-graphs on 4 vertices with 2, 3 and 4 edges respectively. The value of δ(K 3 4 − 2e, n) was found to be n/4 + o(n) by Kühn and Osthus [21]; recently Czygrinow, DeBiasio and Nagle [4] found the exact value for large n to be either n/4 or n/4 + 1 according to the parity of n/4. Lo and Markström [25,27] showed that δ(K 3 4 − e, n) = n/2 + o(n) and that δ(K 3 4 , n) = 3n/4 + o(n).…”

Section: Perfect Packings In Graphsmentioning

confidence: 99%

“…and Osthus [21] and then for p = 1, q = 2k − 2 by Keevash, Kühn, Mycroft and Osthus [15]; the proof given here is essentially identical, but is included for completeness.…”

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confidence: 91%

Packing k-partite k-uniform hypergraphs

Mycroft

2016

Journal of Combinatorial Theory, Series A

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Abstract. Let G and H be k-graphs (k-uniform hypergraphs); then a perfect H-packing in G is a collection of vertex-disjoint copies of H in G which together cover every vertex of G. For any fixed H let δ(H, n) be the minimum δ such that any k-graph G on n vertices with minimum codegree δ(G) ≥ δ contains a perfect H-packing. The problem of determining δ(H, n) has been widely studied for graphs (i.e. 2-graphs), but little is known for k ≥ 3. Here we determine the asymptotic value of δ(H, n) for all complete k-partite k-graphs H, as well as a wide class of other k-partite k-graphs. In particular, these results provide an asymptotic solution to a question of Rödl and Ruciński on the value of δ(H, n) when H is a loose cycle. We also determine asymptotically the codegree threshold needed to guarantee an H-packing covering all but a constant number of vertices of G for any complete k-partite k-graph H.

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“…In [10] they proved an asymptotically sharp bound on the minimum collective degree which ensures the existence of loose Hamilton cycles in 3-uniform hypergraphs. This result was generalised to higher uniformity by the last two authors [4] and independently by Keevash, Kühn, Osthus and Mycroft in [7].…”

Section: Theoremmentioning

confidence: 99%

“…As mentioned above, Theorem 3 is best possible up to the error constant γ as seen by the following construction from [10].…”

Section: Then H Contains a Loose Hamilton Cyclementioning

confidence: 99%

Minimum vertex degree conditions for loose Hamilton cycles in 3-uniform hypergraphs

Buí

Hín

Schacht

2013

Journal of Combinatorial Theory, Series B

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Abstract. We investigate minimum vertex degree conditions for 3-uniform hypergraphs which ensure the existence of loose Hamilton cycles. A loose Hamilton cycle is a spanning cycle in which only consecutive edges intersect and these intersections consist of precisely one vertex.We prove that every 3-uniform n-vertex (n even) hypergraph H with minimum vertex degree δ 1 pHq ě`7 16`o p1q˘`n 2˘c ontains a loose Hamilton cycle. This bound is asymptotically best possible. §1. IntroductionWe consider k-uniform hypergraphs H " pV, Eq with vertex sets V " V pHq and edge sets E " EpHq Ď`V k˘, where`V k˘d enotes the family of all k-element subsets of the set V . We often identify a hypergraph H with its edge set, i.e., H Ď`V k˘, and for an edge tv 1 , . . . , v k u P H we often suppress the enclosing braces and write v 1 . . . v k P H instead. Given a k-uniform hypergraph H " pV, Eq and a set S " tv 1 , . . . , v s u P`V sl et degpSq " degpv 1 , . . . , v s q denote the number of edges of H containing the set S and let N pSq " N pv 1 , . . . , v s q denote the set of those pk´sq-element sets T P`V k´s˘s uch that S Y T forms an edge in H. We denote by δ s pHq the minimum s-degree of H, i.e., the minimum of degpSq over all s-element sets S Ď V . For s " 1 the corresponding minimum degree δ 1 pHq is referred to as minimum vertex degree whereas for s " k´1 we call the corresponding minimum degree δ k´1 pHq the minimum collective degree of H.We study sufficient minimum degree conditions which enforce the existence of spanning, so-called Hamilton cycles. A k-uniform hypergraph C is called an -cycle if there is a cyclic ordering of the vertices of C such that every edge consists of k consecutive vertices, every vertex is contained in an edge and two consecutive edges (where the ordering of the edges 2010 Mathematics Subject Classification. 05C65 (primary), 05C45 (secondary).

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Loose Hamilton cycles in 3-uniform hypergraphs of high minimum degree

Cited by 115 publications

References 19 publications

Computational Complexity of the Hamiltonian Cycle Problem in Dense Hypergraphs

Computational Complexity of the Hamiltonian Cycle Problem in Dense Hypergraphs

Packing k-partite k-uniform hypergraphs

Minimum vertex degree conditions for loose Hamilton cycles in 3-uniform hypergraphs

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