Total perfect codes in Cayley sum graphs of cyclic groups
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909865144745984 |
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| author | Koohestani, Masoumeh Mojdeh, Doost Ali Ghasemi, Mohsen |
| author_facet | Koohestani, Masoumeh Mojdeh, Doost Ali Ghasemi, Mohsen |
| contents | We consider Cayley sum graphs over the cyclic group $\mathbb{Z}_n$ and aim to explore several necessary and sufficient conditions for the existence of total perfect codes in these graphs. Specifically, we examine various cases for the connection set of the graph including when it is periodic, aperiodic, or square-free. To this end, we utilize a correspondence that we first establish between total perfect codes and factorizations of groups, along with their algebraic properties. We then generalize some of these conditions to the direct product of cyclic groups, i.e. $\mathbb{Z}_{n_1} \times \dots \times \mathbb{Z}_{n_d}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_20376 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Total perfect codes in Cayley sum graphs of cyclic groups Koohestani, Masoumeh Mojdeh, Doost Ali Ghasemi, Mohsen Combinatorics We consider Cayley sum graphs over the cyclic group $\mathbb{Z}_n$ and aim to explore several necessary and sufficient conditions for the existence of total perfect codes in these graphs. Specifically, we examine various cases for the connection set of the graph including when it is periodic, aperiodic, or square-free. To this end, we utilize a correspondence that we first establish between total perfect codes and factorizations of groups, along with their algebraic properties. We then generalize some of these conditions to the direct product of cyclic groups, i.e. $\mathbb{Z}_{n_1} \times \dots \times \mathbb{Z}_{n_d}$. |
| title | Total perfect codes in Cayley sum graphs of cyclic groups |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.20376 |