From the Coxeter graph to the Klein graph

TL;DR

This paper constructs the Klein cubic graph from the Coxeter cubic graph using a novel 'zipping' process on cycles, revealing its embedding into a 3-torus and its relation to the Klein quartic graph.

Contribution

It introduces a new method to derive the Klein cubic graph from the Coxeter graph via cycle 'zipping' and explores its embedding and duality properties.

Findings

Klein cubic graph obtained from Coxeter graph through cycle zipping.

Embedding of the Klein cubic graph into a 3-torus as the Klein map.

Identification of the dual graph as the Klein quartic 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 $(3,7)_8$.