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Even Embeddings of the Complete Graphs and Their Cycle Parities
Author(s) -
Noguchi Kenta
Publication year - 2017
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.22055
Subject(s) - mathematics , embedding , combinatorics , homomorphism , parity (physics) , euler characteristic , graph , surface (topology) , graph homomorphism , discrete mathematics , euler's formula , line graph , geometry , graph power , computer science , mathematical analysis , physics , particle physics , artificial intelligence
The complete graph K n on n vertices can be quadrangularly embedded on an orientable (resp. nonorientable) closed surface F 2 with Euler characteristic ε ( F 2 ) = n ( 5 − n ) / 4 if and only if n ≡ 0 , 5 ( mod 8 ) (resp. n ≡ 0 , 1 ( mod 4 ) and n ≠ 5 ). In this article, we shall show that if K n quadrangulates a closed surface F 2 , then K n has a quadrangular embedding on F 2 so that the length of each closed walk in the embedding has the parity specified by any given homomorphism ρ : π 1 ( F 2 ) → Z 2 , called the cycle parity.