Embedding K3,3 and K5 on the Double Torus

TL;DR

This paper explicitly constructs all 2-cell embeddings of the Kuratowski graphs K3,3 and K5 on the double torus, providing new polygonal representations and confirming previous enumeration results.

Contribution

It offers a constructive method to find all embeddings of K3,3 and K5 on the double torus, including explicit representations, advancing understanding of graph embeddings on complex surfaces.

Findings

Unique non-orientable embedding of K3,3 on the double torus

14 orientable embeddings of K5 on the double torus

17 non-orientable embeddings of K5 on the double torus

Abstract

The Kuratowski graphs $K_{3,3}$ and $K_5$ characterize planarity. Counting distinct 2-cell embeddings of these two graphs on orientable surfaces was previously done by using Burnside's Lemma and their automorphism groups, without actually constructing the embeddings. We obtain all 2-cell embeddings of these graphs on the double torus, using a constructive approach. This shows that there is a unique non-orientable 2-cell embedding of $K_{3,3}$, 14 orientable and 17 non-orientable 2-cell embeddings of $K_5$ on the double torus, which explicitly confirms the enumerative results. As a consequence, several new polygonal representations of the double torus are presented.