A pair degree condition for Hamiltonian cycles in $3$-uniform hypergraphs

TL;DR

This paper establishes a new, improved pair degree condition for the existence of tight Hamiltonian cycles in 3-uniform hypergraphs, advancing the understanding of degree conditions that guarantee such cycles.

Contribution

It provides a 3-uniform hypergraph analogue of Pósa's theorem, strengthening the asymptotic condition for Hamiltonian cycles beyond previous results.

Findings

Introduces a new pair degree condition for 3-uniform hypergraphs.

Strengthens the asymptotic version of R"odl, Rucirnski, and Szemerédi's result.

Advances towards a full characterization of degree matrices ensuring Hamiltonian cycles.

Abstract

We prove a new sufficient pair degree condition for tight Hamiltonian cycles in $3$-uniform hypergraphs that (asymptotically) improves the best known pair degree condition due to R\"odl, Ruci\'nski, and Szemer\'edi. For graphs, Chv\'atal characterised all those sequences of integers for which every pointwise larger (or equal) degree sequence guarantees the existence of a Hamiltonian cycle. A step towards Chv\'atal's theorem was taken by P\'osa, who improved on Dirac's tight minimum degree condition for Hamiltonian cycles by showing that a certain weaker condition on the degree sequence of a graph already yields a Hamiltonian cycle. In this work, we take a similar step towards a full characterisation of all pair degree matrices that ensure the existence of tight Hamiltonian cycles in $3$-uniform hypergraphs by proving a $3$-uniform analogue of P\'osa's result. In particular, our result strengthens the asymptotic version of the result by R\"odl, Ruci\'nski, and Szemer\'edi.