Hamiltonian paths and cycles in some 4-uniform hypergraphs

In 1999, Katona and Kierstead conjectured that if a $k$-uniform hypergraph $\mathcal{H}$ on $n$ vertices has minimum co-degree $\lfloor \frac{n-k+3}{2}\rfloor$,i\mathrm{.}e\mathrm{.},eachsetof$k-1$ 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 $V(\mathcal{H})$ has a partition $A$, $B$ such that $\mathrm{\mid }A\mathrm{\mid }=\lceil n\mathrm{/}2\rceil$, $|{e\in E({\cal H}):|e \cap A|=2}| <\epsilon n^4$forafixedsmallconstant$\epsilon>0$.