Normal Cayley digraphs of dihedral groups with CI-property

Abstract

A Cayley (di)graph Cay(G,S) of a group G with respect to a set S ⊆ G is said to be normal if the image of G under its right regular representation is normal in the automorphism group of Cay(G,S), and is called a CI-(di)graph if for every T ⊆ G with Cay(G,S)≅ Cay(G,T), there is α∈ Aut(G) such that Sα = T. A finite group G is called a DCI-group or an NDCI-group if all Cayley digraphs or all normal Cayley digraphs of G are CI-digraphs, respectively, and is called a CI-group or an NCI-group if all Cayley graphs or all 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 took 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 an NCI-group or an 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. As a byproduct, we construct a non-CI Cayley graph of the dihedral group D8, but Holt and Royle [A census of small transitive groups and vertex-transitive graphs, J. Symbolic Comput. 101 (2020) 51-60] claimed that D8 is a CI-group by an algorithm there.