Dihedral groups with the \(m\)-DCI property (Q6604498)

A Cayley (di)graph \(\text{Cay}(G,S)\) on the group \(G\) has the (D)CI property if whenever \(\text{Cay}(G, T) \cong \text{Cay}(G,S)\), there is some automorphism \(\alpha\) of the group \(G\) such that \(S^\alpha=T\). A group has the \(m\)-DCI property if every Cayley digraph \(\text{Cay}(G,S)\) with \(|S|=m\) has the DCI property.\N\NIn this paper, the authors provide necessary conditions for a Cayley graph on the dihedral group of order \(2n\) to have the \(m\)-DCI property. Furthermore, they show that these conditions are sufficient in the case where \(n\) is a prime power. Their proofs involve analysing the Sylow subgroups of the dihedral group.\N\NSimilar results had previously been produced by \textit{C. H. Li} [Eur. J. Comb. 18, No. 6, 655--665 (1997; Zbl 0888.05033)] for cyclic groups.