Concerning multiplier automorphisms of cyclic Steiner triple systems

A cyclic Steiner triple system, presented additively over Z _v as a set B of starter blocks, has a non-trivial multiplier automorphism λ ≠ 1 when λB is a set of starter blocks for the same Steiner triple system. When does a cyclic Steiner triple system of order v having a nontrivial multiplier automorphism exist? Constructions are developed for such systems; of most interest, a novel extension of Netto's classical construction for prime orders congruent to 1 (mod 6) to prime powers is proved. Nonexistence results are then established, particularly in the cases when v = (2^β + 1)^α, when v = 9p with p ≡ 5 (mod 6), and in certain cases when all prime divisors are congruent to 5 (mod 6). Finally, a complete solution is given for all v < 1000, in which the remaining cases are produced by simple computations.     

Embedding Steiner triple systems in hexagon triple systems

A hexagon triple is the graph consisting of the three triangles (triples) {a,b,c},{c,d,e}, and {e,f,a}, where a,b,c,d,e, and f are distinct. The triple {a,c,e} is called an inside triple. A hexagon triple system of order n is a pair (X,H) where H is a collection of edge disjoint hexagon triples which partitions the edge set of K_n with vertex set X. The inside triples form a partial Steiner triple system. We show that any Steiner triple system of order n can be embedded in the inside triples of a hexagon triple system of order approximately 3n.     

Characterization of affine Steiner triple systems and Hall triple systems

It is known that a Steiner triple system is projective if and only if it does not contain the four-triple configuration C₁₄. We find three configurations such that a Steiner triple system is affine if and only if it does not contain any of these configurations. Similarly, we characterize Hall triple systems, a superclass of affine Steiner triple systems, using two forbidden configurations.     

Translation triple systems

We prove the following result. Let S be a Steiner triple system embedded in the projective plane of order n, such that r=n+1, and such that there exists a line l of exterior to S. Let G be a collineation group of fixing S, fixing l and transitive on the blocks of S. Then n=3 and S=l=AG(2, 3), and G contains the group of translations of S with respect to l.This research was supported by NSERC Grant A3485.     

Indecomposable triple systems

A t-design (λ, t, d, n) is a system $\text{B}$ of sets of size d from an n-set S, such that each t subset of S is contained in exactly λ elements of $\text{B}$. A t-design is indecomposable (written IND(λ, t, d, n)) if there does not exist a subset $\text{B}$ ? $\text{B}$ such that $\text{B}$ is a (λ, t, d, n) for some λ, 1 ? λ < λ. A triple system is a (λ; 2, 3, n). Recursive and constructive methods (several due to Hanani) are employed to show that: (1) an IND(2; 2, 3, n) exists for n ≡ 0, 1 (mod 3), n ? 4 and n ≡ 7 (designs of Bhattacharya are used here), (2) an IND(3; 2, 3, n) exists for n odd, n ? 5, (3) if an IND(λ, 2, 3, n) exists, n odd, then there exists an infinite number of indecomposable triple systems with that λ.     

Directed triple systems

Directed triple systems are an example of block designs on directed graphs. A block design on a directed graph can be defined as follows. Let G be a directed graph of k vertices which contain no loops. Let S be a set of υ elements. A collection of k-subsets of S with an assignment of the elements of each k-subset to the vertices of G is called a block design on G of order υ if the following is satisfied. Any ordered pair of elements of S is assigned λ times to an edge of G.For example, if S = {a, b, c, d, e} and

and bae; cad; abc; dbe; acd; bce; adb; cde; aed; bec; is a collection of 3-subsets so written that in each subset the first element is assigned to the vertex 1, the second to 2, and the third to 3, then the collection is a block design on G with λ = 1.In this paper, it is shown that for the graph

if λ = 1, then the graph exists for all υ such that ν ? 2 mod 3.     

Hanani triple systems

Hanani triple systems onv≡1 (mod 6) elements are Steiner triple systems having (v−1)/2 pairwise disjoint almost parallel classes (sets of pairwise disjoint triples that spanv−1 elements), and the remaining triples form a partial parallel class. Hanani triple systems are one natural analogue of the Kirkman triple systems onv≡3 (mod 6) elements, which form the solution of the celebrated Kirkman schoolgirl problem. We prove that a Hanani triple system exists for allv≡1 (mod 6) except forv ∈ {7, 13}.     

Embeddings of pure mendelsohn triple systems and pure directed triple systems

It is proved in this article that the necessary and sufficient conditions for the embedding of a λ-fold pure Mendelsohn triple system of order v in λ-fold pure Mendelsohn triple of order u are λu(u ? 1) ≡ 0 (mod 3) and u ? 2v + 1. Similar results for the embeddings of pure directed triple systems are also obtained. © 1995 John Wiley & Sons, Inc.     

Graph factorization,general triple systems,and cyclic triple systems

In this self-contained exposition, results are developed concerning one-factorizations of complete graphs, and incidence matrices are used to turn these factorization results into embedding theorems on Steiner triple systems. The result is a constructive graphical proof that a Steiner triple system exists for any order congruent to 1 or 3 modulo 6. A pairing construction is then introduced to show that one can also obtain triple systems which are cyclically generated.     

Anti-mitre steiner triple systems

A mitre in a Steiner triple system is a set of five triples on seven points, in which two are disjoint. Recursive constructions for Steiner triple systems containing no mitre are developed, leading to such anti-mitre systems for at least 9/16 of the admissible orders. Computational results for small cyclic Steiner triple systems are also included.     

Planar Steiner triple systems

A regular planar Steiner triple system is a Steiner triple system provided with a family of non-trivial sub-systems of the same cardinality (called planes) such that (i) every set of 3 non collinear points is contained in exactly one plane and (ii) for every plane H and every disjoint block B, there are exactly planes containing B and intersecting H in a block. We prove that a regular planar Steiner triple system is necessarily a projective space of dimension greater than 2 over GF(2), the 3-dimensional affine space over GF(3), an S(2, 3, 2 (6m+7) (3m²+3m+1)+1) with m1, an S(2, 3, 171), an S(2, 3, 183) or an S(2, 3, 2055).     

Perfect dexagon triple systems

We completely determine the spectrum for perfect dexagon triple systems.     

Extended directed triple systems

Let {n;b₂,b₁} denote the class of extended directed triple systems of the order n in which the number of blocks of the form [a,b,a] is b₂ and the number of blocks of the form [b,a,a] or [a,a,b] is b₁. In this paper, we have shown that the necessary and sufficient condition for the existence of the class {n;b₂,b₁} is b₁≠1, 0?b₂+b₁?n and

(1)

for ;

(2)

for .