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. Last semester at this seminar, Ruth Luo presented exact bounds of similar type for Hamiltonian Berge cycles in r-uniform, n-vertex hypergraphs for all 2 < r < n. This talk will present 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 at least n/2, and the bounds differ for r less than n/2 and r at least n/2.

This is joint work with Alexandr Kostochka and Ruth Luo.