# Characterising CCA Sylow cyclic groups whose order is not divisible by four

## TL;DR

This paper characterizes Sylow cyclic groups with orders not divisible by four that admit non-CCA Cayley graphs, and introduces new constructions of such graphs, advancing understanding of symmetry properties in algebraic graph theory.

## Contribution

It provides a complete characterization of non-CCA Cayley graphs on Sylow cyclic groups with certain order restrictions and offers new methods to construct these graphs.

## Key findings

- Characterization of non-CCA graphs on Sylow cyclic groups
- New constructions of non-CCA Cayley graphs
- Insights into automorphism groups of Cayley graphs

## Abstract

A Cayley graph on a group $G$ has a natural edge-colouring. We say that such a graph is CCA if every automorphism of the graph that preserves this edge-colouring is an element of the normaliser of the regular representation of $G$. A group $G$ is then said to be CCA if every Cayley graph on $G$ is CCA. Our main result is a characterisation of non-CCA graphs on groups that are Sylow cyclic and whose order is not divisible by four. We also provide several new constructions of non-CCA graphs.