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 HH with minimum vertex degree δ1(H)⩾(716+o(1))(n2) contains a loose Hamilton cycle. This bound is asymptotically best possible.