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Purely tetrahedral quadruple systems

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Abstract

An oriented tetrahedron is a set of four vertices and four cyclic triples with the property that any ordered pair of vertices is contained in exactly one of the cyclic triples. A tetrahedral quadruple system of order n (briefly TQS(n)) is a pair (X, B), where X is an n-element set and B is a set of oriented tetrahedra such that every cyclic triple on X is contained in a unique member of B. A TQS(n) (X, B) is pure if there do not exist two oriented tetrahedra with the same vertex set. In this paper, we show that there is a pure TQS(n) if and only if n ≡ 2,4 (mod 6), n > 4, or n ≡ 1, 5 (mod 12). One corollary is that there is a simple two-fold quadruple system of order n if and only if n ≡ 2,4 (mod 6) and n > 4, or n ≡ 1,5 (mod 12). Another corollary is that there is an overlarge set of pure Mendelsohn triple systems of order n for n ≡ 1,3 (mod 6), n > 3, or n ≡ 0,4 (mod 12).

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Authors and Affiliations

  1. Department of Mathematics, Suzhou University, Suzhou, 215006, China

    Ji Lijun

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  1. Ji Lijun
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Ji, L. Purely tetrahedral quadruple systems. SCI CHINA SER A 49, 1327–1340 (2006). https://doi.org/10.1007/s11425-006-2013-5

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  • DOI: https://doi.org/10.1007/s11425-006-2013-5

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