TY - JOUR
AU - Steven Schluchter
AU - J. Z. Schroeder
T1 - Self-dual Embeddings of K_{4m,4n} in Different Orientable and Nonorientable Pseudosurfaces with the Same Euler Characteristic
T2 - Electronic Journal of Graph Theory and Applications
JF - Electronic Journal of Graph Theory and Applications
DA - 2017///
PY - 2017
IS - 2
VL - 5
SP - 247
EP - 263
AB - 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}.
DO - 10.5614/ejgta.2017.5.2.8
ER -