A classical theorem of Dirac states that any graph on n vertices with minimum degree at least n/2 contains a Hamiltonian cycle, an edge-traversing cycle that visits each vertex in the graph exactly once. Since Dirac's theorem, researchers have found lower bounds on the number of Hamiltonian cycles a graph contains as a function of its minimum degree. These bounds aren't too far away from what one would expect to see in a typical random graph with appropriate edge probability. Here we contribute to the line of research that seeks to extend these results to k-uniform hypergraphs, where cycles are now chains of edges whose consecutive links overlap with one another in some fixed number, \ell, of vertices. We show that the number of so-called \ell-cycles in hypergraphs with sufficiently high co-degree is at least, up to a sub-exponential factor, what one would expect to see in a typical random hypergraph with appropriate (hyper)edge probability.

Joint work with Asaf Ferber and Adva Mond (University of Cambridge).