Dirac-type theorems for long Berge cycles in hypergraphs

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. In the same paper, Dirac also observed that a graph with minimum degree at least $k\ge 2$ contains a cycle of length at least $k+1$. The purpose of this paper is twofold: we prove exact bounds of similar type for hamiltonian Berge cycles as well as for Berge cycles of length at least k in r-uniform, n-vertex hypergraphs for all combinations of $k,r$ and n with $3\le r,k\le n$. The bounds differ for different ranges of r compared to n and k.