Intersection Hypergraph on D_n
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909493350105088 |
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| author | Ballal, Sachin N, Ardra A |
| author_facet | Ballal, Sachin N, Ardra A |
| contents | Let $G$ be a group and $S$ be the set of all non-trivial proper subgroups of $G$. The intersection hypergraph of $G$, denoted by $\tildeΓ_\mathcal{H}(G)$, is a hypergraph whose vertex set is $\{H \in S \,\, | \,\, H \cap K = \{e\} \,\, \text{for some} \, K \in S \}$ and hyperedges are the maximal subsets of the vertex set with the property that any two vertices in it have a trivial intersection. The aim of this paper is to study the intersection hypergraph of dihedral groups, $\tildeΓ_\mathcal{H}(D_n)$. We examine some of the structural properties, viz., diameter, girth and chromatic number of $\tildeΓ_\mathcal{H}(D_n)$. Also, we provide characterizations for hypertreees, star structures of $\tildeΓ_\mathcal{H}(D_n)$, and investigate the planarity and non-planarity of $\tildeΓ_\mathcal{H}(D_n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_10117 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Intersection Hypergraph on D_n Ballal, Sachin N, Ardra A Combinatorics Group Theory Let $G$ be a group and $S$ be the set of all non-trivial proper subgroups of $G$. The intersection hypergraph of $G$, denoted by $\tildeΓ_\mathcal{H}(G)$, is a hypergraph whose vertex set is $\{H \in S \,\, | \,\, H \cap K = \{e\} \,\, \text{for some} \, K \in S \}$ and hyperedges are the maximal subsets of the vertex set with the property that any two vertices in it have a trivial intersection. The aim of this paper is to study the intersection hypergraph of dihedral groups, $\tildeΓ_\mathcal{H}(D_n)$. We examine some of the structural properties, viz., diameter, girth and chromatic number of $\tildeΓ_\mathcal{H}(D_n)$. Also, we provide characterizations for hypertreees, star structures of $\tildeΓ_\mathcal{H}(D_n)$, and investigate the planarity and non-planarity of $\tildeΓ_\mathcal{H}(D_n)$. |
| title | Intersection Hypergraph on D_n |
| topic | Combinatorics Group Theory |
| url | https://arxiv.org/abs/2502.10117 |