Minimum Degree Conditions for Hamilton $l$-Cycles in $k$-Uniform Hypergraphs

Abstract

We show that for $ \eta>0 $ and sufficiently large $ n $, every 5-graph on $ n $ vertices with $\delta_{2}(H)\ge (91/216+\eta)\binom{n}{3}$ contains a Hamilton 2-cycle. This minimum 2-degree condition is asymptotically best possible. Moreover, we give some related results on minimum $ d $-degree conditions in $ k $-graphs that guarantee the existence of a Hamilton $ \ell $-cycle when $\ell\le d \le k-1$ and $1\le \ell < k/2$.