Perfect codes in Cayley graphs of Hajós groups
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908490838048768 |
|---|---|
| author | Hafidh, Yusuf Xia, Binzhou Zhou, Sanming |
| author_facet | Hafidh, Yusuf Xia, Binzhou Zhou, Sanming |
| contents | A perfect code in a graph $Γ$ is a subset $C$ of the vertex set of $Γ$ such that every vertex of $Γ$ outside $C$ has exactly one neighbour in $C$. A perfect code in a directed graph can be defined similarly by requiring that for every vertex $v$ outside $C$ there exists exactly one vertex $u$ in $C$ such that the arc from $u$ to $v$ exists in $Γ$. A subset $X$ of an abelian group $G$ is said to be periodic if there exists a non-identity element $g$ of $G$ such that $g + X = X$. A factorization of $G$ is a pair of nonempty subsets $(A, B)$ of $G$ such that every element $g$ of $G$ can be expressed uniquely as $g = a+b$ with $a \in A$ and $b \in B$. If for every factorization $(A, B)$ of an abelian group $G$ at least one of $A$ and $B$ is periodic, then $G$ is said to be a Hajós group. In this paper we classify all Cayley graphs (directed or undirected) of Hajós groups which admit perfect codes, and moreover we determine all perfect codes in such Cayley graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_11164 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Perfect codes in Cayley graphs of Hajós groups Hafidh, Yusuf Xia, Binzhou Zhou, Sanming Combinatorics 05C25, 05C69 G.2.2 A perfect code in a graph $Γ$ is a subset $C$ of the vertex set of $Γ$ such that every vertex of $Γ$ outside $C$ has exactly one neighbour in $C$. A perfect code in a directed graph can be defined similarly by requiring that for every vertex $v$ outside $C$ there exists exactly one vertex $u$ in $C$ such that the arc from $u$ to $v$ exists in $Γ$. A subset $X$ of an abelian group $G$ is said to be periodic if there exists a non-identity element $g$ of $G$ such that $g + X = X$. A factorization of $G$ is a pair of nonempty subsets $(A, B)$ of $G$ such that every element $g$ of $G$ can be expressed uniquely as $g = a+b$ with $a \in A$ and $b \in B$. If for every factorization $(A, B)$ of an abelian group $G$ at least one of $A$ and $B$ is periodic, then $G$ is said to be a Hajós group. In this paper we classify all Cayley graphs (directed or undirected) of Hajós groups which admit perfect codes, and moreover we determine all perfect codes in such Cayley graphs. |
| title | Perfect codes in Cayley graphs of Hajós groups |
| topic | Combinatorics 05C25, 05C69 G.2.2 |
| url | https://arxiv.org/abs/2508.11164 |