Embedding K5 and K3,3 on orientable surfaces
Andrei Gagarin, William Lawrence Kocay · ORCA Online Research @Cardiff (Cardiff University) · 2020
The Kuratowski graphs K5 and K3,3 are fundamental non-planar graphs. We are interested in obtaining all their distinct 2-cell embeddings on orientable surfaces. The 2-cell embeddings of K5 and K3,3 on the torus are well-known. Using a constructive approach of expanding from minors, we obtain all 2-cell embeddings of these graphs on the double torus. As a consequence, several new polygonal representations of the double torus are described. Rotation systems for the one-face embeddings of K5 on the triple torus are also found, using an exhaustive search approach.