Article
On the Biembeddability of Certain Steiner Triple Systems of Order 15
Department of Pure Mathematics, The Open University, Walton Hall, Milton Keynes, MK7 6AA, .ukf1
European Journal of Combinatorics (Impact Factor: 0.65). 07/2002; 23(5):499505. DOI: 10.1006/eujc.2002.0559 Source: OAI
ABSTRACT
There are 80 nonisomorphic 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 biembedding 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 biembedding. We also show that the same pair has five nonisomorphic biembeddings in a nonorientable 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 "
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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 selfembeddings. 
 "It was proved in [2] that at least one pair, namely {#1, #2} in the standard numbering, has no biembedding in an orientable surface. "
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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. 
 "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]. "
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ABSTRACT: Constructions due to Ringel show that there exists a nonorientable face 2colourable 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.