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Journal of the London Mathematical Society 1977 s2-15(3):377-383; doi:10.1112/jlms/s2-15.3.377
© 1977 by London Mathematical Society
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© Oxford University Press

Antipodal Embeddings of Graphs

M. Farzan and D. A. Waller

Department of Pure Mathematics, University College of Swansea Swansea, SA2 8PP, Great Britain

An antipodal graph D of diameter d has the property that each vertex {upsilon} has a unique (antipodal) vertex {upsilon} at distance d from {upsilon} 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 {pi} : 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 {pi}.


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