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Antipodal Embeddings of Graphs
Author(s) -
Farzan M.,
Waller D. A.
Publication year - 1977
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-15.3.377
Subject(s) - antipodal point , combinatorics , mathematics , vertex (graph theory) , quotient , morphism , discrete mathematics , graph , geometry
An antipodal graph D of diameter d has the property that each vertex υ has a unique (antipodal) vertex υ at distance d from υ in D . We show that any such D has circuits of length 2 d passing through antipodal pairs of vertices. The identification of antipodal vertex‐pairs in D produces a quotient graph G with a double cover projection morphism p : D → G . Using the two‐fold quotient map of surfaces π : S 2 → RP 2 where the real projective plane is obtained from the sphere, we study the relation between embeddings of a planar graph in S 2 and embeddings of G in RP 2 . In particular, our main theorem establishes that every planar antipodal graph D has an embedding in S 2 such that p is a restriction of the projection π.