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 $k\ge r+2$. In this paper we address the case $k\le r+1$ 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 $\min \{2k,n,\mathrm{\mid }E(H)\mathrm{\mid }\}$. If $\mathrm{\mid }E(H)\mathrm{\mid }\ge n$, this bound coincides with the bound of the Dirac’s Theorem for 2-connected graphs.