On Intersection and Co-maximal Hypergraph of $\mathbb{Z}_n$
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| Format: | Preprint |
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2025
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| _version_ | 1866909794900639744 |
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| author | Ballal, Sachin N, Ardra A |
| author_facet | Ballal, Sachin N, Ardra A |
| contents | The aim of this paper is to study the intersection hypergraph $\tildeΓ_\mathcal{H}(\mathbb{Z}_n)$ and co-maximal hypergraph $Co_\mathcal{H}(\mathbb{Z}_n)$ on the subgroups of $\mathbb{Z}_n$. We prove that the intersection and co-maximal hypergraph of a finite abelian group are isomorphic. Hence, we focus on $\tildeΓ_\mathcal{H}(\mathbb{Z}_n)$ and examine some of the structural properties, viz., diameter, girth and chromatic number of $\tildeΓ_\mathcal{H}(\mathbb{Z}_n)$. Also, we provide characterizations for hypertrees, star structures of $\tildeΓ_\mathcal{H}(\mathbb{Z}_n)$, and investigate the planarity and non-planarity of $\tildeΓ_\mathcal{H}(\mathbb{Z}_n)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_14618 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Intersection and Co-maximal Hypergraph of $\mathbb{Z}_n$ Ballal, Sachin N, Ardra A Combinatorics Group Theory 05C25, 05C65 The aim of this paper is to study the intersection hypergraph $\tildeΓ_\mathcal{H}(\mathbb{Z}_n)$ and co-maximal hypergraph $Co_\mathcal{H}(\mathbb{Z}_n)$ on the subgroups of $\mathbb{Z}_n$. We prove that the intersection and co-maximal hypergraph of a finite abelian group are isomorphic. Hence, we focus on $\tildeΓ_\mathcal{H}(\mathbb{Z}_n)$ and examine some of the structural properties, viz., diameter, girth and chromatic number of $\tildeΓ_\mathcal{H}(\mathbb{Z}_n)$. Also, we provide characterizations for hypertrees, star structures of $\tildeΓ_\mathcal{H}(\mathbb{Z}_n)$, and investigate the planarity and non-planarity of $\tildeΓ_\mathcal{H}(\mathbb{Z}_n)$. |
| title | On Intersection and Co-maximal Hypergraph of $\mathbb{Z}_n$ |
| topic | Combinatorics Group Theory 05C25, 05C65 |
| url | https://arxiv.org/abs/2509.14618 |