On Isomorphisms of Tetravalent Cayley Digraphs over Dihedral Groups

Let m be a positive integer. A group G is said to be an m-DCI-group or an m-CI-group if G has the k-DCI property or k-CI property for all positive integers k at most m, respectively. Let G be a dihedral group of order $2n$ with n≥ 3. Qu and Yu proved that G is an m-DCI-group or m-CI-group, for every m∈ \1,2,3\, if and only if n is odd. In this paper, it is shown that G is a $4$-DCI-group if and only if n is odd and not divisible by $9$, and G is a $4$-CI-group if and only if n is odd.