The DCI and CI classification conjecture for dihedral groups

Let $n$ be a positive integer and let $\mathrm{D}_{2n}$ be the dihedral group of order $2n$. A finite group is a DCI-group if every Cayley digraph on it is a CI-digraph, and a CI-group if every Cayley graph on it is a CI-graph. Dihedral DCI and CI classification conjecture. The group $\mathrm{D}_{2n}$ is a DCI-group if and only if $n=2$ or $n$ is odd-square-free, and $\mathrm{D}_{2n}$ is a CI-group if and only if $n=2,9$ or $n$ is odd-square-free. The preceding corollary establishes the necessity of these conditions; the conjecture asserts the converse.