2021/05/27 by Jin-Hua Xie, Xie, Jin-Hua, Yan‐Quan Feng +3
Engineering · Mathematics · #05C25 #20B25 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2105.12925
openalex publication_date 2021/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 α∈ Aut(G) such that Sα=T, whenever Cay(G,S)≅ 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 Ádám 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 D2n of order 2n is a DCI-group then n=2 or n is odd-square-free, and that if D2n 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.