Hamiltonian paths and cycles in some 4-uniform hypergraphs

TL;DR

This paper proves a conjecture about the existence of Hamiltonian cycles in certain 4-uniform hypergraphs with large vertex sets and specific partition properties, extending previous results for k=3.

Contribution

It confirms the Katona-Kierstead conjecture for 4-uniform hypergraphs under new partition and size conditions, broadening the understanding of Hamiltonian cycles.

Findings

The conjecture holds for large n when k=4 with a specific vertex partition.

The paper establishes conditions under which Hamiltonian cycles exist in 4-uniform hypergraphs.

It extends previous results from k=3 to k=4 for the conjecture.

Abstract

In 1999, Katona and Kierstead conjectured that if a $k$-uniform hypergraph $\cal H$ 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. 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 $V({\cal H})$ has a partition $A$, $B$ such that $|A|=\lceil n/2\rceil$, $|\{e\in E({\cal H}):|e \cap A|=2\}| <\epsilon n^4$ for a fixed small constant $\epsilon>0$.