Finite groups with planar generating graph

Abstract

Given a finite group $G$, the generating graph $\Gamma(G)$ of $G$ has as vertices the non-identity elements of $G$ and two vertices are adjacent if and only if they are distinct and generate $G$ as group elements. Let $G$ be a 2-generated finite group. We prove that $\Gamma(G)$ is planar if and only if $G$ is isomorphic to one of the following groups: $C_2, C_3, C_4, C_5, C_6, C_2 \times C_2, D_3, D_4, Q_8, C_4\times C_2, D_6.$