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 consecutive edges intersect in a single vertex. We prove that every 3-uniform n-vertex (n even) hypergraph H with minimum vertex degree δ 1 ( H ) ⩾ ( 7 16 + o ( 1 ) ) ( n 2 ) contains a loose Hamilton cycle. This bound is asymptotically best possible.