Normal Cayley digraphs of dihedral groups with CI-property

A Cayley (di)graph Cay(G,S) of a group G with respect to set S ⊆ is said be normal if the image under its right regular representation in automorphism Cay(G,S), and called CI-(di)graph for every T Cay(G,S)≅ Cay(G,T), there α∈ Aut(G) such that Sα = T. finite DCI-group or an NDCI-group all digraphs are CI-digraphs, respectively, CI-group NCI-group graphs CI-graphs, respectively. Motivated by conjecture proposed Ádám 1967, CI-groups DCI-groups have been actively studied during last fifty years many researchers algebraic graph theory. It took about thirty obtain classification cyclic DCI-groups, recently, first two authors, among others, classified NCI-groups NDCI-groups. Even though partial results on dihedral their still elusive. In this paper, we prove order 2n only n 2, 4 odd. As direct consequence, D2n then 2 odd-square-free, 9 throwing some new light DCI-groups. byproduct, construct non-CI D8, but Holt Royle [A census small transitive groups vertex-transitive graphs, J. Symbolic Comput. 101 (2020) 51-60] claimed D8 algorithm there.