# Cycles as edge intersection hypergraphs

## TL;DR

This paper constructs specific hypergraphs whose edge intersection hypergraphs form cycles, addressing a problem about the existence of such hypergraphs with particular regularity and uniformity properties.

## Contribution

It proves the existence of 3-regular hypergraphs with cycle edge intersection hypergraphs for all sufficiently large n, solving an open problem.

## Key findings

- Existence of 3-regular hypergraphs with cycle edge intersection hypergraphs for n ≥ 24.
- Construction of hypergraphs with specified regularity and uniformity.
- Addresses an open problem from prior research.

## Abstract

If ${\cal H}=(V,{\cal E})$ is a hypergraph, its edge intersection hypergraph $EI({\cal H})=(V,{\cal E}^{EI})$ has the edge set ${\cal E}^{EI}=\{e_1 \cap e_2 \ |\ e_1, e_2 \in {\cal E} \ \wedge \ e_1 \neq e_2 \ \wedge \ |e_1 \cap e_2 |\geq2\}$. Picking up a problem from arXiv:1901.06292, for $n \ge 24$ we prove that there is a 3-regular (and - if $n$ is even - 6-uniform) hypergraph ${\cal H}=(V,{\cal E})$ with $\lceil \frac{n}{2} \rceil$ hyperedges and $EI({\cal H}) = C_n$.