Embeddings of Resolvable Triple Systems

Recursive constructions for large sets of resolvable hybrid triple systems

Surface embeddings of Steiner triple systems

A Steiner triple system of order n (STS(n)) is said to be embeddable in an orientable surface if there is an orientable embedding of the complete graph Kn whose faces can be properly 2-coloured (say, black and white) in such a way that all black faces are triangles and these are precisely the blocks of the STS(n). If, in addition, all white faces are triangular, then the collection of all white...

Bi-Embeddings of Steiner Triple Systems of Order 15

It was shown by Gerhard Ringel that one of the three non-isomorphic Steiner triple systems of order 15 having an automorphism of order 5 may be biembedded as the faces of a face 2-colourable triangular embedding of K15 in a suitable orientable surface. Ringel’s bi-embedding was obtained from an appropriate current graph. In a recent paper, the present authors showed that a second STS(15) of thi...

Small embeddings for partial triple systems of odd index

It has been conjectured that any partial triple system of order u and index λ can be embedded in a triple system of order v and index λ whenever v ≥ 2u + 1, λ(v − 1) is even and λ ( v 2 ) ≡ 0 (mod 3). This conjecture is known to hold for λ = 1 and for all even λ ≥ 2. Here the conjecture is proven for all remaining values of λ when u ≥ 28.

Self-embeddings of doubled affine Steiner triple systems

Given a properly face two-coloured triangulation of the graph Kn in a surface, a Steiner triple system can be constructed from each of the colour classes. The two Steiner triple systems obtained in this manner are said to form a biembedding. If the systems are isomorphic to each other it is a self-embedding. In the following, for each k ≥ 2, we construct a self-embedding of the doubled affine S...