Loose Hamilton cycles in hypergraphs

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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“…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: 72%

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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“…The following proposition allows us to do this without difficulty; we shall only delete vertices which do not lie in the sets X j . [15], which was proved by a straightforward application of Azuma's inequality, states that for any v ∈ W this inequality holds with probability at least 1 − 1/n 2 , so taking a union bound over all vertices of W proves the result. 5.5.…”

Section: Theorem 54 ([13]) Suppose Thatmentioning

confidence: 94%

“…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: 90%

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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“…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%

“…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%

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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Loose Hamilton cycles in hypergraphs

Cited by 67 publications

References 16 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

Packing k-partite k-uniform hypergraphs

Eulerian and Hamiltonian Dicycles in Directed Hypergraphs

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