Point Code Minimum Steiner Triple Systems

The point code of a Steiner triple system uniquely determines the system when the number of vectors whose weight equals the replication number agrees with the number of points. The existence of a Steiner triple system with this minimum point code property is established for all v 1,3 (mod 6) with v 15.     

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.     

Simple Signed Steiner Triple Systems

Let X be a v‐set, be a set of 3‐subsets (triples) of X, and be a partition of with . The pair is called a simple signed Steiner triple system, denoted by ST, if the number of occurrences of every 2‐subset of X in triples is one more than the number of occurrences in triples . In this paper, we prove that exists if and only if , , and , where and for , . © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 332–343, 2012     

Trinal Decompositions of Steiner Triple Systems into Triangles

It is well known that when or , there exists a Steiner triple system (STS) of order n decomposable into triangles (three pairwise intersecting triples whose intersection is empty). A triangle in an STS determines naturally two more triples: the triple of “vertices” , and the triple of “midpoints” . The number of these triples in both cases, that of “vertex” triples (inner) or that of “midpoint triples” (outer), equals one‐third of the number of triples in the STS. In this paper, we consider a new problem of trinal decompositions of an STS into triangles. In this problem, one asks for three distinct decompositions of an STS of order n into triangles such that the union of the three collections of inner triples (outer triples, respectively) from the three decompositions form the set of triples of an STS of the same order. These decompositions are called trinal inner and trinal outer decompositions, respectively. We settle the existence question for trinal inner decompositions completely, and for trinal outer decompositions with two possible exceptions.     

组大小为10的广义Steiner三元系

广义Steiner三元系GS(2,3,n,g)等价于g+1元最优常重量码(n,3,3)。本文证明了GS(2,3,n,10)存在的必要条件n≡0,1(mod3),n≥12也是充分的。     

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.     

Colouring Cubic Graphs by Small Steiner Triple Systems

Given a Steiner triple system , we say that a cubic graph G is -colourable if its edges can be coloured by points of in such way that the colours of any three edges meeting at a vertex form a triple of . We prove that there is Steiner triple system of order 21 which is universal in the sense that every simple cubic graph is -colourable. This improves the result of Grannell et al. [J. Graph Theory 46 (2004), 15–24] who found a similar system of order 381. On the other hand, it is known that any universal Steiner triple system must have order at least 13, and it has been conjectured that this bound is sharp (Holroyd and Škoviera [J. Combin. Theory Ser. B 91 (2004), 57–66]). Research partially supported by APVT, project 51-027604. Research partially supported by VEGA, grant 1/3022/06.     

Caps and Colouring Steiner Triple Systems

Hill [6] showed that the largest cap in PG(5,3) has cardinality 56. Using this cap it is easy to construct a cap of cardinality 45 in AG(5,3). Here we show that the size of a cap in AG(5,3) is bounded above by 48. We also give an example of three disjoint 45-caps in AG(5,3). Using these two results we are able to prove that the Steiner triple system AG(5,3) is 6-chromatic, and so we exhibit the first specific example of a 6-chromatic Steiner triple system.     

The Fine Intersection Problem for Steiner Triple Systems

The intersection of two Steiner triple systems and is the set . The fine intersection problem for Steiner triple systems is to determine for each v, the set I(v), consisting of all possible pairs (m, n) such that there exist two Steiner triple systems of order v whose intersection satisfies and . We show that for v ≡ 1 or 3 (mod 6), |I(v)| = Θ(v ³), where previous results only imply that |I(v)| = Ω(v ²). Received: January 23, 2006. Final Version received: September 2, 2006     

From Squashed 6‐Cycles to Steiner Triple Systems

Squashed 6‐cycle systems are introduced as a natural counterpart to 2‐perfect 6‐cycle systems. The spectrum for the latter has been determined previously in [5]. We determine completely the spectrum for squashed 6‐cycle systems, and also for squashed 6‐cycle packings.     

The Automorphism Groups of Steiner Triple Systems Obtained by the Bose Construction

The automorphism group of the Steiner triple system of order v 3 (mod 6), obtained from the Bose construction using any Abelian Group G of order 2s + 1, is determined. The main result is that if G is not isomorphic to Z ₃ ⁿ × Z ₉ ^m , n 0, m 0, the full automorphism group is isomorphic to Hol(G) × Z ₃, where Hol(G) is the Holomorph of G. If G is isomorphic to Z ₃ ⁿ × Z ₉ ^m , further automorphisms occur, and these are described in full.     

Equitable Specialized Block-Colourings for Steiner Triple Systems

We continue the study of specialized block-colourings of Steiner triple systems initiated in [2] in which the triples through any element are coloured according to a given partition π of the replication number. Such colourings are equitable if π is an equitable partition (i.e., the difference between any two parts of π is at most one). Our main results deal with colourings according to equitable partitions into two, and three parts, respectively.     

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.     

Rigid Steiner Triple Systems Obtained from Projective Triple Systems

It was shown by Babai in 1980 that almost all Steiner triple systems are rigid; that is, their only automorphism is the identity permutation. Those Steiner triple systems with the largest automorphism groups are the projective systems of orders . In this paper, we show that each such projective system may be transformed to a rigid Steiner triple system by at most n Pasch trades whenever .     

On the Intersection Problem for Steiner Triple Systems of Different Orders

A Steiner triple system of order v, or STS(v), is a pair (V, ) with V a set of v points and a set of 3-subsets of V called blocks or triples, such that every pair of distinct elements of V occurs in exactly one triple. The intersection problem for STS is to determine the possible numbers of blocks common to two Steiner triple systems STS(u), (U, ), and STS(v), (V, ), with U⊆V. The case where U=V was solved by Lindner and Rosa in 1975. Here, we let U⊂V and completely solve this question for v−u=2,4 and for v≥2u−3. supported by NSERC research grant #OGP0170220. supported by NSERC postdoctoral fellowship. supported by NSERC research grant #OGP007621.     

Partitioning 3‐arcs into Steiner Triple Systems

In 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.     

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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Bi-Embeddings of Steiner Triple Systems of Order 15

 It was shown by Gerhard Ringel that one of the three non-isomorphic Steiner triple systems of order 15 having an automorphism of order 5 may be bi-embedded as the faces of a face 2-colourable triangular embedding of K ₁₅ in a suitable orientable surface. Ringel's bi-embedding was obtained from an appropriate current graph. In a recent paper, the present authors showed that a second STS(15) of this type may also be bi-embedded. In the present paper we show that this second bi-embedding may also be obtained from a current graph. Furthermore, we exhibit a third current graph which yields a bi-embedding of the third STS(15) of this type. Received: October 5, 1998     

Large Cross‐Free Sets in Steiner Triple Systems

A cross‐free set of size m in a Steiner triple system is three pairwise disjoint m‐element subsets such that no intersects all the three ‐s. We conjecture that for every admissible n there is an STS(n) with a cross‐free set of size which if true, is best possible. We prove this conjecture for the case , constructing an STS containing a cross‐free set of size 6k. We note that some of the 3‐bichromatic STSs, constructed by Colbourn, Dinitz, and Rosa, have cross‐free sets of size close to 6k (but cannot have size exactly 6k). The constructed STS shows that equality is possible for in the following result: in every 3‐coloring of the blocks of any Steiner triple system STS(n) there is a monochromatic connected component of size at least (we conjecture that equality holds for every admissible n). The analog problem can be asked for r‐colorings as well, if and is a prime power, we show that the answer is the same as in case of complete graphs: in every r‐coloring of the blocks of any STS(n), there is a monochromatic connected component with at least points, and this is sharp for infinitely many n.     

There are 1239 Steiner Triple Systems STS(31) of 2-rank 27

A computer search over the words of weight 3 in the code of blocks of a classical Steiner triple system (STS) on 31 points is carried out to classify all STS(31) whose incidence matrix has 2-rank equal to 27, one more than the possible minimum of 26. There is a total of 1239 nonisomorphic STS(31) of 2-rank 27.