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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | https://arxiv.org/abs/2504.14942 |
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| _version_ | 1866915251914539008 |
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| author | Parveen, Parveen Bhattacharjya, Bikash |
| author_facet | Parveen, Parveen Bhattacharjya, Bikash |
| contents | Let \( G \) be a finite non-cyclic group. Define \( \mathrm{Cyc}(G) \) as the set of all elements \( a \in G \) such that for any $b\in G$, the subgroup \( \langle a, b \rangle \) is cyclic. The \emph{non-cyclic graph} $Γ(G)$ of \( G \) is a simple undirected graph with vertex set \( G \setminus \mathrm{Cyc}(G) \), where two distinct vertices \( x \) and \( y \) are adjacent if the subgroup \( \langle x, y \rangle \) is not cyclic. An independent subset $C$ of the vertex set of a graph $Γ$ is called a perfect code of $Γ$ if every vertex of $V(Γ)\setminus C$ is adjacent to exactly one vertex in $C$. A subset \( T \) of the vertex set a graph \( Γ\) is said to be a \emph{total perfect code} if every vertex of \( Γ\) is adjacent to exactly one vertex in \( T \). In this paper, we prove that the graph $Γ(G)$ is Hamiltonian for any finite non-cyclic nilpotent group $G$. Also, we characterize all finite groups such that their non-cyclic graphs admit a perfect code. Finally, we prove that for a non-cyclic nilpotent group $G$, the non-cyclic graph $Γ(G)$ does not admit total perfect code. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14942 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Hamiltonicity and Perfect Codes in Non-Cyclic Graphs of Finite Groups Parveen, Parveen Bhattacharjya, Bikash Combinatorics 05C25 Let \( G \) be a finite non-cyclic group. Define \( \mathrm{Cyc}(G) \) as the set of all elements \( a \in G \) such that for any $b\in G$, the subgroup \( \langle a, b \rangle \) is cyclic. The \emph{non-cyclic graph} $Γ(G)$ of \( G \) is a simple undirected graph with vertex set \( G \setminus \mathrm{Cyc}(G) \), where two distinct vertices \( x \) and \( y \) are adjacent if the subgroup \( \langle x, y \rangle \) is not cyclic. An independent subset $C$ of the vertex set of a graph $Γ$ is called a perfect code of $Γ$ if every vertex of $V(Γ)\setminus C$ is adjacent to exactly one vertex in $C$. A subset \( T \) of the vertex set a graph \( Γ\) is said to be a \emph{total perfect code} if every vertex of \( Γ\) is adjacent to exactly one vertex in \( T \). In this paper, we prove that the graph $Γ(G)$ is Hamiltonian for any finite non-cyclic nilpotent group $G$. Also, we characterize all finite groups such that their non-cyclic graphs admit a perfect code. Finally, we prove that for a non-cyclic nilpotent group $G$, the non-cyclic graph $Γ(G)$ does not admit total perfect code. |
| title | On Hamiltonicity and Perfect Codes in Non-Cyclic Graphs of Finite Groups |
| topic | Combinatorics 05C25 |
| url | https://arxiv.org/abs/2504.14942 |