Steiner Triple Systems Of Order Research Articles

Steiner triple systems of order 21 with subsystems

The smallest open case for classifying Steiner triple systems is order 21. A Steiner triple system of order 21, an STS\((21)\), can have subsystems of orders 7 and 9, and it is known that there are 12,661,527,336 isomorphism classes of STS\((21)\)s with sub-STS\((9)\)s. Here, the classification of STS\((21)\)s with subsystems is completed by settling the case of STS\((21)\)s with sub-STS\((7)\)s. There are 116,635,963,205,551 isomorphism classes of such systems. An estimation of the number of isomorphism classes of STS\((21)\)s is given.

A visual representation of the Steiner triple systems of order 13

Steiner triple systems (STSs) are a basic topic in combinatorics. In an STS the elements can be collected in threes in such a way that any pair of elements is contained in a unique triple. The two smallest nontrivial STSs, with 7 and 9 elements, arise in the context of finite geometry and nonsingular cubic curves, and have well-known pictorial representations. On the contrary, an STS with 13 elements does not have an intrinsic geometric nature, nor a natural pictorial illustration. In this paper we present a visual representation of the two non-isomorphic Steiner triple systems of order 13 by means of a regular hexagram. The thirteen points of each system are the vertices of the twelve equilateral triangles inscribed in the hexagram. In the case of the non-cyclic system, our representation allows one to visualize in a simple, elegant and highly symmetric way the twenty-six triples, the six automorphisms and their orbits, the eight quadrilaterals, the ten mitres, the thirteen grids, the four 3-colouring patterns, the block-colouring and some distinguished ovals. Our construction is based on a very simple idea (seeing the blocks as much as possible as equilateral triangles), which can be further extended to get new representations of the STSs of order 7 and 9, and of one of the STSs of order 15.

An update on the existence of Kirkman triple systems with steiner triple systems as subdesigns

AbstractA Kirkman triple system of order , KTS, is a resolvable Steiner triple system on elements. In this paper, we investigate an open problem posed by Doug Stinson, namely the existence of KTS which contain as a subdesign a Steiner triple system of order , an STS. We present several different constructions for designs of this form. As a consequence, we completely settle the extremal case , for which a list of possible exceptions had remained for close to 30 years. Our new constructions also provide the first infinite classes for the more general problem. We reduce the other maximal case to (at present) three possible exceptions. In addition, we obtain results for other cases of the form and also near . Our primary method introduces a new type of Kirkman frame which contains group divisible design subsystems. These subsystems can occur with different configurations, and we use two different varieties in our constructions.

Novák's conjecture on cyclic Steiner triple systems and its generalization

Novák conjectured in 1974 that for any cyclic Steiner triple systems of order v with v≡1(mod6), it is always possible to choose one block from each block orbit so that the chosen blocks are pairwise disjoint. We consider the generalization of this conjecture to cyclic (v,k,λ)-designs with 1⩽λ⩽k−1. Superimposing multiple copies of a cyclic symmetric design shows that the generalization cannot hold for all v, but we conjecture that it holds whenever v is sufficiently large compared to k. We confirm that the generalization of the conjecture holds when v is prime and λ=1 and also when λ⩽(k−1)/2 and v is sufficiently large compared to k. As a corollary, we show that for any k⩾3, with the possible exception of finitely many composite orders v, every cyclic (v,k,1)-design without short orbits is generated by a (v,k,1)-disjoint difference family.

An algebraic representation of Steiner triple systems of order 13

In this paper we construct an incidence structure isomorphic to a Steiner triple system of order 13 by defining a set B of twentysix vectors in the 13-dimensional vector space V=GF(5)13, with the property that there exist precisely thirteen 6-subsets of B whose elements sum up to zero in V, which can also be characterized as the intersections of B with thirteen linear hyperplanes of V.

Determination of the size of defining set for Steiner triple systems

The coloring of hypergraph and its special case Steiner triple systems, STS(v), is studied extensively. But the defining set of the coloring of hypergraph even its special case STS(v), has not been explored yet. We study minimum defining set and the largest minimal defining set for 3-coloring of STS(v). We determined minimum defining set and the largest minimal defining set, for all non-isomorphic STS(v), v £ 15. Also we have found the defining number for all Steiner triple systems of order v, and some lower bounds for the size of the largest minimal defining set for all Steiner triple systems of order v, for each admissible v.

Sparse Steiner triple systems of order 21

AbstractA ‐configuration is a set of blocks on points. For Steiner triple systems, ‐configurations are of particular interest. The smallest nontrivial such configuration is the Pasch configuration, which is a ‐configuration. A Steiner triple system of order , an STS, is ‐sparse if it does not contain any ‐configuration for . The existence problem for anti‐Pasch Steiner triple systems has been solved, but these have been classified only up to order 19. In the current work, a computer‐aided classification shows that there are 83,003,869 isomorphism classes of anti‐Pasch STS(21)s. Exploration of the classified systems reveals that there are three 5‐sparse STS(21)s but no 6‐sparse STS(21)s. The anti‐Pasch STS(21)s lead to 14 Kirkman triple systems, none of which is doubly resolvable.

On the maximum double independence number of Steiner triple systems

AbstractThe maximum independence number of Steiner triple systems of order is well‐known. Motivated by questions of access balancing in storage systems, we determine the maximum total cardinality of a pair of disjoint independent sets of Steiner triple systems of order for all admissible orders.

Kirkman triple systems with subsystems

A Steiner triple system of order v, STS(v), together with a resolution of its blocks is called a Kirkman triple system of order v, KTS(v). A KTS(v) exists if and only if v≡3(mod6). The smallest order for which the KTS(v) have not been classified is v=21, which is also the smallest order for which the existence of a doubly resolvable STS(v) is open. Here, KTS(21) with STS(7) and STS(9) subsystems are classified, leading to more than 13 million KTS(21). In this process, systems missing from an earlier classification of KTS(21) with nontrivial automorphisms are encountered, so such a classification is redone.

On a Conjecture of Erdős on Locally Sparse Steiner Triple Systems

A famous theorem of Kirkman says that there exists a Steiner triple system of order n if and only if n ≡ 1,3 mod 6. In 1973, Erdős conjectured that one can find so-called ‘sparse’ Steiner triple systems. Roughly speaking, the aim is to have at most j−3 triples on every set of j points, which would be best possible. (Triple systems with this sparseness property are also referred to as having high girth.) We prove this conjecture asymptotically by analysing a natural generalization of the triangle removal process. Our result also solves a problem posed by Lefmann, Phelps and Rodl as well as Ellis and Linial in a strong form, and answers a question of Krivelevich, Kwan, Loh and Sudakov. Moreover, we pose a conjecture which would generalize the Erdős conjecture to Steiner systems with arbitrary parameters and provide some evidence for this.

On determining when small embeddings of partial Steiner triple systems exist

AbstractA partial Steiner triple system of order is a pair , where is a set of elements and is a set of triples of elements of such that any two elements of occur together in at most one triple. If each pair of elements occur together in exactly one triple it is a Steiner triple system. An embedding of a partial Steiner triple system is a (complete) Steiner triple system such that and . For a given partial Steiner triple system of order it is known that an embedding of order exists whenever satisfies the obvious necessary conditions. Determining whether “small” embeddings of order exist is a more difficult task. Here we extend a result of Colbourn on the ‐completeness of these problems. We also exhibit a family of counterexamples to a conjecture concerning when small embeddings exist.

Steiner Triple Systems of Order 21 with a Transversal Subdesign TD(3, 6)

We prove several structural properties of Steiner triple systems (STS) of order 3w+3 that include one or more transversal subdesigns TD(3,w). Using an exhaustive search, we find that there are 2004720 isomorphism classes of STS(21) including a subdesign TD(3,6), or, equivalently, a 6-by-6 latin square.

Sequencing partial Steiner triple systems

AbstractA partial Steiner triple system of order is sequenceable if there is a sequence of length of its distinct points such that no proper segment of the sequence is a union of point‐disjoint blocks. We prove that if a partial Steiner triple system has at most three point‐disjoint blocks, then it is sequenceable.

Extended Bicolorings of Steiner Triple Systems of Order $2^{h}-1$

Extended Bicolorings of Steiner Triple Systems of Order $2^{h}-1$

Orthogonal representations of Steiner triple system incidence graphs

The unique Steiner triple system of order 7 has a point-block incidence graph known as the Heawood graph. Motivated by questions in combinatorial matrix theory, we consider the problem of constructing a faithful orthogonal representation of this graph, i.e., an assignment of a vector in Cd to each vertex such that two vertices are adjacent precisely when assigned nonorthogonal vectors. We show that d=10 is the smallest number of dimensions in which such a representation exists, a value known as the minimum semidefinite rank of the graph, and give such a representation in 10 real dimensions. We then show how the same approach gives a lower bound on this parameter for the incidence graph of any Steiner triple system, and highlight some questions concerning the general upper bound.

Partitioning 3‐arcs into Steiner Triple Systems

AbstractIn this article, it is shown that there is a partitioning of the set of 3‐arcs in a projective plane of order three into nine pairwise disjoint Steiner triple systems of order 13.

3-pyramidal Steiner triple systems

A design is said to be f -pyramidal when it has an automorphism group which fixes f points and acts sharply transitively on all the others. The problem of establishing the set of values of v for which there exists an f -pyramidal Steiner triple system of order v has been deeply investigated in the case f = 1 but it remains open for a special class of values of v . The same problem for the next possible f , which is f = 3 , is here completely solved: there exists a 3 -pyramidal Steiner triple system of order v if and only if v ≡ 7, 9, 15 (mod 24) or v ≡ 3, 19 (mod 48).

Steiner Triple Systems with High Chromatic Index

It is conjectured that every Steiner triple system of order $v \neq 7$ has chromatic index at most $(v+3)/2$ when $v \equiv 3 \pmod{6}$ and at most $(v+5)/2$ when $v \equiv 1 \pmod{6}$. Herein, we construct a Steiner triple system of order $v$ with chromatic index at least $(v+3)/2$ for each integer $v \equiv 3 \pmod{6}$ such that $v \geq 15$, with four possible exceptions. We further show that the maximum number of disjoint parallel classes in the systems constructed is sublinear in $v$. Finally, we establish for each order $v \equiv 15 \pmod{18}$ that there are at least $v^{v^2(1/6+o(1))}$ non-isomorphic Steiner triple systems with chromatic index at least $(v+3)/2$ and that some of these systems are cyclic.

Sylow Theory for Quasigroups II: Sectional Action

AbstractThe first paper in this series initiated a study of Sylow theory for quasigroups and Latin squares based on orbits of the left multiplication group. The current paper is based on so‐called pseudo‐orbits, which are formed by the images of a subset under the set of left translations. The two approaches agree for groups, but differ in the general case. Subsets are described as sectional if the pseudo‐orbit that they generate actually partitions the quasigroup. Sectional subsets are especially well behaved in the newly identified class of conflatable quasigroups, which provides a unified treatment of Moufang, Bol, and conjugacy closure properties. Relationships between sectional and Lagrangean properties of subquasigroups are established. Structural implications of sectional properties in loops are investigated, and divisors of the order of a finite quasigroup are classified according to the behavior of sectional subsets and pseudo‐orbits. An upper bound is given on the size of a pseudo‐orbit. Various interactions of the Sylow theory with design theory are discussed. In particular, it is shown how Sylow theory yields readily computable isomorphism invariants with the resolving power to distinguish each of the 80 Steiner triple systems of order 15.

Triple Systems are Eulerian

AbstractAn Euler tour of a hypergraph (also called a rank‐2 universal cycle or 1‐overlap cycle in the context of designs) is a closed walk that traverses every edge exactly once. In this paper, using a graph‐theoretic approach, we prove that every triple system with at least two triples is eulerian, that is, it admits an Euler tour. Horan and Hurlbert have previously shown that for every admissible order >3, there exists a Steiner triple system with an Euler tour, while Dewar and Stevens have proved that every cyclic Steiner triple system of order >3 and every cyclic twofold triple system admits an Euler tour.