$\mathbb{G}$-planar abelian groups

Abstract

For a group $G$ with generating set $S=\left\{s_1,s_2,\dots ,s_k\right\}$, the $\mathbb{G}$-graph of $G$, denoted by $\Gamma \left(G,S\right)$, is the graph whose vertices are distinct cosets of $\langle s_i\rangle$ in $G$. Two distinct vertices are joined by an edge when the set intersection of the cosets is nonempty. In this paper, we explore the planarity of $\Gamma \left(G,S\right)$.