Hamiltonian Paths and Cycles in Some 4-Uniform Hypergraphs

In 1999, Katona and Kierstead conjectured that if a k -uniform hypergraph ℋ on n vertices has minimum co-degree ⌊n-k+3/2⌋ , i.e., each set of k-1 vertices is contained in at least ⌊n-k+3/2⌋ 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(ℋ) has a partition A , B such that |A|=⌈ n/2⌉ , |{e∈ E(ℋ):|e ∩ A|=2}| <ϵ n^4 for a fixed small constant ϵ >0 .