Hamiltonicity and σ-hypergraphs

We define and study a special type of hypergraph. A σ-hypergraph H = H(n, r, q | σ), where σ is a partition of r, is an r-uniform hypergraph having nq vertices partitioned into n classes of q vertices each. If the classes are denoted by V1, V2,...,Vn, then a subset K of V (H) of size r is an edge if the partition of r formed by the non-zero cardinalities | K ∩ Vi |, 1 ≤ i ≤ n, is σ. The non-empty intersections K ∩ Vi are called the parts of K, and s(σ) denotes the number of parts. We consider various types of cycles in hypergraphs such as Berge cycles and sharp cycles in which only consecutive edges have a nonempty intersection. We show that most σ-hypergraphs contain a Hamiltonian Berge cycle and that, for n ≥ s+ 1 and q ≥ r(r − 1), a σ-hypergraph H always contains a sharp Hamiltonian cycle. We also extend this result to k-intersecting cycles.