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.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. Rödl, Ruciński and Szemerédi 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 \(|A|=\lceil n/2\rceil \), \(|\{e\in E({\mathcal {H}}):|e \cap A|=2\}| <\epsilon n^{4}\) for a fixed small constant \(\epsilon >0\).