Embedding K3,3 and K5 on the Double Torus

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.