Results 1  10
of
16
Embedding large subgraphs into dense graphs
, 2009
"... What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac’s theorem on Hamilton cycles and Tutte’s theorem on perfect matchings. Perfect matchings are generalized by perfect Fpackings, where instead of covering ..."
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Cited by 34 (11 self)
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What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac’s theorem on Hamilton cycles and Tutte’s theorem on perfect matchings. Perfect matchings are generalized by perfect Fpackings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect Fpacking, so as in the case of Dirac’s theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect Fpacking. The Regularity lemma of Szemerédi and the Blowup lemma of Komlós, Sárközy and Szemerédi have proved to be powerful tools in attacking such problems and quite recently, several longstanding problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on Fpackings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved.
Perfect matchings and Hamilton cycles in hypergraphs with large degrees
, 2009
"... We establish a new lower bound on the lwise collective minimum degree which guarantees the existence of a perfect matching in a kuniform hypergraph, where 1 ≤ l < k/2. For l = 1, this improves a long standing bound by Daykin and Häggkvist [4]. Our proof is a modification of the approach of Han ..."
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Cited by 15 (1 self)
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We establish a new lower bound on the lwise collective minimum degree which guarantees the existence of a perfect matching in a kuniform hypergraph, where 1 ≤ l < k/2. For l = 1, this improves a long standing bound by Daykin and Häggkvist [4]. Our proof is a modification of the approach of Han, Person, and Schacht from [8]. In addition, we fill a gap left by the results solving a similar question for the existence of Hamilton cycles.
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.
Hamilton ℓcycles in uniform hypergraphs
 JOURNAL OF COMBINATORIAL THEORY. SERIES A
"... We say that a kuniform hypergraph C is an ℓcycle if there exists a cyclic ordering of the vertices of C such that every edge of C consists of k consecutive vertices and such that every pair of consecutive edges (in the natural ordering of the edges) intersects in precisely ℓ vertices. We prove th ..."
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Cited by 12 (3 self)
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We say that a kuniform hypergraph C is an ℓcycle if there exists a cyclic ordering of the vertices of C such that every edge of C consists of k consecutive vertices and such that every pair of consecutive edges (in the natural ordering of the edges) intersects in precisely ℓ vertices. We prove that if 1 ≤ ℓ < k and k − ℓ does not divide k then any kuniform hypergraph on n vertices with minimum degree at least nd k k− ` e(k−`) +o(n) contains a Hamilton ℓcycle. This confirms a conjecture of Hàn and Schacht. Together with results of Rödl, Ruciński and Szemerédi, our result asymptotically determines the minimum degree which forces an `cycle for any ` with 1 ≤ ℓ < k.
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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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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Hamilton cycles in graphs and hypergraphs: an extremal perspective
"... As one of the most fundamental and wellknown NPcomplete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion and quasirandomness. These concepts and other recent techn ..."
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Cited by 2 (0 self)
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As one of the most fundamental and wellknown NPcomplete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion and quasirandomness. These concepts and other recent techniques have led to the solution of several longstanding problems in the area. New aspects have also emerged, such as resilience, robustness and the study of Hamilton cycles in hypergraphs. We survey these developments and highlight open problems, with an emphasis on extremal and probabilistic approaches.