@article{10.5614/ejgta.2017.5.2.8,
author = {Steven Schluchter and J. Z. Schroeder},
title = {Self-dual Embeddings of K_{4m,4n} in Different Orientable and Nonorientable Pseudosurfaces with the Same Euler Characteristic},
number = {2},
volume = {5},
journal = {Electronic Journal of Graph Theory and Applications},
year = {2017},
abstract = {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 $m, n\ge 1$; 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}.},
doi = {10.5614/ejgta.2017.5.2.8},
pages = {247--263},
}