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

摘要

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 δ_1(H)≥(7/16+o(1))(n/2) contains a loose Hamilton cycle. This bound is asymptotically best possible.