Non-cyclic graphs of (non)orientable genus one

TL;DR

This paper classifies all finite non-cyclic groups whose non-cyclic graphs have orientable or non-orientable genus one, providing a complete characterization of these groups based on their graph embeddings.

Contribution

It offers a complete classification of finite non-cyclic groups with non-cyclic graphs of genus one, advancing understanding of the interplay between group structure and graph topology.

Findings

Identifies all finite non-cyclic groups with non-cyclic graphs of genus one.

Determines conditions under which these graphs are embeddable on surfaces of genus one.

Provides a characterization linking group properties to graph topological features.

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 $Cyc(G)=\{a\in G: \langle a,b\rangle\ \text{is cyclic for each } b\in G\}$. In this paper, we classify all finite non-cyclic groups $G$ such that $\Gamma_G$ has (non)orientable genus one.