Antipodal Embeddings of Graphs

An antipodal graph D of diameter d has the property that each vertex v has a unique (antipodal) vertex v at distance d from v in D. We show that any such D has circuits of length Id 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 n : S->RP where the real projective plane is obtained from the sphere, we study the relation between embeddings of a planar graph in S and embeddings of G in RP. In particular, our main theorem establishes that every planar antipodal graph D has an embedding in S such that p is a restriction of the projection 77.