Cycles as edge intersection hypergraphs of $k$-uniform hypergraphs ($k \le 6$) -- a constructive approach

TL;DR

This paper explores how cycles can be represented as edge intersection hypergraphs of k-uniform hypergraphs for k up to 6, filling a gap in existing research between 3- and 6-uniform cases.

Contribution

It provides a constructive approach to characterize cycles as edge intersection hypergraphs of 4- and 5-uniform hypergraphs, extending previous work on 3- and 6-uniform hypergraphs.

Findings

Characterization of cycles as edge intersection hypergraphs for 4- and 5-uniform hypergraphs.

Bridging the gap between known cases for 3- and 6-uniform hypergraphs.

Constructive methods for representing cycles in these hypergraph classes.

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 ${\cal E}^{EI}=\{e_1 \cap e_2 \ |\ e_1, e_2 \in {\cal E} \ \wedge \ e_1 \neq e_2 \ \wedge \ |e_1 \cap e_2 |\geq 2\}$. In the present paper, we consider 4- and 5-uniform hypergraphs ${\cal H}$, respectively, with $EI({\cal H}) = C_n$. Our results fill the gap between the 3- and the 6-uniform case considered in arXiv:1902.00396.