Non-cyclic graphs of (non)orientable genus one

Abstract

Let $G$ be a finite non-cyclic group. The non-cyclic graph ${\Gamma }_G$ of $G$ is the graph whose vertex set is $G\setminus Cyc(G)$, two distinct vertices being adjacent if they do not generate a cyclic subgroup, where $Cyc(G)=\{a\in G:⟨a,b⟩\text{is cyclic for each }b\in G\}$. In this paper, we classify all finite non-cyclic groups $G$ such that ${\Gamma }_G$ has (non)orientable genus one.

9 pages, 3 figures

References in corpus (3)

• Commutative rings with toroidal zero-divisor graphs
• Embeddings of (proper) power graphs of finite groups
• Toroidality and projective-planarity of intersection graphs of subgroups of finite groups