Graphs and Combinatorics

, Volume 26, Issue 4, pp 537–548 | Cite as

Polyhedral Suspensions of Arbitrary Genus

Original Paper
  • 51 Downloads

Abstract

A new class of polyhedra is discovered—bipyramids of arbitrarily prescribed genus. A two-dimensional generalization of Fáry’s Theorem is established. A purely combinatorial definition of a polyhedral suspension is given. A new regular two-dimensional polyhedron is constructed in four dimensions.

Keywords

Regular polyhedra Simplicial complexes Planarity Graph embedding Maximum genus 

Mathematics Subject Classification (2000)

Primary 57M20 Secondary 05C10 90C57 52B05 52B70 57M15 51M20 51M04 55P40 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bushmelev, A.V., Lawrencenko, S.: Polyhedral tori (in Russian). Kvant 2, 3–5 (1985). http://kvant.mirror1.mccme.ru/1985/02/mnogogranniki–tory.htm
  2. 2.
    Coxeter H.S.M.: Introduction to Geometry, 2nd edn. Wiley, New York (1969)MATHGoogle Scholar
  3. 3.
    Császár, A.: A polyhedron without diagonals. Acta Sci. Math. (Szeged) 13, 140–142 (1949–1950)Google Scholar
  4. 4.
    Fáry I.: On straight line representations of planar graphs. Acta Sci. Math. (Szeged) 11, 229–233 (1948)Google Scholar
  5. 5.
    Gross J.L., Rosen R.H.: A linear-time planarity algorithm for 2-complexes. J. Assoc. Comput. Mach. 20, 611–617 (1979)MathSciNetGoogle Scholar
  6. 6.
    Gross J.L., Rosen R.H.: A combinatorial characterization of planar 2-complexes. Colloq. Math. 44, 241–247 (1981)MathSciNetGoogle Scholar
  7. 7.
    Gross J.L., Tucker T.W.: Topological Graph Theory. Dover, New York (2001)MATHGoogle Scholar
  8. 8.
    Harary F., Lawrencenko S., Korzhik V.: Realizing the chromatic numbers of triangulations of surfaces. Discrete Math. 122, 197–204 (1993)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hliněný, P., Salazar, G.: Approximating the crossing number of toroidal graphs. In: Tokuyama, T., et al. (eds.) Algorithms and Computation. Proceedings of ISAAC 2007. Lecture Notes in Computer Science, vol. 4835, pp. 148–159. Springer, Berlin (2007)Google Scholar
  10. 10.
    Kuratowski K.: Sur le problème des courbes gauches en topologie. Fund. Math. 15, 271–283 (1930)MATHGoogle Scholar
  11. 11.
    Lawrencenko, S.: All the twenty-one irreducible triangulations of the torus are realizable as polyhedra in E 3 (in Russian)] (Second-place-winning Manuscript, Student Research Paper Contest, Faculty of Mech. & Math.). Moscow State University, Moscow (1983)Google Scholar
  12. 12.
    Lawrencenko, S.: All irreducible triangulations of the torus are realizable as polyhedra in E 3 (in Russian) (Master’s Thesis). Moscow State University, Moscow (1984)Google Scholar
  13. 13.
    Lawrencenko, S.: Explicit enumeration of all automorphisms of the toroidal irreducible triangulations and all toroidal embeddings of the labeled graphs of those triangulations (in Russian) (No. 2779 Uk87. 1 Oct 1987). UkrNIINTI, Kharkov (1987)Google Scholar
  14. 14.
    Mohar B.: On the minimal genus of 2-complexes. J. Graph Theory 24, 281–290 (1997)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Mohar, B.: Personal communication to S. Lawrencenko (31 Mar 2009)Google Scholar
  16. 16.
    Negami, S.: A verbal comment on a talk given by S. Lawrencenko at 20th (International) Workshop on Topological Graph Theory in Yokohama (27 Nov 2008)Google Scholar
  17. 17.
    Pach, J., Tóth, G.: Crossing number of toroidal graphs. In: Klazar, M., et al. (eds.): Topics in Discrete Mathematics, Dedicated to Jarik Nešetoil on the occasion of his 60th birthday (Algorithms and Combinatorics, vol. 26, pp. 581–590). Springer, Berlin (2006)Google Scholar
  18. 18.
    Ringeisen R.D.: Graphs of given genus and arbitrarily large maximum genus. Discrete Math. 6, 169–174 (1973)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Ringel G., Youngs J.W.T.: Solution of the Heawood map-coloring problem. Proc. Natl. Acad. Sci. USA 60, 438–445 (1968)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Steinitz E., Rademacher H.: Vorlesungen über die Theorie der Polyeder. Springer, Berlin (1934)Google Scholar

Copyright information

© Springer 2010

Authors and Affiliations

  1. 1.Department of MathematicsMoscow Institute of Steel and AlloysMoscowRussia

Personalised recommendations