From the Coxeter graph to the Klein graph

Abstract

We show that the 56-vertex Klein cubic graph $\G'$ can be obtained from the 28-vertex Coxeter cubic graph $\G$ by 'zipping' adequately the squares of the 24 7-cycles of $\G$ endowed with an orientation obtained by considering $\G$ as a $\mathcal C$-ultrahomogeneous digraph, where $\mathcal C$ is the collection formed by both the oriented 7-cycles $\vec{C}_7$ and the 2-arcs $\vec{P}_3$ that tightly fasten those $\vec{C}_7$ in $\G$. In the process, it is seen that $\G'$ is a ${\mathcal C}'$-ultrahomogeneous (undirected) graph, where ${\mathcal C}'$ is the collection formed by both the 7-cycles $C_7$ and the 1-paths $P_2$ that tightly fasten those $C_7$ in $\G'$. This yields an embedding of $\G'$ into a 3-torus $T_3$ which forms the Klein map of Coxeter notation $(7,3)_8$. The dual graph of $\G'$ in $T_3$ is the distance-regular Klein quartic graph, with corresponding dual map of Coxeter notation $(3,7)_8$.