Applications of ordinary voltage graph theory to graph embeddability

摘要

Let p be a prime greater than 5. We show that, while the generalized Petersen graphs of the form GP(2p,2) have cellular toroidal embeddings, they have no such embeddings having the additional property that a free action of a group on the graph extends to a cellular automorphism of the torus. Such an embedding is called a derived embedding. We also show that GP(6,2) does have a derived embedding in the torus, and we show that for any odd q, each generalized Petersen graph of the form GP(2q,2) has a derived embedding in the Klein bottle, which has the same Euler characteristic as the torus. We close with some comments that frame these results in the light of Abrams and Slilaty's recent work on graphs featuring group actions that extend to spherical embeddings of those graphs.