Paths and circuits in $\mathbb{G}$-graphs

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 $\Gamma \left(G,S\right)$, 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 study the existence of Hamiltonian and Eulerian paths and circuits in $\Gamma \left(G,S\right)$.