Dirac's Theorem for hamiltonian Berge cycles in uniform hypergraphs

TL;DR

This paper extends Dirac's Theorem to hypergraphs, establishing exact minimum degree bounds for the existence of Hamiltonian Berge cycles and long Berge cycles in r-uniform hypergraphs.

Contribution

It provides the first exact degree bounds for Hamiltonian Berge cycles in hypergraphs, differentiating cases based on the uniformity parameter r.

Findings

Exact bounds for Hamiltonian Berge cycles in hypergraphs.

Bounds differ for r<n/2 and r≥n/2 cases.

Degree conditions for long Berge cycles with length ≥ n/2.

Abstract

The famous Dirac's Theorem gives an exact bound on the minimum degree of an $n$-vertex graph guaranteeing the existence of a hamiltonian cycle. We prove exact bounds of similar type for hamiltonian Berge cycles in $r$-uniform, $n$-vertex hypergraphs for all $3\leq r< n$. The bounds are different for $r<n/2$ and $r\geq n/2$. We also give bounds on the minimum degree guaranteeing existence of Berge cycles of length at least $k$ in such hypergraphs; the bounds are exact for all $k\geq n/2$.