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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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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DOI: https://doi.org/10.1007/s00493-008-2295-z