Normal Cayley digraphs of cyclic groups with CI-property

TL;DR

This paper characterizes when cyclic groups have the property that all their normal Cayley digraphs or graphs are CI-structures, revealing a precise divisibility condition related to the number 8.

Contribution

It provides a complete characterization of NDCI and NCI properties for cyclic groups based on divisibility by 8, expanding understanding of Cayley digraph symmetries.

Findings

Cyclic groups of order not divisible by 8 are NDCI-groups.

Cyclic groups of order 8 or not divisible by 8 are NCI-groups.

The divisibility by 8 determines the CI-property for normal Cayley (di)graphs.

Abstract

A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to a subset $S$ of $G$ is called normal if the right regular representation of $G$ is a normal subgroup in the full automorphism group of $Cay(G,S)$, and is called a CI-(di)graph if for every $T\subseteq G$, $Cay(G,S)\cong Cay(G,T)$ implies that there is $\sigma\in Aut(G)$ such that $S^\sigma=T$. We call a group $G$ a NDCI-group if all normal Cayley digraphs of $G$ are CI-digraphs, and a NCI-group if all normal Cayley graphs of $G$ are CI-graphs, respectively. In this paper, we prove that a cyclic group of order $n$ is a NDCI-group if and only if $8\nmid n$, and is a NCI-group if and only if either $n=8$ or $8\nmid n$.