Minimum vertex degree conditions for loose Hamilton cycles in 3-uniform hypergraphs

We investigate minimum vertex degree conditions for 3-uniform hypergraphs which ensure the existence of loose Hamilton cycles. A loose Hamilton cycle is a spanning cycle in which only consecutive edges intersect and these intersections consist of precisely one vertex. We prove that every 3-uniform n-vertex (n even) hypergraph H with minimum vertex degree @d"1(H)>=(716+o(1))(n2) contains a loose Hamilton cycle. This bound is asymptotically best possible.

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