A Hypergraph Analog of Dirac’s Theorem for Long Cycles in 2-Connected graphs, II: Large Uniformities

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, the authors 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 ≥ r + 2. In this paper we address the case k ≤ 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)| ≥ n, this bound coincides with the bound of the Dirac’s Theorem for 2-connected graphs.