Intersection Hypergraph on D_n

TL;DR

This paper investigates the intersection hypergraph of dihedral groups, analyzing its structural properties such as diameter, girth, chromatic number, and planarity, and characterizes special structures like hypertrees and stars.

Contribution

It provides a detailed study of the intersection hypergraph of dihedral groups, including structural properties and characterizations of specific hypergraph configurations.

Findings

Determined the diameter, girth, and chromatic number of the hypergraph.

Characterized hypertrees and star structures within the hypergraph.

Analyzed the planarity and non-planarity conditions of the hypergraph.

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 $\tilde{\Gamma}_\mathcal{H}(D_n)$.