Subgroup perfect codes of $ S_n $ in Cayley graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910010919878656 |
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| author | Shaw, Ankan Kotal, Shibesh Bagchi, Satya |
| author_facet | Shaw, Ankan Kotal, Shibesh Bagchi, Satya |
| 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 there exists a Cayley graph of $G$ which admits $H$ as a perfect code. In this work, we present a classification of cyclic 2-subgroup perfect codes in $ S_n$. We analyze these subgroup codes, detailing their structure and properties. We extend our discussion to various classes of subgroup codes in the symmetric group $ S_n $, encompassing both commutative and non-commutative cases. We provide numerous examples to illustrate and support our findings. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_03867 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Subgroup perfect codes of $ S_n $ in Cayley graphs Shaw, Ankan Kotal, Shibesh Bagchi, Satya Combinatorics Discrete Mathematics 05C25, 05C69, 94B99, 20B30 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 there exists a Cayley graph of $G$ which admits $H$ as a perfect code. In this work, we present a classification of cyclic 2-subgroup perfect codes in $ S_n$. We analyze these subgroup codes, detailing their structure and properties. We extend our discussion to various classes of subgroup codes in the symmetric group $ S_n $, encompassing both commutative and non-commutative cases. We provide numerous examples to illustrate and support our findings. |
| title | Subgroup perfect codes of $ S_n $ in Cayley graphs |
| topic | Combinatorics Discrete Mathematics 05C25, 05C69, 94B99, 20B30 |
| url | https://arxiv.org/abs/2602.03867 |