(Total) Perfect codes in (extended) subgroup sum graphs
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916539999977472 |
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| author | Ma, Xuanlong Yang, Yuefeng Zhai, Liangliang |
| author_facet | Ma, Xuanlong Yang, Yuefeng Zhai, Liangliang |
| contents | Given a finite group $G$ with identity $e$ and a normal subgroup $H$ of $G$, the subgroup sum graph $Γ_{G,H}$ (resp. extended subgroup sum graph $Γ_{G,H}^+$) of $G$ with respect to $H$ is the graph with vertex set $G$, in which distinct vertices $x$ and $y$ are adjacent whenever $xy\in H\setminus \{e\}$ (resp. $xy\in H$). A group $G$ is said to be {\em code-perfect} if for any normal subgroup $H$ of $G$, $Γ_{G,H}$ admits a perfect code. In this paper, we give a necessary and sufficient condition for which normal subgroups $H$ of $G$ satisfy that a (extended) subgroup sum graph of $G$ with respect to $H$ admits a (total) perfect code, and classify all code-perfect Dedekind groups. As an application, we classify all normal subgroups such that the subgroup sum graph of a cyclic group, a dihedral group or a dicyclic group with respect to such a normal subgroup admits perfect codes, respectively. We also determine all abelian groups $A$ and subgroups $H$ of $A$ such that $Γ_{A,H}$ admits a total perfect code. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17509 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | (Total) Perfect codes in (extended) subgroup sum graphs Ma, Xuanlong Yang, Yuefeng Zhai, Liangliang Combinatorics Given a finite group $G$ with identity $e$ and a normal subgroup $H$ of $G$, the subgroup sum graph $Γ_{G,H}$ (resp. extended subgroup sum graph $Γ_{G,H}^+$) of $G$ with respect to $H$ is the graph with vertex set $G$, in which distinct vertices $x$ and $y$ are adjacent whenever $xy\in H\setminus \{e\}$ (resp. $xy\in H$). A group $G$ is said to be {\em code-perfect} if for any normal subgroup $H$ of $G$, $Γ_{G,H}$ admits a perfect code. In this paper, we give a necessary and sufficient condition for which normal subgroups $H$ of $G$ satisfy that a (extended) subgroup sum graph of $G$ with respect to $H$ admits a (total) perfect code, and classify all code-perfect Dedekind groups. As an application, we classify all normal subgroups such that the subgroup sum graph of a cyclic group, a dihedral group or a dicyclic group with respect to such a normal subgroup admits perfect codes, respectively. We also determine all abelian groups $A$ and subgroups $H$ of $A$ such that $Γ_{A,H}$ admits a total perfect code. |
| title | (Total) Perfect codes in (extended) subgroup sum graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2412.17509 |