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 ? > 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 + ?)n edges.