Article

# On the Bi-embeddability of Certain Steiner Triple Systems of Order 15

Department of Pure Mathematics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, .ukf1
(Impact Factor: 0.65). 07/2002; 23(5):499-505. DOI: 10.1006/eujc.2002.0559
Source: OAI

ABSTRACT

There are 80 non-isomorphic Steiner triple systems of order 15. A standard listing of these is given in Mathon et al.(1983, Ars Combin., 15, 3–110). We prove that systems #1 and #2 have no bi-embedding together in an orientable surface. This is the first known example of a pair of Steiner triple systems of ordern , satisfying the admissibility condition n ≡ 3 or 7(mod 12), which admits no orientable bi-embedding. We also show that the same pair has five non-isomorphic bi-embeddings in a non-orientable surface.

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• "It was shown in [5] that every pair, including isomorphic pairs, has a biembedding in a nonorientable surface. However, it was proved in [4] that at least one pair, namely {#1, #2} in the standard "
##### Article: Orientable biembeddings of Steiner triple systems of order 15
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ABSTRACT: A complete enumeration is given of orientable biembeddings involving five of the 80 Steiner triple systems of order 15. As a consequence, it follows that each of the 80 systems has a biembedding in an orientable surface, and precisely 78 of the systems have orientable self-embeddings.
Journal of Combinatorial Mathematics and Combinatorial Computing 02/2009; 68.
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• "It was proved in [2] that at least one pair, namely {#1, #2} in the standard numbering, has no biembedding in an orientable surface. "
##### Article: A census of the orientable biembeddings of Steiner triple systems of order 15
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ABSTRACT: A complete census is given of the orientable biembeddings of Steiner triple systems of order 15. There are 80 Steiner triple systems of order 15 and these generate a total of 9530 orientable biembeddings.
01/2008; 42.
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• "Constructions given by Ringel [15] for n ≡ 3 (mod 12), and by Youngs [16] for n ≡ 7 (mod 12) prove that there is at least one orientable biembedding for each such value of n. However, it is known that there are pairs of STS(15)s which admit no orientable biembedding [1]. "
##### Article: Nonorientable biembeddings of Steiner triple systems
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ABSTRACT: Constructions due to Ringel show that there exists a nonorientable face 2-colourable triangular embedding of the complete graph on n vertices (equivalently a nonorientable biembedding of two Steiner triple systems of order n) for all with n⩾9. We prove the corresponding existence theorem for with n⩾13.
Discrete Mathematics 08/2004; 285(1-3-285):121-126. DOI:10.1016/j.disc.2004.01.013 · 0.56 Impact Factor