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Tilings of nonorientable surfaces by Steiner triple systems

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Abstract

A Steiner triple system of order n (for short, STS(n)) is a system of three-element blocks (triples) of elements of an n-set such that each unordered pair of elements occurs in precisely one triple. Assign to each triple (i,j,k) ∊ STS(n) a topological triangle with vertices i, j, and k. Gluing together like sides of the triangles that correspond to a pair of disjoint STS(n) of a special form yields a black-and-white tiling of some closed surface. For each n ≡ 3 (mod 6) we prove that there exist nonisomorphic tilings of nonorientable surfaces by pairs of Steiner triple systems of order n. We also show that for half of the values n ≡ 1 (mod 6) there are nonisomorphic tilings of nonorientable closed surfaces.

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References

  1. Ringel, G., Map Color Theorem, New York: Springer, 1974.

    MATH  Google Scholar 

  2. Grannell, M.J., Griggs, T.S., and Širáň, J., Face 2-Colourable Triangular Embeddings of Complete Graphs, J. Combin. Theory, Ser. B, 1998, vol. 74, no. 1, pp. 8–19.

    Article  MATH  MathSciNet  Google Scholar 

  3. Grannell, M.J., Griggs, T.S., and Širáň, J., Surface Embeddings of Steiner Triple Systems, J. Combin. Des., 1998, vol. 6, no. 5, pp. 325–336.

    Article  MATH  MathSciNet  Google Scholar 

  4. Bonnigton, C.P., Grannell, M.J., Griggs, T.S., and Širáň, J., Exponential Families of Non-Isomorphic Triangulations of Complete Graphs, J. Combin. Theory, Ser. B, 2000, vol. 78, no. 2, pp. 169–184.

    Article  MathSciNet  Google Scholar 

  5. Kühnel, W., Topological Aspects of Twofold Triple Systems, Expo. Math., 1998, vol. 16, no. 4, pp. 289–332.

    MATH  Google Scholar 

  6. Seifert, H. and Threlfall, W., Lehrbuch der Topologie, Leipzig: Teubner, 1934. Translated under the title Topologiya, Moscow-Leningrad: GONTI, 1938.

    Google Scholar 

  7. Youngs, J.W.T., The Mistery of the Heawood Conjecture, Graph Theory and Its Applications, Harris, B., Ed., New York: Academic, 1970, pp. 17–50.

    Google Scholar 

  8. Hall, M., Jr., Combinatorial Theory, Waltham: Blaisdell, 1967. Translated under the title Kombinatorika, Moscow: Mir, 1970.

    MATH  Google Scholar 

  9. Ducrocq, P.M. and Sterboul, F., Les G-systèmes triples, Ann. Discrete Math., 1980, vol. 9, pp. 141–145.

    Article  MATH  MathSciNet  Google Scholar 

  10. Solov’eva, F.I., Steiner Triple Systems and the Rope Problem, in Proc. 2nd Siberian Congr. on Industrial and Applied Mathematics (INPRIM-96), Novosibirsk, 1996, pp. 125–126.

  11. Bennett, G.K., Grannell, M.J., and Griggs, T.S., Bi-Embeddings of Steiner Triple Systems of Order 15, Graphs Combin., 2001, vol. 17, no. 2, pp. 193–197.

    Article  MATH  MathSciNet  Google Scholar 

  12. Solov’eva, F.I., Structure of i-Components of Perfect Binary Codes, Discrete Appl. Math., 2001, vol. 111, no. 1–2, pp. 189–197.

    Article  MATH  MathSciNet  Google Scholar 

  13. Levin, A.I., Constructions of Steiner Systems, Master Thesis, Yakutsk. Gos. Univ., Yakutsk, 1977.

    Google Scholar 

  14. Solov’eva, F.I., Tilings of Closed Surfaces by Steiner Triple Systems, in Proc. Int. Workshop on Coding and Cryptography (WCC’2003), Versailles, France, 2003, pp. 425–431.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Siberian Branch of the RAS, Novosibirsk, Russia

    F. I. Solov’eva

  2. Novosibirsk State University, Russia

    F. I. Solov’eva

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  1. F. I. Solov’eva
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Correspondence to F. I. Solov’eva.

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Original Russian Text © F.I. Solov’eva, 2007, published in Problemy Peredachi Informatsii, 2007, Vol. 43, No. 3, pp. 54–65.

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Solov’eva, F.I. Tilings of nonorientable surfaces by Steiner triple systems. Probl Inf Transm 43, 213–224 (2007). https://doi.org/10.1134/S0032946007030040

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  • DOI: https://doi.org/10.1134/S0032946007030040

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