Cyclic $m$-DCI-groups and $m$-CI-groups

Abstract

Based on the earlier work of Li (European J. Combin. 1997) and Dobson (Discrete Math. 2008), in this paper we complete the classification of cyclic $m$-DCI-groups and $m$-CI-groups. For a positive integer $m$ such that $m\ge 3$, we show that the group ${\mathbb{Z}}_n$ is an $m$-DCI-group if and only if $n$ is not divisible by $8$ nor by $p^2$ for any odd prime $p<m$. Furthermore, if $m\ge 6$, then we show that ${\mathbb{Z}}_n$ is an $m$-CI-group if and only if either $n\in \{8,9,18\}$, or $n\notin \{8,9,18\}$ and $n$ is not divisible by $8$ nor by $p^2$ for any odd prime $p<\frac{m-1}{2}$.