Even Embeddings of the Complete Graphs and Their Cycle Parities |
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| Authors: | Kenta Noguchi |
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| Affiliation: | DEPARTMENT OF INFORMATION ENVIRONMENT, TOKYO DENKI UNIVERSITY, INZAI, JAPAN |
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| Abstract: | The complete graph  on n vertices can be quadrangularly embedded on an orientable (resp. nonorientable) closed surface F2 with Euler characteristic  if and only if  (resp.  and  ). In this article, we shall show that if  quadrangulates a closed surface F2, then  has a quadrangular embedding on F2 so that the length of each closed walk in the embedding has the parity specified by any given homomorphism  , called the cycle parity. |
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| Keywords: | current graph cycle parity even embedding quadrangulation |
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on n vertices can be quadrangularly embedded on an orientable (resp. nonorientable) closed surface F2 with Euler characteristic 
if and only if 
(resp. 
and 
). In this article, we shall show that if 
quadrangulates a closed surface F2, then 
has a quadrangular embedding on F2 so that the length of each closed walk in the embedding has the parity specified by any given homomorphism 
, called the cycle parity.