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}. We consider a hypergraph version of this result. A Berge cycle in a hypergraph is an alternating sequence of distinct vertices and edges v₁,e₂,v₂,…,e_c,v₁ such that {v_i,v_i+1}⊆e_i for all i (with indices taken modulo c). We prove that for n≥k≥r+2≥5, every 2-connected r-uniform n-vertex hypergraph with minimum degree at least k-1r-1+1 has a Berge cycle of length at least min{2k,n}. The bound is exact for all k≥r+2≥5.