Intersection Hypergraph on D_n
Sachin Ballal, Ardra A N
TL;DR
This work studies the intersection hypergraph $\tilde{\Gamma}_\mathcal{H}(D_n)$ on the proper nontrivial subgroups of the dihedral group $D_n$, focusing on structural properties such as diameter, girth, and chromatic numbers, and on topological aspects like planarity. It establishes detailed relationships between $n$ and graph hypergraph parameters, proving connectedness with diameter depending on $n$, and showing that the chromatic number is $2$ while the chromatic index depends on whether $n$ is a multiple of $4$. It provides a complete planarity classification: planar iff $n$ is prime or $n=4$, and non-planar for other $n$, supported by incidence-graph genus bounds and explicit subgraph constructions such as $K_{3,3}$. The results give a comprehensive picture of how subgroup interactions in $D_n$ influence hypertree structure, planarity, and genus, contributing to the theory of hypergraphs on algebraic structures.
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 $\tildeΓ_\mathcal{H}(G)$, 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 $\tildeΓ_\mathcal{H}(D_n)$.