Two new families of non-CCA groups

TL;DR

This paper identifies two new infinite families of Cayley graphs and groups that lack the CCA property, expanding the known classes of non-CCA groups with explicit constructions.

Contribution

It introduces two new infinite families of non-CCA groups formed by dihedral groups combined with themselves or cyclic groups, including the smallest previously unknown non-CCA group.

Findings

Two new infinite non-CCA families identified

Explicit constructions of non-CCA Cayley graphs provided

Includes the smallest non-CCA group known prior to this work

Abstract

We determine two new infinite families of Cayley graphs that admit colour-preserving automorphisms that do not come from the group action. By definition, this means that these Cayley graphs fail to have the CCA (Cayley Colour Automorphism) property, and the corresponding infinite families of groups also fail to have the CCA property. The families of groups consist of the direct product of any dihedral group of order $2n$ where $n \ge 3$ is odd, with either itself, or the cyclic group of order $n$. In particular, this family of examples includes the smallest non-CCA group that had not fit into any previous family of known non-CCA groups.