Georgia Institute of Technology College of Sciences

Dirac proved that every $n$-vertex graph with minimum degree at least $n/2$ contains a hamiltonian cycle. Moreover, every graph with minimum degree $k \geq 2$ contains a cycle of length at least $k+1$, and this can be further improved if the graph is 2-connected. In this talk, we prove analogs of these theorems for hypergraphs. That is, we give sharp minimum degree conditions that imply the existence of long Berge cycles in uniform hypergraphs. This is joint work with Alexandr Kostochka and Grace McCourt.