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 Mull (1999) and Mull et al. (2008), using Burnside's Lemma and automorphism groups of K 3 , 3 and K 5 , 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 , and 14 orientable and 17 non -orientable 2-cell embeddings of K 5 on the double torus, which are explicitly obtained using an algorithmic procedure of expanding from minors. Therefore we confirm the numbers of embeddings obtained by Mull (1999) and Mull et al. (2008). As a consequence, several new polygonal representations of the double torus are presented. Rotation systems for the one -face embeddings of K 5 on the triple torus are also found, using exhaustive search. (c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY -NC -ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).