Co-maximal Hypergraph on Dn
Sachin Ballal, Ardra A N
TL;DR
The paper studies the co-maximal hypergraph $Co_{\\mathcal{H}}(D_n)$ of the dihedral group, deriving its structural properties such as diameter, girth, and chromatic number, and characterizing when it forms hypertrees or star hypergraphs and when it is 3-uniform. It classifies vertices into Type (1) and Type (2) subgroups, identifies exact conditions for vertex inclusion, and proves the universal bound $\\mathrm{diam}(Co_{\\mathcal{H}}(D_n))\\le 3$ with connectivity. It then investigates planar, toroidal, and projective embeddings via incidence graphs, establishing planarity only for prime-power $n$, and toroidal/projective embeddability criteria with concrete obstructions (e.g., $K_{3,3}$ subdivisions) and explicit constructions, including the case $n=6$. The results highlight notable differences from the co-maximal graph and enrich the understanding of hypergraph structures arising from group theory and surface embeddings.
Abstract
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.