Hamilton ℓ-cycles in uniform hypergraphs

Kühn, Daniela; Mycroft, Richard; Osthus, Deryk

doi:10.1016/j.jcta.2010.02.010

Cited by 76 publications

(99 citation statements)

References 18 publications

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“…Later this result was generalised to all 0 ă ă k by Kühn, Mycroft, and Osthus [9]. These results are asymptotically best possible for all k and 0 ă ă k. Hence, asymptotically, the problem of finding Hamilton -cycles in uniform hypergraphs with large minimum collective degree is solved.…”

Section: Theoremmentioning

confidence: 87%

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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Section: Theoremmentioning

confidence: 87%

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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“…In [2] t-tight paths have been studied in other settings. A similar, but more restrictive notion called ℓ-cycle appears in [11] yet in other context.…”

Section: Theorem 14 Fix R ≥ K > 2 If H Is An R-uniform Hypergraph mentioning

confidence: 91%

Hypergraph extensions of the Erdős-Gallai Theorem

Gyźri

Katona

Lemons

2016

European Journal of Combinatorics

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“…In fact, if Hamiltonian hypergraphs have received some attention (see [5,6,7]), Eulerian hypergraphs seem to have been considered in their full generality only recently in [4]. A particular case of Eulerian cycles in 3-uniform hypergraphs (called triangulated irregular networks) has been considered in [8,9,10] motivated by applications in geographic systems or in computer graphics.…”

Section: Introductionmentioning

confidence: 99%

“…For example, an undirected hypergraph H is called Hamiltonian if there exists a Hamiltonian-l cycle C in H, that is a cycle C where any two consecutive (hyper)edges intersect themselves in exactly l vertices and every vertex of H belongs to exactly one of those intersections [11,6,7]. Such a notion can also be generalized to dihypergraphs.…”

Section: Introductionmentioning

confidence: 99%

Eulerian and Hamiltonian Dicycles in Directed Hypergraphs

Araújo

Bermond

Ducoffe

2014

Discrete Math. Algorithm. Appl.

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In this article, we generalize the concepts of Eulerian and Hamiltonian digraphs to directed hypergraphs. A dihypergraph H is a pair (V(H), E(H)), where V(H) is a non-empty set of elements, called vertices, and E(H) is a collection of ordered pairs of subsets of V(H), called hyperarcs. It is Eulerian (resp. Hamiltonian) if there is a dicycle containing each hyperarc (resp. each vertex) exactly once. We first present some properties of Eulerian and Hamiltonian dihypergraphs. For example, we show that deciding whether a dihypergraph is Eulerian is an NP-complete problem. We also study when iterated line dihypergraphs are Eulerian and Hamiltonian. Finally, we study when the generalized de Bruijn dihypergraphs are Eulerian and Hamiltonian. In particular, we determine when they contain a complete Berge dicycle, i.e. an Eulerian and Hamiltonian dicycle.

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Hamilton ℓ-cycles in uniform hypergraphs

Cited by 76 publications

References 18 publications

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

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

Hypergraph extensions of the Erdős-Gallai Theorem

Eulerian and Hamiltonian Dicycles in Directed Hypergraphs

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