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

Abstract

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 $\mathcal{H}$ with minimum vertex degree ${\delta }_1(\mathcal{H})\ge \left(\frac{7}{16}+o(1)\right)\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ contains a loose Hamilton cycle. This bound is asymptotically best possible.