A Hypergraph Analog of Dirac's Theorem for Long Cycles in 2-Connected Graphs, II: 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, 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 $k \geq r+2$. In this paper 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 the Dirac's Theorem for 2-connected graphs.