Biembedding Steiner Triple Systems in Surfaces Using the Bose Construction

A uniform framework is presented for biembedding Steiner triple systems obtained from the Bose construction using a cyclic group of odd order, in both orientable and nonorientable surfaces. Within this framework, in the nonorientable case, a formula is given for the number of isomorphism classes and the particular biembedding of Ducrocq and Sterboul (preprint 18pp., 1978) is identified. In the orientable case, it is shown that the biembedding of Grannell et al. (J Combin Des 6 ( 7 ), 325–336) is, up to isomorphism, the unique biembedding of its type. Automorphism groups of the biembeddings are also given.