Hamiltonian paths and cycles in some 4-uniform hypergraphs

In 1999, Katona and Kierstead conjectured that if a $k$-uniform hypergraph $cal H$ on $n$ vertices has minimum co-degree $lfloor frac{n-k+3}{2}rfloor$, i.e., each set of $k-1$ vertices is contained in at least $lfloor frac{n-k+3}{2}rfloor$ edges, then it has a Hamiltonian cycle. R{o}dl, Ruci{n}ski and Szemer{e}di in 2011 proved that the conjecture is true when $k=3$ and $n$ is large. We show that this Katona-Kierstead conjecture holds if $k=4$, $n$ is large, and $V({cal H})$ has a partition $A$, $B$ such that $|A|=lceil n/2rceil$, $|{ein E({cal H}):|e cap A|=2}| <epsilon n^4$.