Finite groups with planar generating graph
arXiv:1908.01649
summary
The paper classifies all finite 2‑generated groups whose generating graph is planar, showing that only a short list of small groups (e.g., C₂, C₃, C₄, C₅, C₆, C₂×C₂, D₃, D₄, Q₈, C₄×C₂, D₆) have this property.
Abstract
Given a finite group $G$, the generating graph $Î(G)$ 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 $Î(G)$ is planar if and only if $G$ is isomorphic to one of the following groups: $C_2, C_3, C_4, C_5, C_6, C_2 \times C_2, D_3, D_4, Q_8, C_4\times C_2, D_6.$
Topics & keywords
#finite groups#generating graph#planar graphs#graph theory#group classificationgenerating graphplanarity2‑generated groupfinite groupC₂D₃Q₈