Normal Cayley digraphs of dihedral groups with CI-property

TL;DR

This paper characterizes when dihedral groups are normal Cayley CI-graphs and DCI-graphs, providing a complete classification based on the order of the group, which advances understanding in algebraic graph theory.

Contribution

It proves that dihedral groups are NCI or NDCI if and only if their order meets specific conditions, and classifies dihedral CI and DCI groups based on their order.

Findings

Dihedral groups are NCI or NDCI if and only if n=2,4, or n is odd.

D_{2n} is a DCI-group if n=2 or n is odd-square-free.

D_{2n} is a CI-group if n=2, 9, or n is odd-square-free.

Abstract

A Cayley (di)graph $Cay(G,S)$ of a group $G$ with respect to $S$ is said to be normal if the right regular representation of $G$ is normal in the automorphism group of $Cay(G,S)$, and is called a CI-(di)graph if there is $\alpha\in Aut(G)$ such that $S^\alpha=T$, whenever $Cay(G,S)\cong Cay(G,T)$ for a Cayley (di)graph $Cay(G,T)$. A finite group $G$ is called a DCI-group or a NDCI-group if all Cayley digraphs or normal Cayley digraphs of $G$ are CI-digraphs, and is called a CI-group or a NCI-group if all Cayley graphs or normal Cayley graphs of $G$ are CI-graphs, respectively. Motivated by a conjecture proposed by \'Ad\'am in 1967, CI-groups and DCI-groups have been actively studied during the last fifty years by many researchers in algebraic graph theory. It takes about thirty years to obtain the classification of cyclic CI-groups and DCI-groups, and recently, the first two authors, among others, classified cyclic NCI-groups and NDCI-groups. Even though there are many partial results on dihedral CI-groups and DCI-groups, their classification is still elusive. In this paper, we prove that a dihedral group of order $2n$ is a NCI-group or a NDCI-group if and only if $n=2,4$ or $n$ is odd. As a direct consequence, we have that if a dihedral group $D_{2n}$ of order $2n$ is a DCI-group then $n=2$ or $n$ is odd-square-free, and that if $D_{2n}$ is a CI-group then $n=2,9$ or $n$ is odd-square-free, throwing some new light on classification of dihedral CI-groups and DCI-groups.