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.e., each set of \(k-1\) is contained in at least \) edges, then it Hamiltonian cycle. Rödl, Ruciński Szemerédi 2011 proved the conjecture true when \(k=3\) large. We show this Katona-Kierstead holds \(k=4\), large, \(V({\mathcal {H}})\) partition A, B such \(|A|=\lceil n/2\rceil \(|\{e\in E({\mathcal {H}}):|e \cap A|=2\}| <\epsilon n^{4}\) for fixed small constant \(\epsilon >0\).