An approximate Dirac-type theorem for k-uniform hypergraphs

Abstract

A k-uniform hypergraph is hamiltonian if for some cyclic ordering of its vertex set, every k consecutive vertices form an edge. In 1952 Dirac proved that if the minimum degree in an n-vertex graph is at least n/2 then the graph is hamiltonian.

We prove an approximate version of an analogous result for uniform hypergraphs: For every K ≥ 3 and γ > 0, and for all n large enough, a sufficient condition for an n-vertex k-uniform hypergraph to be hamiltonian is that each (k − 1)-element set of vertices is contained in at least (1/2 + γ)n edges.

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Correspondence to Vojtěch Rödl.

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Research supported by NSF grant DMS-0300529.

Research supported by KBN grant 2P03A 015 23 and N201036 32/2546. Part of research performed at Emory University, Atlanta.

Research supported by NSF grant DMS-0100784.

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Rödl, V., Szemerédi, E. & Ruciński, A. An approximate Dirac-type theorem for k-uniform hypergraphs. Combinatorica 28, 229–260 (2008). https://doi.org/10.1007/s00493-008-2295-z

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Mathematics Subject Classification (2000)

  • 05C65
  • 05C45
  • 05D05