A. D. Forbes

On 6-sparse Steiner triple systems

We give the first known examples of 6-sparse Steiner triple systems by constructing 29 such systems in the residue class 7 modulo 12, with orders ranging from 139 to 4447. We then present a recursive construction which establishes the existence of 6-sparse systems for an infinite set of orders. Observations are also made concerning existing construction methods for perfect Steiner triple systems, and we give a further example of such a system. This has order 135,859 and is only the fourteenth known. Finally, we present a uniform Steiner triple system of order 180,907.

On colourings of Steiner triple systems

A Steiner triple system, STS(v), is said to be χ-chromatic if the points can be coloured using χ colours, but no fewer, such that no block is monochromatic. All known 3-chromatic STS(v) are also equitably colourable, i.e. there exists a 3-colouring in which the cardinalities of the colour classes differ by at most one. We present examples of 3-chromatic STS(v) which do not admit equitable 3-colourings. We also present further examples of systems with unique and balanced colourings.

Further 6-sparse Steiner Triple Systems

We give a construction that produces 6-sparse Steiner triple systems of order v for all sufficiently large v of the form 3p, p prime and p ≡ 3 (mod 4). We also give a complete list of all 429 6-sparse systems with v < 10000 produced by this construction.

The design of the century

We construct a 2-chromatic Steiner system S(2, 4, 100) in which every block contains three points of one colour and one point of the other colour. The existence of such a design has been open for over 25 years.

Archimedean graph designsII

We examine the design spectra for the vertexedge graphs of some Archimedean solids. In particular, we complete the computation of the spectrum for the truncated cuboctahedron (1 or 64 modulo 144). We extend the known spectrum of the rhombicosidodecahedron by showing that there exist designs of order 81 modulo 240. We add residue class 81 modulo 120 to the known spectra of the icosidodecahedron and the snub cube, each with one possible exception. We add residue class 145 modulo 180 to the known spectra of the truncated dodecahedron and the truncated icosahedron, each with two possible exceptions. Finally, we exhibit the first explicit examples of snub dodecahedron designs.

Biembeddings of Steiner triple systems in orientable pseudosurfaces with one pinch point

We prove that for all n ≡ 13 or 37 (mod 72), there exists a biembedding of a pair of Steiner triple systems of order n in an orientable pseudosurface having precisely one regular pinch point of multiplicity 2.

Uniquely 3-colourable Steiner triple systems

A Steiner triple system (STS(υ)) is said to be 3-balanced if every 3-colouring of it is equitable; that is, if the cardinalities of the colour classes differ by at most one. A 3-colouring, φ, of an STS(υ) is unique if there is no other way of 3-colouring the STS(υ) except possibly by permuting the colours of φ. We show that for every admissible υ ≥ 25, there exists a 3-balanced STS(υ) with a unique 3-colouring and an STS(υ) which has a unique, non-equitable 3- colouring.

Distance and fractional isomorphism in Steiner triple systems

In [8], Quattrochi and Rinaldi introduced the idea ofn−1-isomorphism between Steiner systems. In this paper we study this concept in the context of Steiner triple systems. The main result is that for any positive integerN, there existsv0(N) such that for all admissiblev≥v0(N) and for each STS(v) (sayS), there exists an STS(v) (sayS′) such that for somen>N, S is strictlyn−1-isomorphic toS′. We also prove that for all admissiblev≥13, there exist two STS(v)s which are strictly 2−1-isomorphic.Define the distance between two Steiner triple systemsS andS′ of the same order to be the minimum volume of a tradeT which transformsS into a system isomorphic toS′. We determine the distance between any two Steiner triple systems of order 15 and, further, give a complete classification of strictly 2−1-isomorphic and 3−1-isomorphic pairs of STS(15)s.