A Hypergraph Analog of Dirac’s Theorem for Long Cycles in 2-Connected Graphs

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