501 research outputs found
On extremal hypergraphs for hamiltonian cycles
We study sufficient conditions for Hamiltonian cycles in hypergraphs, and
obtain both Tur\'an- and Dirac-type results. While the Tur\'an-type 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
Dirac-type result yields a sufficient condition relying solely on the minimum
vertex degree.Comment: 13 page
Loose Hamiltonian cycles forced by large -degree - sharp version
We prove for all and the sharp minimum
-degree bound for a -uniform hypergraph on vertices
to contain a Hamiltonian -cycle if divides and is
sufficiently large. This extends a result of Han and Zhao for -uniform
hypegraphs.Comment: 14 pages, second version addresses changes arising from the referee
report
Hamilton cycles in graphs and hypergraphs: an extremal perspective
As one of the most fundamental and well-known NP-complete 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 quasi-randomness. These concepts and other recent
techniques have led to the solution of several long-standing 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.Comment: to appear in the Proceedings of the ICM 2014; due to given page
limits, this final version is slightly shorter than the previous arxiv
versio
Hamilton cycles in hypergraphs below the Dirac threshold
We establish a precise characterisation of -uniform hypergraphs with
minimum codegree close to which contain a Hamilton -cycle. As an
immediate corollary we identify the exact Dirac threshold for Hamilton
-cycles in -uniform hypergraphs. Moreover, by derandomising the proof of
our characterisation we provide a polynomial-time algorithm which, given a
-uniform hypergraph with minimum codegree close to , either finds a
Hamilton -cycle in or provides a certificate that no such cycle exists.
This surprising result stands in contrast to the graph setting, in which below
the Dirac threshold it is NP-hard to determine if a graph is Hamiltonian. We
also consider tight Hamilton cycles in -uniform hypergraphs for , giving a series of reductions to show that it is NP-hard to determine
whether a -uniform hypergraph with minimum degree contains a tight Hamilton cycle. It is therefore
unlikely that a similar characterisation can be obtained for tight Hamilton
cycles.Comment: v2: minor revisions in response to reviewer comments, most pseudocode
and details of the polynomial time reduction moved to the appendix which will
not appear in the printed version of the paper. To appear in Journal of
Combinatorial Theory, Series
Hipergráfok = Hypergraphs
A projekt célkitűzéseit sikerült megvalósÃtani. A négy év során több mint száz kiváló eredmény született, amibÅ‘l eddig 84 dolgozat jelent meg a téma legkiválóbb folyóirataiban, mint Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, stb. Számos régóta fennálló sejtést bebizonyÃtottunk, egész régi nyitott problémát megoldottunk hipergráfokkal kapcsolatban illetve kapcsolódó területeken. A problémák némelyike sok éve, olykor több évtizede nyitott volt. Nem egy közvetlen kutatási eredmény, de szintén bizonyos értékmérÅ‘, hogy a résztvevÅ‘k egyike a Norvég Királyi Akadémia tagja lett és elnyerte a Steele dÃjat. | We managed to reach the goals of the project. We achieved more than one hundred excellent results, 84 of them appeared already in the most prestigious journals of the subject, like Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, etc. We proved several long standing conjectures, solved quite old open problems in the area of hypergraphs and related subjects. Some of the problems were open for many years, sometimes for decades. It is not a direct research result but kind of an evaluation too that a member of the team became a member of the Norvegian Royal Academy and won Steele Prize
The Tur\'an number of sparse spanning graphs
For a graph , the {\em extremal number} is the maximum number of
edges in a graph of order not containing a subgraph isomorphic to . Let
and denote the minimum degree and maximum degree of
, respectively. We prove that for all sufficiently large, if is any
graph of order with , then . The condition on the maximum degree is tight up to a
constant factor. This generalizes a classical result of Ore for the case
, and resolves, in a strong form, a conjecture of Glebov, Person, and
Weps for the case of graphs. A counter-example to their more general conjecture
concerning the extremal number of bounded degree spanning hypergraphs is also
given
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