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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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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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