# Super-pancyclic hypergraphs and bipartite graphs

## TL;DR

This paper establishes Dirac-type conditions for hypergraphs to be hamiltonian and super-pancyclic, extending results to bipartite graphs and proving a longstanding conjecture on long cycles.

## Contribution

It introduces new Dirac-type criteria for hypergraphs to be super-pancyclic and extends Jackson's results on bipartite graphs, including a proof of a 1981 conjecture.

## Key findings

- Hypergraphs with few edges can be hamiltonian under certain conditions.
- Conditions imply hypergraphs are super-pancyclic, containing Berge cycles for large vertex subsets.
- Proved Jackson's conjecture on long cycles in 2-connected 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 $|A| \geq 3$, $\mathcal H$ 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.