Super-pancyclic hypergraphs and bipartite graphs

Abstract

We find Dirac-type sufficient conditions for a hypergraph $\mathcal{H}$ with few edges to be hamiltonian. We also show that these conditions provide that $\mathcal{H}$ is {\em super-pancyclic}, i.e., for each $A\subseteq V(\mathcal{H})$ with $|A|\ge 3$, $\mathcal{H}$ contains a Berge cycle with vertex set $A$. We mostly use the language of bipartite graphs, because every bipartite graph is the incidence graph of a multihypergraph. In particular, we extend some results of Jackson on the existence of long cycles in bipartite graphs where the vertices in one part have high minimum degree. Furthermore, we prove a conjecture of Jackson from 1981 on long cycles in 2-connected bipartite graphs.