Co-maximal Hypergraph on Dn
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908343937794048 |
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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 $G$. \textit{The co-maximal hypergraph of $G$}, denoted by $Co_\mathcal{H}(G)$, is a hypergraph whose vertex set is $\{H \in S \,\, | \,\, H K = G \,\, \text{for some} \, K \in S \}$ and hyperedges are the maximal subsets of the vertex set with the property that the product of any two vertices is equal to $G$. The aim of this paper is to study the co-maximal hypergraph of dihedral groups, $Co_\mathcal{H}(D_n)$. We examine some of the structural properties, viz., diameter, girth and chromatic number of $Co_\mathcal{H}(D_n)$. Also, we provide characterizations for hypertrees, star structures and 3-uniform hypergraphs of $Co_\mathcal{H}(D_n)$. Further, we discuss the possibilities of $Co_\mathcal{H}(D_n)$ which can be embedded on the plane, torus and projective plane. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_21554 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Co-maximal Hypergraph on Dn Ballal, Sachin N, Ardra A Combinatorics 05C25, 05C65 Let $G$ be a group and $S$ be the set of all non-trivial proper subgroups of $G$. \textit{The co-maximal hypergraph of $G$}, denoted by $Co_\mathcal{H}(G)$, is a hypergraph whose vertex set is $\{H \in S \,\, | \,\, H K = G \,\, \text{for some} \, K \in S \}$ and hyperedges are the maximal subsets of the vertex set with the property that the product of any two vertices is equal to $G$. The aim of this paper is to study the co-maximal hypergraph of dihedral groups, $Co_\mathcal{H}(D_n)$. We examine some of the structural properties, viz., diameter, girth and chromatic number of $Co_\mathcal{H}(D_n)$. Also, we provide characterizations for hypertrees, star structures and 3-uniform hypergraphs of $Co_\mathcal{H}(D_n)$. Further, we discuss the possibilities of $Co_\mathcal{H}(D_n)$ which can be embedded on the plane, torus and projective plane. |
| title | Co-maximal Hypergraph on Dn |
| topic | Combinatorics 05C25, 05C65 |
| url | https://arxiv.org/abs/2504.21554 |