Longest cycles in 3-connected hypergraphs and bipartite graphs

In the language of hypergraphs, our main result is a Dirac-type bound: we prove that every $3$-connected hypergraph $H$ with $ \delta(H)\geq \max\{|V(H)|, \frac{|E(H)|+10}{4}\}$ has a hamiltonian Berge cycle. This is sharp and refines a conjecture by Jackson from 1981 (in the language of bipartite graphs). Our proofs are in the language of bipartite graphs, since the incidence graph of each hypergraph is bipartite.