Intersection Hypergraph on D_n

Abstract

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 $\tilde{\Gamma}_\mathcal{H}(G)$, 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, $\tilde{\Gamma}_\mathcal{H}(D_n)$. We examine some of the structural properties, viz., diameter, girth and chromatic number of $\tilde{\Gamma}_\mathcal{H}(D_n)$. Also, we provide characterizations for hypertreees, star structures of $\tilde{\Gamma}_\mathcal{H}(D_n)$, and investigate the planarity and non-planarity of $\tilde{\Gamma}_\mathcal{H}(D_n)$.