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Packing Hamilton Cycles in Random and Pseudorandom Hypergraphs
, 2010
"... We say that a kuniform hypergraph C is a Hamilton cycle of type ℓ, for some 1 ≤ ℓ ≤ k, if there exists a cyclic ordering of the vertices of C such that every edge consists of k consecutive vertices and for every pair of consecutive edges Ei−1, Ei in C (in the natural ordering of the edges) we have ..."
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Cited by 14 (6 self)
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We say that a kuniform hypergraph C is a Hamilton cycle of type ℓ, for some 1 ≤ ℓ ≤ k, if there exists a cyclic ordering of the vertices of C such that every edge consists of k consecutive vertices and for every pair of consecutive edges Ei−1, Ei in C (in the natural ordering of the edges) we have Ei−1 − Ei  = ℓ. We prove that for ℓ ≤ k ≤ 2ℓ, with high probability almost all edges of a random kuniform hypergraph H(n, p, k) with p(n) ≫ log 2 n/n can be decomposed into edge disjoint type ℓ Hamilton cycles. We also provide sufficient conditions for decomposing almost all edges of a pseudorandom kuniform hypergraph into type ℓ Hamilton cycles, for ℓ ≤ k ≤ 2ℓ. For the case ℓ = k these results show that almost all edges of corresponding random and pseudorandom hypergraphs can be packed into disjoint perfect matchings.
Packing tight Hamilton cycles in 3uniform hypergraphs
"... Let H be a 3uniform hypergraph with n vertices. A tight Hamilton cycle C ⊂ H is a collection of n edges for which there is an ordering of the vertices v1,..., vn such that every triple of consecutive vertices {vi, vi+1, vi+2} is an edge of C (indices are considered modulo n). We develop new techniq ..."
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Cited by 8 (4 self)
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Let H be a 3uniform hypergraph with n vertices. A tight Hamilton cycle C ⊂ H is a collection of n edges for which there is an ordering of the vertices v1,..., vn such that every triple of consecutive vertices {vi, vi+1, vi+2} is an edge of C (indices are considered modulo n). We develop new techniques which enable us to prove that under certain natural pseudorandom conditions, almost all edges of H can be covered by edgedisjoint tight Hamilton cycles, for n divisible by 4. Consequently, we derive the corollary that random 3uniform hypergraphs can be almost completely packed with tight Hamilton cycles whp, for n divisible by 4 and p not too small. Along the way, we develop a similar result for packing Hamilton cycles in pseudorandom digraphs with even numbers of vertices. 1
MINIMUM VERTEX DEGREE CONDITIONS FOR LOOSE HAMILTON CYCLES IN 3UNIFORM HYPERGRAPHS
"... We investigate minimum vertex degree conditions for 3uniform hypergraphs which ensure the existence of loose Hamilton cycles. A loose Hamilton cycle is a spanning cycle in which consecutive edges intersect in a single vertex. We prove that every 3uniform nvertex (n even) hypergraph H with minim ..."
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Cited by 5 (0 self)
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We investigate minimum vertex degree conditions for 3uniform hypergraphs which ensure the existence of loose Hamilton cycles. A loose Hamilton cycle is a spanning cycle in which consecutive edges intersect in a single vertex. We prove that every 3uniform nvertex (n even) hypergraph H with minimum vertex degree δ1(H) ≥ ( 7 16 + o(1)) ( n) contains a loose Hamil2 ton cycle. This bound is asymptotically best possible.
Loose Hamilton cycles in hypergraphs
, 2008
"... We prove that any kuniform hypergraph on n vertices with minimum degree n at least + o(n) contains a loose Hamilton cycle. The proof strategy is similar to that 2(k−1) used by Kühn and Osthus for the 3uniform case. Though some additional difficulties arise in the kuniform case, our argument her ..."
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Cited by 5 (1 self)
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We prove that any kuniform hypergraph on n vertices with minimum degree n at least + o(n) contains a loose Hamilton cycle. The proof strategy is similar to that 2(k−1) used by Kühn and Osthus for the 3uniform case. Though some additional difficulties arise in the kuniform case, our argument here is considerably simplified by applying the recent hypergraph blowup lemma of Keevash.
On extremal hypergraphs for Hamiltonian cycles
 European J. Combin
"... Abstract. We study sufficient conditions for Hamiltonian cycles in hypergraphs, and obtain both Turán and Diractype results. While the Turántype result gives an exact threshold for the appearance of a Hamiltonian cycle in a hypergraph depending only on the extremal number of a certain path, the ..."
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Cited by 5 (1 self)
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Abstract. We study sufficient conditions for Hamiltonian cycles in hypergraphs, and obtain both Turán and Diractype results. While the Turántype result gives an exact threshold for the appearance of a Hamiltonian cycle in a hypergraph depending only on the extremal number of a certain path, the Diractype result yields a sufficient condition relying solely on the minimum vertex degree.
Minimum vertex degree threshold for loose Hamilton cycles
 in 3graphs. Journal of Combinatorial Theory, Series B, accepted
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Minimum codegree threshold for hamilton cycles in kuniform hypergraphs
 Journal of Combinatorial Theory, Series A
, 2015
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