Which maximal subgroups are perfect codes?
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908473892012032 |
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| author | Qiao, Shouhong Su, Ning Xia, Binzhou Zhang, Zhishuo Zhou, Sanming |
| author_facet | Qiao, Shouhong Su, Ning Xia, Binzhou Zhang, Zhishuo Zhou, Sanming |
| contents | A perfect code in a graph $Γ=(V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A subgroup $H$ of a group $G$ is called a subgroup perfect code of $G$ if it is a perfect code in some Cayley graph of $G$. In this paper, we undertake a systematic study of which maximal subgroups of a group can be perfect codes. Our approach highlights a characterization of subgroup perfect codes in terms of their ``local'' complements. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_23635 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Which maximal subgroups are perfect codes? Qiao, Shouhong Su, Ning Xia, Binzhou Zhang, Zhishuo Zhou, Sanming Combinatorics A perfect code in a graph $Γ=(V, E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V \setminus C$ is adjacent to exactly one vertex in $C$. A subgroup $H$ of a group $G$ is called a subgroup perfect code of $G$ if it is a perfect code in some Cayley graph of $G$. In this paper, we undertake a systematic study of which maximal subgroups of a group can be perfect codes. Our approach highlights a characterization of subgroup perfect codes in terms of their ``local'' complements. |
| title | Which maximal subgroups are perfect codes? |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.23635 |