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DOI: 10.4230/LIPIcs.STACS.2016.38
URN: urn:nbn:de:0030-drops-57392
URL: http://drops.dagstuhl.de/opus/volltexte/2016/5739/
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Garbe, Frederik ; Mycroft, Richard

The Complexity of the Hamilton Cycle Problem in Hypergraphs of High Minimum Codegree

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Abstract

We consider the complexity of the Hamilton cycle decision problem when restricted to k-uniform hypergraphs H of high minimum codegree delta(H). We show that for tight Hamilton cycles this problem is NP-hard even when restricted to k-uniform hypergraphs H with delta(H) >= n/2 - C, where n is the order of H and C is a constant which depends only on k. This answers a question raised by Karpinski, Rucinski and Szymanska. Additionally we give a polynomial-time algorithm which, for a sufficiently small constant epsilon > 0, determines whether or not a 4-uniform hypergraph H on n vertices with delta(H) >= n/2 - epsilon * n contains a Hamilton 2-cycle. This demonstrates that some looser Hamilton cycles exhibit interestingly different behaviour compared to tight Hamilton cycles. A key part of the proof is a precise characterisation of all 4-uniform hypergraphs H on n vertices with delta(H) >= n/2 - epsilon * n which do not contain a Hamilton 2-cycle; this may be of independent interest. As an additional corollary of this characterisation, we obtain an exact Dirac-type bound for the existence of a Hamilton 2-cycle in a large 4-uniform hypergraph.

BibTeX - Entry

@InProceedings{garbe_et_al:LIPIcs:2016:5739,
  author =	{Frederik Garbe and Richard Mycroft},
  title =	{{The Complexity of the Hamilton Cycle Problem in Hypergraphs of High Minimum Codegree}},
  booktitle =	{33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)},
  pages =	{38:1--38:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-001-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{47},
  editor =	{Nicolas Ollinger and Heribert Vollmer},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/5739},
  URN =		{urn:nbn:de:0030-drops-57392},
  doi =		{10.4230/LIPIcs.STACS.2016.38},
  annote =	{Keywords: Hamilton cycles, hypergraphs, graph algorithms}
}

Keywords: Hamilton cycles, hypergraphs, graph algorithms
Seminar: 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)
Issue Date: 2016
Date of publication: 16.02.2016


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