# Some Structure Theory for Cayley Graphs and Associated Hypergraphs

## TL;DR

This paper develops a structural theory for Cayley graphs avoiding certain cyclic patterns, linking their properties to hypergraph acyclicity and characterizing their tree-like structure.

## Contribution

It introduces new characterizations of Cayley graphs with specific cyclic pattern restrictions and connects these to $ ext{alpha}$-acyclic hypergraphs, highlighting their structural properties.

## Key findings

- Cayley graphs avoiding specific cyclic coset patterns exhibit tree-like structures.
- Short paths in these graphs correspond to chordless paths in hypergraphs.
- The work establishes a connection between Cayley graph structure and hypergraph acyclicity.

## Abstract

We expand the structural theory of \ca graphs that avoid specific cyclic coset patterns. We present several characterisations of tree-likeness for these structures and show a close connection to $\alpha$-acyclic hypergraphs. A focus lies on the behaviour of short paths of overlapping cosets in these \ca graphs, and their relation to short chordless paths in hypergraphs that are locally acyclic.