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Applications of ordinary voltage graph theory to graph embeddability
Author(s) -
Schluchter Steven
Publication year - 2018
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.22172
Subject(s) - mathematics , embedding , torus , combinatorics , klein bottle , graph embedding , graph , discrete mathematics , automorphism , topological graph theory , voltage graph , line graph , computer science , geometry , artificial intelligence
Let p be a prime greater than 5. We show that, while the generalized Petersen graphs of the form G P ( 2 p , 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 G P ( 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 G P ( 2 q , 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.