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

Abstract

We show that for sufficiently large $n$, every 3-uniform hypergraph on $n$ vertices with minimum vertex degree at least $\binom{n-1}2 - \binom{\lfloor\frac34 n\rfloor}2 + c$, where $c=2$ if $n\in 4\mathbb{N}$ and $c=1$ if $n\in 2\mathbb{N}\setminus 4\mathbb{N}$, contains a loose Hamilton cycle. This degree condition is best possible and improves on the work of Buß, Hà n and Schacht who proved the corresponding asymptotical result.

23 pages, 1 figure, Accepted for publication in JCTB