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

Abstract

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 consider a hypergraph version of this result. A Berge cycle in a hypergraph is an alternating sequence of distinct vertices and edges $v_1,e_2,v_2,\dots ,e_c,v_1$ such that $\{v_i,v_{i+1}\}\subseteq e_i$ for all $i$ (with indices taken modulo $c$). We prove that for $n\ge k\ge r+2\ge 5$, every $2$-connected $r$-uniform $n$-vertex hypergraph with minimum degree at least $\left(\genfrac{}{}{0ex}{}{k-1}{r-1}\right)+1$ has a Berge cycle of length at least $\min \{2k,n\}$. The bound is exact for all $k\ge r+2\ge 5$.

23 pages, 2 figures