Surface embeddings of the Klein and the Möbius–Kantor graphs

Abstract

This paper describes an invariant representation for finite graphs embedded on orientable tori of arbitrary genus, with working examples of embeddings of the Möbius–Kantor graph on the torus, the genus-2 bitorus and the genus-3 tritorus, as well as the two-dimensional, 7-valent Klein graph on the tritorus (and its dual: the 3-valent Klein graph). The genus-2 and -3 embeddings describe quotient graphs of 2- and 3-periodic reticulations of hyperbolic surfaces. This invariant is used to identify infinite nets related to the Möbius–Kantor and 7-valent Klein graphs.