标题： Dn上的共极大超图 显示英文标题

标题： Co-maximal Hypergraph on Dn

摘要： 设$G$是一个群，$S$是$G$的所有非平凡真子群的集合。\textit{共极超图 of $G$}，记作$Co_\mathcal{H}(G)$，是一个超图，其顶点集为$\{H \in S \,\, | \,\, H K = G \,\, \text{for some} \, K \in S \}$，超边是顶点集的最大子集，具有这样的性质：任意两个顶点的乘积等于$G$。 本文的目的是研究二面体群的共极大超图，$Co_\mathcal{H}(D_n)$。 我们考察了一些结构特性，即直径、环长和色数，$Co_\mathcal{H}(D_n)$。 此外，我们提供了$Co_\mathcal{H}(D_n)$的超树、星形结构和3一致超图的特征。 进一步地，我们讨论了可以嵌入平面、环面和射影平面上的$Co_\mathcal{H}(D_n)$的可能性。 显示英文摘要

摘要： 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.