Self-dual embeddings of K_4m,4n in different orientable and nonorientable pseudosurfaces with the same Euler characteristic - Steven Schluchter; J. Z. Schroeder (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2017)
A proper embedding of a graph G in a pseudosurface P is an embedding in which the regions of the complement of G in P are homeomorphic to discs and a vertex of G appears at each pinchpoint in P; we say that a proper embedding of G in P is self dual if there exists an isomorphism from G to its dual graph. We give an explicit construction of a self-dual embedding of the complete bipartite graph K_4m,4n in an orientable pseudosurface for all m, n1; we show that this embedding maximizes the number of umbrellas of each vertex and has the property that for any vertex v of K_4m,4n, there are two faces of the constructed embedding that intersect all umbrellas of v. Leveraging these properties and applying a lemma of Bruhn and Diestel, we apply a surgery introduced here or a different known surgery of Edmonds to each of our constructed embeddings for which at least one of m or n is at least 2. The result of these surgeries is that there exist distinct orientable and nonorientable pseudosurfaces with the same Euler characteristic that feature a self-dual embedding of K_4m,4n.