Ordinary Voltage Graphs In Pseudosurfaces And Derived Cellular Homology With Applications To Graph Embeddability Open Access
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We develop a global homological understanding of cellular graph embeddings in orientable and nonorientable surfaces and pseudosurfaces from local intersection data. We apply our new theorems to show that homological data of a graph embedding described as the derived graph of an ordinary voltage graph embedding with voltage group A can be derived by considering cosets of specially constructed subgroups of A.For a prime p greater than 5, we apply our theorems, while making distinctly homological arguments, to show that there is no embedding in the torus of a Generalized Petersen Graph of the form GP(2p,2) that can be derived from an ordinary voltage graph embedding, even though we can produce embeddings of such graphs in the torus. We conclude that no free group action on GP(2p,2) embedded in the torus extends to a cellular automorphism of the torus.For distinct primes p,q at least 5, we use similar argumentation to prove corresponding conclusions about all embeddings of the Cartesian product of two circle graphs Cp and Cq in Dyck's surface.