Cyclic $m$-DCI-groups and $m$-CI-groups
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909460640825344 |
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| author | Kovács, István Šinkovec, Luka |
| author_facet | Kovács, István Šinkovec, Luka |
| contents | 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 $p < \frac{m - 1}{2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10723 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cyclic $m$-DCI-groups and $m$-CI-groups Kovács, István Šinkovec, Luka Combinatorics 05C25, 20B25 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 $p < \frac{m - 1}{2}$. |
| title | Cyclic $m$-DCI-groups and $m$-CI-groups |
| topic | Combinatorics 05C25, 20B25 |
| url | https://arxiv.org/abs/2501.10723 |