On colour-preserving automorphisms of Cayley graphs

TL;DR

This paper investigates the structure of automorphisms of Cayley graphs that preserve or permute edge colours, identifying groups with simple automorphism forms and classifying groups lacking this property.

Contribution

It characterizes groups for which all colour-preserving automorphisms of Cayley graphs are composed of left-translations and automorphisms, and classifies groups without this property.

Findings

Identified classes of groups with simple automorphism structures

Determined orders of groups lacking the property

Extended results to colour-permuting automorphisms

Abstract

We study the automorphisms of a Cayley graph that preserve its natural edge-colouring. More precisely, we are interested in groups G, such that every such automorphism of every connected Cayley graph on G has a very simple form: the composition of a left-translation and a group automorphism. We find classes of groups that have the property, and we determine the orders of all groups that do not have the property. We also have analogous results for automorphisms that permute the colours, rather than preserving them.