Embedding K3,3 and K5 on the Double Torus

03/02/2022

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by   William L. Kocay, et al.

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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.