Antipodal Embeddings of Graphs
M. Farzan, Derek A. Waller · Journal of the London Mathematical Society · 1977
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 2d 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 : S2→RP2 where the real projective plane is obtained from the sphere, we study the relation between embeddings of a planar graph in S2 and embeddings of G in RP2. In particular, our main theorem establishes that every planar antipodal graph D has an embedding in S2 such that p is a restriction of the projection π.