Hamilton cycles in hypergraphs below the Dirac threshold

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F18%3A00493792" target="_blank" >RIV/67985840:_____/18:00493792 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1016/j.jctb.2018.04.010" target="_blank" >http://dx.doi.org/10.1016/j.jctb.2018.04.010</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1016/j.jctb.2018.04.010" target="_blank" >10.1016/j.jctb.2018.04.010</a>

Result language
angličtina
Original language name
Hamilton cycles in hypergraphs below the Dirac threshold
Original language description
We establish a precise characterisation of 4-uniform hypergraphs with minimum codegree close to n/2 which contain a Hamilton 2-cycle. As an immediate corollary we identify the exact Dirac threshold for Hamilton 2-cycles in 4-uniform hypergraphs. Moreover, by derandomising the proof of our characterisation we provide a polynomial-time algorithm which, given a 4-uniform hypergraph H with minimum codegree close to n/2, either finds a Hamilton 2-cycle in H 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 k-uniform hypergraphs H for k≥3, giving a series of reductions to show that it is NP-hard to determine whether a k-uniform hypergraph H with minimum degree δ(H)≥[Formula presented]|V(H)|−O(1) contains a tight Hamilton cycle.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2018
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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