# k-Hypergraphs with regular automorphism groups

## Main Article Content

## Abstract

Regular representations of finite groups, as introduced by Cayley,

are among the most natural permutation representations of finite

groups. Thus, the question which regular representations appear as

full automorphism groups of combinatorial structures has been

addressed and resolved for several classes of structures, notably for

graphs (where they are called Graphical Regular Representations,

GRR's), digraphs (Digraphical Regular Representations, DRR's)

as well as for hypergraphs allowing for hyperedges of

varying sizes.

In the present paper, we focus on $k$-hypergraphs, which are hypergraphs

in which all hyperedges are of the same size $k$, and address the question

which $k$-regular hypergraphs possess full automorphism groups

acting regularly on the vertices. We rely on the concept of a Cayley

hypergraph (defined here) and show that all sufficiently large finite

groups admit a regular representation as the full automorphism group

of a $3$-hypergraph.

are among the most natural permutation representations of finite

groups. Thus, the question which regular representations appear as

full automorphism groups of combinatorial structures has been

addressed and resolved for several classes of structures, notably for

graphs (where they are called Graphical Regular Representations,

GRR's), digraphs (Digraphical Regular Representations, DRR's)

as well as for hypergraphs allowing for hyperedges of

varying sizes.

In the present paper, we focus on $k$-hypergraphs, which are hypergraphs

in which all hyperedges are of the same size $k$, and address the question

which $k$-regular hypergraphs possess full automorphism groups

acting regularly on the vertices. We rely on the concept of a Cayley

hypergraph (defined here) and show that all sufficiently large finite

groups admit a regular representation as the full automorphism group

of a $3$-hypergraph.

## Article Details

How to Cite

Jajcay, R., & Jajcayová, T.
(2019).
k-Hypergraphs with regular automorphism groups.

*Acta Mathematica Universitatis Comenianae, 88*(3), 835-840. Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/1257/773
Issue

Section

EUROCOMB 2019