Speaker: Grace McCourt
Title: A hypergraph analog of Dirac's Theorem for long cycles in 2-connected graphs: Large uniformities
Abstract: Dirac proved that each n-vertex 2-connected graph with minimum degree k contains a cycle of length at least min{2k, n}. We obtain analogous results for Berge cycles in hypergraphs. Recently, we proved an exact lower bound on the minimum degree ensuring a Berge cycle of length at least min{2k, n} in n-vertex r-uniform 2-connected hypergraphs when k \geq r+2. In this talk, we address the case k \leq r+1 in which the bounds have a different behavior. We prove that each n-vertex r-uniform 2-connected hypergraph H with minimum degree k contains a Berge cycle of length at least min{2k, n, |E(H)|}. If |E(H)| \geq n, this bound coincides with the bound of Dirac's theorem for 2-connected graphs. This is joint work with Alexandr Kostochka and Ruth Luo.