Dirac-type theorems for long Berge cycles in hypergraphs (Q6564608)

The famous Dirac theorem gives an exact bound on the minimum degree of an \(n\)-vertex graph guaranteeing the existence of a Hamiltonian cycle. The purpose of this paper is twofold: the authors prove exact bounds of a similar type for Hamiltonian Berge cycles as well as for Berge cycles of length at least \(k\) in \(r\)-uniform, \(n\)-vertex hypergraphs for all combinations of \(k\), \(r\) and \(n\) with \(3 \leq r\), \(k \leq n\). The bounds differ for different ranges of \(r\) compared to \(n\) and \(k\). For example, let \(t = t(n) = \lfloor(n-1)/2 \rfloor\), and suppose \(3 \leq r < n\). Let \(H\) be an \(n\)-vertex \(r\)-graph. If (a) \(r \leq t\) and \(\delta(H) \geq \binom{t}{r-1}+1\) or (b) \(r \geq n/2\) and \(\delta(H) \geq r\), then \(H\) contains a Hamiltonian Berge cycle. Let \(n\), \(k\), and \(r\) be positive integers such that \(n \geq k \geq r \geq 3\) and \(r > t\). If \(H\) is an \(n\)-vertex \(r\)-graph with \(\delta(H) \geq \lfloor r(k-1)/n\rfloor +1\), then \(H\) contains a Berge cycle of length \(k\) or longer.