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Planarity of Cayley diagrams

Part of the Lecture Notes in Mathematics book series (LNM,volume 303)

Abstract

Use has been made of the notion of point symmetric embeddings and the bounding properties of Jordan curves in the plane. We may relax the notion as follows. An embedding of a Cayley diagram in an orientable compact 2-manifold is called weakly point symmetric if the succession of edges either clockwise or counterclockwise is the same at each vertex, to within a cyclic permutation (or an inverse of a cyclic permutation). Under the conditions of Theorem 1, with point symmetry replaced by weak point symmetry, the local graph L* may be continued. Can it be embedded in the projective plane, or in some suitable "minimal" non-orientable compact 2-manifold? Since bounding properties of Jordan curves no longer apply in non-orientable surfaces, the problem of the projective planarity of a presentation with weakly point symmetric local graph remains.

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References

  1. J. Edmonds, A combinatorial representation for polyhedral surfaces, Notices Amer. Math. Soc. 7 (1960), 646.

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© 1972 Springer-Verlag

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Levinson, H.W., Rapaport, E.S. (1972). Planarity of Cayley diagrams. In: Alavi, Y., Lick, D.R., White, A.T. (eds) Graph Theory and Applications. Lecture Notes in Mathematics, vol 303. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0067370

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  • DOI: https://doi.org/10.1007/BFb0067370

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-06096-3

  • Online ISBN: 978-3-540-38114-3

  • eBook Packages: Springer Book Archive