Dihedral groups with the m-DCI property

A Cayley digraph \(\textrm{Cay}(G,S)\) $\text{Cay}(G,S)$ of a group G with respect to a subset S of G is called a CI-digraph if for every Cayley digraph \(\textrm{Cay}(G,T)\) $\text{Cay}(G,T)$ isomorphic to \(\textrm{Cay}(G,S)\) $\text{Cay}(G,S)$ , there exists an \(\alpha \in \textrm{Aut}(G)\) $\alpha \in \text{Aut}(G)$ such that \(S^\alpha =T\) $S^{\alpha }=T$ . For a positive integer m, G is said to have the m-DCI property if all Cayley digraphs of G with out-valency m are CI-digraphs. Li (European J Combin 18:655–665, 1997) gave a necessary condition for cyclic groups to have the m-DCI property, and in this paper, we find a necessary condition for dihedral groups to have the m-DCI property. Let \(\textrm{D}_{2n}\) ${\text{D}}_{2n}$ be the dihedral group of order 2n, and assume that \(\textrm{D}_{2n}\) ${\text{D}}_{2n}$ has the m-DCI property for some \(1 \le m\le n-1\) $1\le m\le n-1$ . It is shown that n is odd, and if further \(p+1\le m\le n-1\) $p+1\le m\le n-1$ for an odd prime divisor p of n, then \(p^2\not \mid n\) $p^2\nmid n$ . Furthermore, if n is a power of a prime q, then \(\textrm{D}_{2n}\) ${\text{D}}_{2n}$ has the m-DCI property if and only if either \(n=q\) $n=q$ , or q is odd and \(1\le m\le q\) $1\le m\le q$ .