Co-maximal Hypergraph on Dn

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 $C{o}_{\mathcal{H}}(G)$, is a hypergraph whose vertex set is $\{H\in S|HK=G\text{for some}K\in S\}$ 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, $C{o}_{\mathcal{H}}(D_n)$. We examine some of the structural properties, viz., diameter, girth and chromatic number of $C{o}_{\mathcal{H}}(D_n)$. Also, we provide characterizations for hypertrees, star structures and 3-uniform hypergraphs of $C{o}_{\mathcal{H}}(D_n)$. Further, we discuss the possibilities of $C{o}_{\mathcal{H}}(D_n)$ which can be embedded on the plane, torus and projective plane.