A hypergraph analog of Dirac's Theorem for long cycles in 2-connected graphs

Alexandr Kostochka, Ruth Luo, Grace McCourt

Published 2022-12-30Version 1

Dirac proved that each $n$-vertex $2$-connected graph with minimum degree at least $k$ contains a cycle of length at least $\min\{2k, n\}$. We prove a hypergraph version of this result: for $n \geq k \geq r+2 \geq 5$, every $2$-connected $r$-uniform $n$-vertex hypergraph with minimum degree at least ${k-1 \choose r-1} + 1$ has a Berge cycle of length at least $\min\{2k, n\}$. The bound is exact for all $k\geq r+2\geq 5$.