Partial Steiner Triple Systems Research Articles

Block avoiding point sequencings of partial Steiner systems

A partial(n,k,t)_lambda -system is a pair (X,{mathcal {B}}) where X is an n-set of vertices and {mathcal {B}} is a collection of k-subsets of X called blocks such that each t-set of vertices is a subset of at most lambda blocks. A sequencing of such a system is a labelling of its vertices with distinct elements of {0,ldots ,n-1}. A sequencing is ell -block avoiding or, more briefly, ell -good if no block is contained in a set of ell vertices with consecutive labels. Here we give a short proof that, for fixed k, t and lambda , any partial (n,k,t)_lambda -system has an ell -good sequencing for some ell =Theta (n^{1/t}) as n becomes large. This improves on results of Blackburn and Etzion, and of Stinson and Veitch. Our result is perhaps of most interest in the case k=t+1 where results of Kostochka, Mubayi and Verstraëte show that the value of ell cannot be increased beyond Theta ((n log n)^{1/t}). A special case of our result shows that every partial Steiner triple system (partial (n,3,2)_1-system) has an ell -good sequencing for each positive integer ell leqslant 0.0908,n^{1/2}.

Countable homogeneous Steiner triple systems avoiding specified subsystems

In this article we construct uncountably many new homogeneous locally finite Steiner triple systems of countably infinite order as Fraïssé limits of classes of finite Steiner triple systems avoiding certain subsystems. The construction relies on a new embedding result: any finite partial Steiner triple system has an embedding into a finite Steiner triple system that contains no nontrivial proper subsystems that are not subsystems of the original partial system. Fraïssé's construction and its variants are rich sources of examples that are central to model-theoretic classification theory, and recently infinite Steiner systems obtained via Fraïssé-type constructions have received attention from the model theory community.

Block‐avoiding point sequencings

AbstractLet and be positive integers. Recent papers by Kreher, Stinson, and Veitch have explored variants of the problem of ordering the points in a triple system (such as a Steiner triple system [STS], directed triple system, or Mendelsohn triple system) on points so that no block occurs in a segment of consecutive entries (thus the ordering is locally block‐avoiding). We describe a greedy algorithm which shows that such an ordering exists, provided that is sufficiently large when compared with . This algorithm leads to improved bounds on the number of points in cases where this was known, but also extends the results to a significantly more general setting (which includes, e.g., orderings that avoid the blocks of a design). Similar results for a cyclic variant of this situation are also established. We construct STSs and quadruple systems where can be large, showing that a bound of Stinson and Veitch is reasonable. Moreover, we generalize the Stinson–Veitch bound to a wider class of block designs and to the cyclic case. The results of Kreher, Stinson, and Veitch were originally inspired by results of Alspach, Kreher, and Pastine, who (motivated by zero‐sum avoiding sequences in abelian groups) were interested in orderings of points in a partial Steiner triple system where no segment is a union of disjoint blocks. Alspach et al. show that, when the system contains at most pairwise disjoint blocks, an ordering exists when the number of points is more than . By making use of a greedy approach, the paper improves this bound to .

On partial parallel classes in partial Steiner triple systems

For an integer ρ such that 1≤ρ≤v∕3, define β(ρ,v) to be the maximum number of blocks in any partial Steiner triple system on v points in which the maximum partial parallel class has size ρ. We obtain lower bounds on β(ρ,v) by giving explicit constructions, and upper bounds on β(ρ,v) result from counting arguments. We show that β(ρ,v)∈Θ(v) if ρ is a constant, and β(ρ,v)∈Θ(v2) if ρ=v∕c, where c is a constant. When ρ is a constant, our upper and lower bounds on β(ρ,v) differ by a constant that depends on ρ. Finally, we apply our results on β(ρ,v) to obtain infinite classes of sequenceable partial Steiner triple systems.

Completion and deficiency problems

Given a partial Steiner triple system (STS) of order n, what is the order of the smallest complete STS it can be embedded into? The study of this question goes back more than 40 years. In this paper we answer it for relatively sparse STSs, showing that given a partial STS of order n with at most r≤εn2 triples, it can always be embedded into a complete STS of order n+O(r), which is asymptotically optimal. We also obtain similar results for completions of Latin squares and other designs.This suggests a new, natural class of questions, called deficiency problems. Given a global spanning property P and a graph G, we define the deficiency def(G) of the graph G with respect to the property P to be the smallest positive integer t such that the join G⁎Kt has property P. To illustrate this concept we consider deficiency versions of some well-studied properties, such as having a Kk-decomposition, Hamiltonicity, having a triangle-factor and having a perfect matching in hypergraphs.The main goal of this paper is to propose a systematic study of these problems; thus several future research directions are also given.

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.

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.

Large girth approximate Steiner triple systems

In 1973 Erdos asked whether there are n-vertex partial Steiner triple systems with arbitrary high girth and quadratically many triples. (Here girth is defined as the smallest integer g \ge 4 for which some g-element vertex-set contains at least g-2 triples.) We answer this question, by showing existence of approximate Steiner triple systems with arbitrary high girth. More concretely, for any fixed \ell \ge 4 we show that a natural constrained random process typically produces a partial Steiner triple system with (1/6-o(1))n^2 triples and girth larger than \ell. The process iteratively adds random triples subject to the constraint that the girth remains larger than \ell. Our result is best possible up to the o(1)-term, which is a negative power of n.

Two monads on the category of graphs

AbstractWe introduce two monads on the category of graphs and prove that their Eilenberg-Moore categories are isomorphic to the category of perfect matchings and the category of partial Steiner triple systems, respectively. As a simple application of these results, we describe the product in the categories of perfect matchings and partial Steiner triple systems.

Ramsey theory on Steiner triples

AbstractWe call a partial Steiner triple system C (configuration) t‐Ramsey if for large enough n (in terms of ), in every t‐coloring of the blocks of any Steiner triple system STS(n) there is a monochromatic copy of C. We prove that configuration C is t‐Ramsey for every t in three cases: C is acyclic every block of C has a point of degree one C has a triangle with blocks 123, 345, 561 with some further blocks attached at points 1 and 4 This implies that we can decide for all but one configurations with at most four blocks whether they are t‐Ramsey. The one in doubt is the sail with blocks 123, 345, 561, 147.

Binomial partial Steiner triple systems containing complete graphs

We propose a new approach to studies on partial Steiner triple systems consisting in determining complete graphs contained in them. We establish the structure which complete graphs yield in a minimal PSTS that contains them. As a by-product we introduce the notion of a binomial PSTS as a configuration with parameters of a minimal PSTS with a complete subgraph. A representation of binomial PSTS with at least a given number of its maximal complete subgraphs is given in terms of systems of perspectives. Finally, we prove that for each admissible integer there is a binomial PSTS with this number of maximal complete subgraphs.

A complete classification of the [formula omitted]-configurations with at least three [formula omitted]-graphs

The class of ((n+12)n−1(n+13)3)-configurations which contain at least n−2Kn-graphs coincides with the class of so called systems of triangle perspectives i.e. of configurations which contain a bundle of n−2 Pasch configurations with a common line. For n=5 the class consists of all binomial partial Steiner triple systems on 15 points, that contain at least three K5-graphs. In this case a complete classification of respective configurations is given and their automorphisms are determined.

Embedding Partial Steiner Triple Systems with Few Triples

In 2009 it was established that any partial Steiner triple system of order $u$ has an embedding of order $v$ for each $v\geq 2u+1$ such that $v \equiv 1,3$ (mod 6), in accordance with a conjecture of Lindner. It is known that for each $u\geq 9$, there exists a partial Steiner triple system of order $u$ that does not have an embedding of order $v$ for any $v<2u+1$, so this result is best possible in one sense. Many partial Steiner triple systems do have embeddings of orders smaller than $2u+1$, however, although little is known about when such embeddings exist. In this paper we construct embeddings of orders less than $2u+1$ for partial Steiner triple systems with few triples. In particular, we show that a partial Steiner triple system of order $u \geq 62$ with at most $\frac{u^2}{50}-\frac{11u}{100}-\frac{116}{75}$ triples has an embedding of order $v$ for each admissible integer $v \geq \frac{8u+17}{5}$.

Small Embeddings of Partial Steiner Triple Systems

AbstractIt was proved in 2009 that any partial Steiner triple system of order u has an embedding of order v for each admissible . This result is best possible in the sense that, for each , there exists a partial Steiner triple system of order u that does not have an embedding of order v for any . Many partial Steiner triple systems do have embeddings of orders smaller than , but much less is known about when these embeddings exist. In this paper, we detail a method for constructing such embeddings. We use this method to show that each member of a wide class of partial Steiner triple systems has an embedding of order v for at least half (or nearly half) of the orders for which an embedding could exist. For some members of this class we are able to completely determine the set of all orders for which the member has an embedding.

Uniform two‐class regular partial Steiner triple systems

AbstractA 2‐class regular partial Steiner triple system is a partial Steiner triple system whose points can be partitioned into 2‐classes such that no triple is contained in either class and any two points belonging to the same class are contained in the same number of triples. It is uniform if the two classes have the same size. We provide necessary and sufficient conditions for the existence of uniform 2‐class regular partial Steiner triple systems.

Trails of triples in partial triple systems

Given v, t, and m, does there exist a partial Steiner triple system of order v with t triples whose triples can be ordered so that any m consecutive triples are pairwise disjoint? Given v, t, and m 1, m 2, . . . , m s with $${t = \sum_{i=1}^s m_i}$$ , does there exist a partial Steiner triple system with t triples whose triples can be partitioned into partial parallel classes of sizes m 1, . . . , m s ? An affirmative answer to the first question gives an affirmative answer to the second when m i ? m for each $${i \in \{1,2,\ldots,s\}}$$ . These questions arise in the analysis of erasure codes for disk arrays and that of codes for unipolar communication, respectively. A complete solution for the first problem is given when m is at most $${\frac{1}{3}\left(v-(9v)^{2/3}\right)+{O}\left(v^{1/3}\right)}$$ .

Spanning trees of 3-uniform hypergraphs

Masbaum and Vaintrobʼs “Pfaffian matrix-tree theorem” implies that counting spanning trees of a 3-uniform hypergraph (abbreviated to 3-graph) can be done in polynomial time for a class of “3-Pfaffian” 3-graphs, comparable to and related to the class of Pfaffian graphs. We prove a complexity result for recognizing a 3-Pfaffian 3-graph and describe two large classes of 3-Pfaffian 3-graphs – one of these is given by a forbidden subgraph characterization analogous to Littleʼs for bipartite Pfaffian graphs, and the other consists of a class of partial Steiner triple systems for which the property of being 3-Pfaffian can be reduced to the property of an associated graph being Pfaffian. We exhibit an infinite set of partial Steiner triple systems that are not 3-Pfaffian, none of which can be reduced to any other by deletion or contraction of triples.We also find some necessary or sufficient conditions for the existence of a spanning tree of a 3-graph (much more succinct than can be obtained by the currently fastest polynomial-time algorithm of Gabow and Stallmann for finding a spanning tree) and a superexponential lower bound on the number of spanning trees of a Steiner triple system.

Semi-Pappus configurations; combinatorial generalizations of the Pappus configuration

A class of partial Steiner triple systems generalizing a reduct of the classical Pappus configuration and thus called semi-Pappus configurations is defined. Fundamental geometric properties, in particular, representations and the automorphisms of semi-Pappus configurations and of the convolutions of semi-Pappus configurations and the group C 2 are established.

Completing partial commutative quasigroups constructed from partial Steiner triple systems is NP-complete

Deciding whether an arbitrary partial commutative quasigroup can be completed is known to be NP-complete. Here, we prove that it remains NP-complete even if the partial quasigroup is constructed, in the standard way, from a partial Steiner triple system. This answers a question raised by Rosa in [A. Rosa, On a class of completable partial edge-colourings, Discrete Appl. Math. 35 (1992) 293–299]. To obtain this result, we prove necessary and sufficient conditions for the existence of a partial Steiner triple system of odd order having a leave L such that E ( L ) = E ( G ) where G is any given graph.

A proof of Lindner's conjecture on embeddings of partial Steiner triple systems

AbstractLindner's conjecture that any partial Steiner triple system of order u can be embedded in a Steiner triple system of order v if $v\equiv 1,3 \; ({\rm mod}\; 6)$ and $v\geq 2u+1$ is proved. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 63–89, 2009