Subgroup perfect codes of $S_n$ in Cayley sum graphs
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| Format: | Preprint |
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2025
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| author | Shaw, Ankan Mondal, Biswajit Bagchi, Satya |
| author_facet | Shaw, Ankan Mondal, Biswajit 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 $C$. Let $ G $ be a finite group, and let $ S $ be a square-free normal subset of $ G $. The Cayley sum graph of $ G $ with respect to $ S $ is a simple graph with vertex set $ G $ and two vertices $ x $ and $ y $ are adjacent if $ xy\in S .$ A subset $ C $ of $ G $ is called perfect code of $ G $ if there exists a Cayley sum graph of $ G $ that admits $ C $ as a perfect code. In particular, if a subgroup of $ G $ is a perfect code of $ G $, then the subgroup is called a subgroup perfect code of $ G $. In this work, we prove that there does not exist any proper perfect subgroup code of symmetric group $ S_n $. Using this result, we provide a complete characterization of the perfect subgroup code of the alternating group $A_n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_05069 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Subgroup perfect codes of $S_n$ in Cayley sum graphs Shaw, Ankan Mondal, Biswajit Bagchi, Satya Combinatorics Discrete Mathematics Information Theory Group Theory 05C25, 05C69, 94B99, 20B30, 20D06 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 $C$. Let $ G $ be a finite group, and let $ S $ be a square-free normal subset of $ G $. The Cayley sum graph of $ G $ with respect to $ S $ is a simple graph with vertex set $ G $ and two vertices $ x $ and $ y $ are adjacent if $ xy\in S .$ A subset $ C $ of $ G $ is called perfect code of $ G $ if there exists a Cayley sum graph of $ G $ that admits $ C $ as a perfect code. In particular, if a subgroup of $ G $ is a perfect code of $ G $, then the subgroup is called a subgroup perfect code of $ G $. In this work, we prove that there does not exist any proper perfect subgroup code of symmetric group $ S_n $. Using this result, we provide a complete characterization of the perfect subgroup code of the alternating group $A_n$. |
| title | Subgroup perfect codes of $S_n$ in Cayley sum graphs |
| topic | Combinatorics Discrete Mathematics Information Theory Group Theory 05C25, 05C69, 94B99, 20B30, 20D06 |
| url | https://arxiv.org/abs/2509.05069 |