Localised codegree conditions for tight Hamiltonian cycles in 3-uniform hypergraphs

Abstract

We study sufficient conditions for the existence of Hamiltonian cycles in uniformly dense $3$-uniform hypergraphs. Problems of this type were first considered by Lenz, Mubayi, and Mycroft for loose Hamiltonian cycles and Aigner-Horev and Levy considered it for tight Hamiltonian cycles for a fairly strong notion of uniformly dense hypergraphs. Wefocus on tight cycles and obtain optimal results for a weaker notion of uniformly dense hypergraphs. We show that if an $n$-vertex $3$-uniform hypergraph $H=(V,E)$ has the property that for any set of vertices $X$ and for any collection $P$ of pairs of vertices, the number of hyperedges composed by a pair belonging to $P$ and one vertex from $X$is at least $(1/4+o(1))|X||P| - o(|V|^3)$ and $H$ has minimum vertex degree at least $\Omega(|V|^2)$, then $H$ contains a tight Hamiltoniancycle. A probabilistic construction shows that the constant $1/4$ is optimal in this context.