Hamiltonian paths and cycles in some 4-uniform hypergraphs

Abstract

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 $\epsilon >0$.