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.     

Dimension of the space generated by Steiner systems

To every Steiner system with parameters on and blocks , we can assign its characteristic vector , which is a ‐vector whose entries are indexed by the ‐subsets of such that for each ‐subset of if and only if . In this paper, we show that the dimension of the vector space generated by all of the characteristic vectors of Steiner systems with parameters is , provided that and there is at least one such system.     

Generalized Steiner systems with block size three and group size g ≡ 3(mod 6)

Generalized Steiner Systems, GS(2, 3, n, g), are equivalent to maximum constant weight codes over an alphabet of size g + 1 with distance 3 and weight 3 in which each codeword has length n. We construct Generalized Steiner Triple Systems, GS(2, 3, n, g), when g ≡ 3(mod 6). © 1997 John Wiley & Sons, Inc. J Combin Designs 5:417–432, 1997     

On list coloring Steiner triple systems

We study the list chromatic number of Steiner triple systems. We show that for every integer s there exists n₀=n₀(s) such that every Steiner triple system on n points STS(n) with n≥n₀ has list chromatic number greater than s. We also show that the list chromatic number of a STS(n) is always within a log n factor of its chromatic number. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 314–322, 2009     

Some constructions for block sequences of Steiner quadruple systems with error correcting consecutive unions

In this article, we investigate a block sequence of a Steiner quadruple system which contains the blocks exactly once such that the collection of all blocks together with all unions of two consecutive blocks of the sequence forms an error correcting code with minimum distance four. In particular, we give two recursive constructions and obtain infinitely many such sequences by utilizing individual sequences as starters of the recursions. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 152–163, 2008     

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.     

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.     

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     

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     

Block transitive Steiner systems with more than one point orbit

For all ‘reasonable’ finite t, k, and s, we construct a t‐(?₀, k, 1) design and a group of automorphisms which is transitive on blocks and has s orbits on points. In particular, there is a 2‐(?0, 4, 1) design with a block‐transitive group of automorphisms having two point orbits. This answers a question of P. J. Cameron and C. E. Praeger. The construction is presented in a purely combinatorial way, but is a by‐product of a new way of looking at a model‐theoretic construction of E. Hrushovski. © 2004 Wiley Periodicals, Inc.     

Steiner centers and Steiner medians of graphs

Let G be a connected graph and S a set of vertices of G. The Steiner distance of S is the smallest number of edges in a connected subgraph of G that contains S and is denoted by d_G(S) or d(S). The Steiner n-eccentricity e_n(v) and Steiner n-distance d_n(v) of a vertex v in G are defined as e_n(v)=max{d(S)| SV(G), |S|=n and vS} and d_n(v)=∑{d(S)| SV(G), |S|=n and vS}, respectively. The Steiner n-center C_n(G) of G is the subgraph induced by the vertices of minimum n-eccentricity. The Steiner n-median M_n(G) of G is the subgraph induced by those vertices with minimum Steiner n-distance. Let T be a tree. Oellermann and Tian [O.R. Oellermann, S. Tian, Steiner centers in graphs, J. Graph Theory 14 (1990) 585–597] showed that C_n(T) is contained in C_n+1(T) for all n2. Beineke et al. [L.W. Beineke, O.R. Oellermann, R.E. Pippert, On the Steiner median of a tree, Discrete Appl. Math. 68 (1996) 249–258] showed that M_n(T) is contained in M_n+1(T) for all n2. Then, Oellermann [O.R. Oellermann, On Steiner centers and Steiner medians of graphs, Networks 34 (1999) 258–263] asked whether these containment relationships hold for general graphs. In this note we show that for every n2 there is an infinite family of block graphs G for which C_n(G)C_n+1(G). We also show that for each n2 there is a distance–hereditary graph G such that M_n(G)M_n+1(G). Despite these negative examples, we prove that if G is a block graph then M_n(G) is contained in M_n+1(G) for all n2. Further, a linear time algorithm for finding the Steiner n-median of a block graph is presented and an efficient algorithm for finding the Steiner n-distances of all vertices in a block graph is described.     

On the chromatic numbers of Steiner triple systems

Geometric properties are used to determine the chromatic number of AG(4, 3) and to derive some important facts on the chromatic number of PG(n, 2). It is also shown that a 4-chromatic STS(v) exists for every admissible order v ≥ 21. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 1–10, 1999     

Cliques in Steiner systems

A partial Steiner (k,l)-system is a k-uniform hypergraph with the property that every l-element subset of V is contained in at most one edge of . In this paper we show that for given k,l and t there exists a partial Steiner (k,l)-system such that whenever an l-element subset from every edge is chosen, the resulting l-uniform hypergraph contains a clique of size t. As the main result of this note, we establish asymptotic lower and upper bounds on the size of such cliques with respect to the order of Steiner systems. Research of the second author partially supported by NSERC grant OGP0025112.     

Some new perfect Steiner triple systems

In a Steiner triple system STS(v) = (V, B), for each pair {a, b} ⊂ V, the cycle graph G_a,b can be defined as follows. The vertices of G_a,b are V \ {a, b, c} where {a, b, c} ∈ B. {x, y} is an edge if either {a, x, y} or {b, x, y} ∈ B. The Steiner triple system is said to be perfect if the cycle graph of every pair is a single (v − 3)-cycle. Perfect STS(v) are known only for v = 7, 9, 25, and 33. We construct perfect STS (v) for v = 79, 139, 367, 811, 1531, 25771, 50923, 61339, and 69991. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 327–330, 1999     

Piotrowski's Infinite Series of Steiner Quadruple SystemsRevisited

The construction of Bays and deWeck [1] of a SteinerQuadruple System SQS(14) was generalized by Piotrowskiin his dissertation ([7], p. 34) to an SQS(2p), p 7 mod 12 with a group transitive on thepoints. However he gave no proof of his construction and hispresesntation was open to misinterpretation. So Hanfried Lenzsuggested to analyse Piotrowski's construction and to supplyit with a proof. In the following we will present Piotrowski'sideas somewhat differently and will furnish a proof of the construction.     

Nilpotent Steiner skeins with nilpotent derived Steiner loops

Armanious and Guelzow obtained the structure theorem of finite nilpotent Steiner skeins. Guelzow gave a construction of a Steiner skein of nilpotence class n with all its derived Steiner loops of nilpotence class 1. Armanious gave a construction for Steiner skeins of nilpotence class n with all its derived Steiner loops of nilpotence class n. In this article we survey the main results on nilpotent Steiner skeins and give a new and simple construction, in the form of polynomials, for Steiner skeins of nilpotence class n with all its derived Steiner loops of nilpotence class n. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 232–238, 2000     

An improved recursive construction for disjoint Steiner quadruple systems

Let $D\left(n\right)$ be the number of pairwise disjoint Steiner quadruple systems (SQS) of order $n$. A simple counting argument shows that $D\left(n\right)\le n?3$ and a set of $n?3$ such systems is called a large set. No nontrivial large set was constructed yet, although it is known that they exist if  $n\equiv 2$ or $4\left(\mathrm{mod}6\right)$ is large enough. When $n\ge 7$ and $n\equiv 1$ or $5\left(\mathrm{mod}6\right)$, we present a recursive construction and prove a recursive formula on $D\left(4n\right)$, as follows: $$D\left(4n\right)\ge 2n+\min \left\{D\left(2n\right),2n?7\right\}.$$ The related construction has a few advantages over some of the previously known constructions for pairwise disjoint SQSs.     

Holey Steiner pentagon systems

In this article, it is shown that the necessary conditions for the existence of a holey Steiner pentagon system (HSPS) of type hⁿ are also sufficient, except possibly for the following cases: (1) when n = 15, and h ≡ 1 or 5 (mod 6) where h ≢ 0 (mod 5), or h = 9; and (2) (h, n) ∈ {(6, 6), (6, 36), (15, 19), (15, 23), (15, 27), (30, 18), (30, 22), (30, 24)}. Moreover, the results of this article guarantee the analogous existence results for group divisible designs (GDDs) of type hⁿ with block-size k = 5 and index λ = 2. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 41–56, 1999