A conjecture on small embeddings of partial Steiner triple systems

A well‐known, and unresolved, conjecture states that every partial Steiner triple system of order u can be embedded in a Steiner triple system of order υ for all υ ≡ 1 or 3, (mod 6), υ ≥ 2u + 1. However, some partial Steiner triple systems of order u can be embedded in Steiner triple systems of order υ <2u + 1. A more general conjecture that considers these small embeddings is presented and verified for some cases. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 313–321, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10017     

Sequencing partial Steiner triple systems

A partial Steiner triple system of order $n$ is sequenceable if there is a sequence of length $n$ 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.     

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.     

There exist Steiner triple systems of order 15 that do not occur in a perfect binary one‐error‐correcting code

The codewords at distance three from a particular codeword of a perfect binary one‐error‐correcting code (of length 2^m?1) form a Steiner triple system. It is a longstanding open problem whether every Steiner triple system of order 2^m?1 occurs in a perfect code. It turns out that this is not the case; relying on a classification of the Steiner quadruple systems of order 16 it is shown that the unique anti‐Pasch Steiner triple system of order 15 provides a counterexample. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 465–468, 2007     

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

Lindner's conjecture that any partial Steiner triple system of order u can be embedded in a Steiner triple system of order v if and is proved. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 63–89, 2009     

Coloring Cubic Graphs by Point‐Intransitive Steiner Triple Systems

An ‐coloring of a cubic graph G is an edge coloring of G by points of a Steiner triple system such that the colors of any three edges meeting at a vertex form a block of . A Steiner triple system that colors every simple cubic graph is said to be universal. It is known that every nontrivial point‐transitive Steiner triple system that is neither projective nor affine is universal. In this article, we present the following results.

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Super‐simple holey Steiner pentagon systems and related designs

A Steiner pentagon system of order v (SPS(v)) is said to be super‐simple if its underlying (v, 5, 2)‐BIBD is super‐simple; that is, any two blocks of the BIBD intersect in at most two points. It is well known that the existence of a holey Steiner pentagon system (HSPS) of type T implies the existence of a (5, 2)‐GDD of type T. We shall call an HSPS of type T super‐simple if its underlying (5, 2)‐GDD of type T is super‐simple; that is, any two blocks of the GDD intersect in at most two points. The existence of HSPSs of uniform type hⁿ has previously been investigated by the authors and others. In this article, we focus our attention on the existence of super‐simple HSPSs of uniform type hⁿ. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 301–328, 2008     

Embedding partial Steiner triple systems so that their automorphisms extend

It is shown that there is a function g on the natural numbers such that a partial Steiner triple system U on u points can be embedded in a Steiner triple system V on ν points, in such a way that all automorphisms of U can be extended to V, for every admissible ν satisfying ν > g(u). We find exponential upper and lower bounds for g. © 2005 Wiley Periodicals, Inc. J Combin Designs.     

Linearly Derived Steiner Triple Systems

We attach a graph to every Steiner triple system. The chromatic number of this graph is related to the possibility of extending the triple system to a quadruple system. For example, the triple systems with chromatic number one are precisely the classical systems of points and lines of a projective geometry over the two-element field, the Hall triple systems have chromatic number three (and, as is well-known, are extendable) and all Steiner triple systems whose graph has chromatic number two are extendable. We also give a configurational characterization of the Hall triple systems in terms of mitres.     

A Steiner triple system which colors all cubic graphs

We prove that there is a Steiner triple system ?? such that every simple cubic graph can have its edges colored by points of ?? in such a way that for each vertex the colors of the three incident edges form a triple in ??. This result complements the result of Holroyd and ?koviera that every bridgeless cubic graph admits a similar coloring by any Steiner triple system of order greater than 3. The Steiner triple system employed in our proof has order 381 and is probably not the smallest possible. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 15–24, 2004     

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.     

Ramsey theory on Steiner triples

We call a partial Steiner triple system C (configuration) t‐Ramsey if for large enough n (in terms of $C,t$), 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.     

Small Embeddings of Partial Steiner Triple Systems

It 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.     

A Construction of Almost Steiner Systems

Let n, k, and t be integers satisfying . A Steiner system with parameters t, k, and n is a k‐uniform hypergraph on n vertices in which every set of t distinct vertices is contained in exactly one edge. An outstanding problem in Design Theory is to determine whether a nontrivial Steiner system exists for . In this note we prove that for every and sufficiently large n, there exists an almost Steiner system with parameters t, k, and n; that is, there exists a k‐uniform hypergraph on n vertices such that every set of t distinct vertices is covered by either one or two edges.     

Infinite classes of anti‐mitre and 5‐sparse Steiner triple systems

In this paper, we present three constructions for anti‐mitre Steiner triple systems and a construction for 5‐sparse ones. The first construction for anti‐mitre STSs settles two of the four unsettled admissible residue classes modulo 18 and the second construction covers such a class modulo 36. The third construction generates many infinite classes of anti‐mitre STSs in the remaining possible orders. As a consequence of these three constructions we can construct anti‐mitre systems for at least 13/14 of the admissible orders. For 5‐sparse STS(υ), we give a construction for υ ≡ 1, 19 (mod 54) and υ ≠ 109. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 237–250, 2006     

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).     

Self‐embeddings of cyclic and projective Steiner quasigroups

It is shown that for every admissible order v for which a cyclic Steiner triple system exists, there exists a biembedding of a cyclic Steiner quasigroup of order v with a copy of itself. Furthermore, it is shown that for each n≥2 the projective Steiner quasigroup of order 2ⁿ?1 has a biembedding with a copy of itself. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:16‐27, 2010     

The Steiner tree problem II: Properties and classes of facets

This is the second part of two papers addressing the study of the facial structure of the Steiner tree polyhedron. In this paper we identify several classes of facet defining inequalities and relate them to special classes of graphs on which the Steiner tree problem is known to be NP-hard.Corresponding author.The author appreciates partial support from National Science Foundation Grants Nos. DSM-8606188 and ECS 8800281.     

Steiner triple systems with two disjoint subsystems

It is shown that there exists a triangle decomposition of the graph obtained from the complete graph of order v by removing the edges of two vertex disjoint complete subgraphs of orders u and w if and only if u,w, and v are odd, (mod 3), and . Such decompositions are equivalent to group divisible designs with block size 3, one group of size u, one group of size w, and v – u – w groups of size 1. This result settles the existence problem for Steiner triple systems having two disjoint specified subsystems, thereby generalizing the well‐known theorem of Doyen and Wilson on the existence of Steiner triple systems with a single specified subsystem. © 2005 Wiley Periodicals, Inc. J Combin Designs