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All Journal Electronic Journal of Graph Theory and Applications (EJGTA)
J. Z. Schroeder
Mosaic Center Radstock, Kej Bratstvo Edinstvo 45, 1230 Gostivar, Republic of Macedonia.
Electrical & Electronics Engineering

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Self-dual embeddings of K_{4m,4n} in different orientable and nonorientable pseudosurfaces with the same Euler characteristic Steven Schluchter; J. Z. Schroeder
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 5, No 2 (2017): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2017.5.2.8

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}.
Co-Authors Steven Schluchter