Cyclic Steiner Triple Systems Research Articles

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.

On the resolutions of cyclic Steiner triple systems with small parameters

The paper presents useful invariants of resolutions of cyclic $STS(v)$ with $v\le 39$, namely of all resolutions of cyclic $STS(15)$, $STS(21)$ and $STS(27)$, of the resolutions with nontrivial automorphisms of cyclic $STS(33)$ and of resolutions with automorphisms of order $13$ of cyclic $STS(39)$.

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.

Nonorientable biembeddings of cyclic Steiner triple systems generated by Skolem sequences

We describe a class of Skolem sequences of order n such that for the cyclic Steiner triple system of order 6n+1 generated by any Skolem sequence from the class we can construct a bipartite index one current graph generating a nonorientable face 2-colorable triangular embedding of the complete graph K6n+1 such that the triangular faces of one of the two color classes form the blocks of the cyclic Steiner triple system.

FROM STEINER TRIPLE SYSTEMS TO 3-SUN SYSTEMS

An $n$-$sun$ is the graph with $2n$ vertices consisting of an $n$-cycle with $n$ pendent edges which form a 1-factor. In this paper we show that the necessary and sufficient conditions for the decomposition of complete tripartite graphs with at least two partite sets having the same size into $3$-suns and give another construction to get a $3$-sun system of order $n$, for $n\equiv 0,1,4,9$ (mod 12). In the construction we metamorphose a Steiner triple system into a $3$-sun system. We then embed a cyclic Steiner triple system of order $n$ into a $3$-sun system of order $2n-1$, for $n\equiv 1$ (mod 6).

Cyclic block designs with block size 3 from Skolem-type sequences

A Skolem-type sequence is a sequence (s 1, . . . , s t ) of positive integers $${i\in D}$$ such that for each $${i\in D}$$ there is exactly one $${j\in \{1, \ldots , t - i\}}$$ such that s j = s j+i = i. Positions in the sequence not occupied by integers $${i\in D}$$ contain null elements. In 1939, Peltesohn solved the existence problem for cyclic Steiner triple systems for v ? 1, 3(mod 6), v ? 9. Using the same technique in 1981, Colbourn and Colbourn extended the solution to all admissible ? > 1. It is known that Skolem-type sequences may be used to construct cyclic Steiner triple systems as well as cyclic triple systems with ? = 2. The main result of this paper is an extension of former results to cyclic triple systems with ? > 2. In addition we introduce a new kind of Skolem-type sequence.

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 2(n) - 1 has a biembedding with a copy of itself. (C) 2010 Wiley Periodicals, Inc. J Combin Designs 19: 16-27, 2010

Orientable biembeddings of cyclic Steiner triple systems from current assignments on Möbius ladder graphs

We give a characterization of a current assignment on the bipartite Möbius ladder graph with 2 n + 1 rungs. Such an assignment yields an index one current graph with current group Z 12 n + 7 that generates an orientable face 2-colorable triangular embedding of the complete graph K 12 n + 7 or, equivalently, an orientable biembedding of two cyclic Steiner triple systems of order 12 n + 7 . We use our characterization to construct Skolem sequences that give rise to such current assignments. These produce many nonisomorphic orientable biembeddings of cyclic Steiner triple systems of order 12 n + 7 .

Conflict-avoiding codes and cyclic triple systems

The paper deals with the problem of constructing a code of the maximum possible cardinality consisting of binary vectors of length n and Hamming weight 3 and having the following property: any 3 × n matrix whose rows are cyclic shifts of three different code vectors contains a 3 × 3 permutation matrix as a submatrix. This property (in the special case w = 3) characterizes conflict-avoiding codes of length n for w active users, introduced in [1]. Using such codes in channels with asynchronous multiple access allows each of w active users to transmit a data packet successfully in one of w attempts during n time slots without collisions with other active users. An upper bound on the maximum cardinality of a conflict-avoiding code of length n with w = 3 is proved, and constructions of optimal codes achieving this bound are given. In particular, there are found conflict-avoiding codes for w = 3 which have much more vectors than codes of the same length obtained from cyclic Steiner triple systems by choosing a representative in each cyclic class.

Complete sets of disjoint difference families and their applications

Let G be an abelian group. A collection of ( G, k, λ) disjoint difference families, { F 0, F 1,…, F s−1} , is a complete set of disjoint difference families if ⋃ 0⩽i⩽s−1⋃ B∈ F i B form a partition of G−{0}. In this paper, several construction methods are provided for complete sets of disjoint difference families. Applications to one-factorizations of complete graphs and to cyclically resolvable cyclic Steiner triple systems are also described.

Cyclically resolvable cyclic Steiner triple systems of order 21 and 39

All cyclically resolvable cyclic Steiner triple systems of order 21 and 39 are enumerated. Up to isomorphism, there are exactly 2 such triple systems of order 21 and 528 of order 39. As a by-product, there are at least 616 non-isomorphic resolvable Steiner triple systems of order 39.

The existence of near-Skolem and hooked near-Skolem sequences

Let m, n be integers, m <n. A near-Skolem sequence of order n and defect m is a sequence S=(s1,s2, . . . . s,,_,)ofintegers.~~{1,2 )..., m-l,m+l, . . . . n) whichsatisfies the following conditions: (1) For every ke{1,2 ,..., m-l,m+l,..., n) there are exactly two elements si, Sj~S, such that si = sj = k. (2) If si=sj=k then j-i=k. A hooked near-Skolem sequence of order n and defect m is a sequence HS = (si , s2 ,. . . , szn _ 1) of integers Sip{ 1,2, . . . , m 1, m + 1, . . . , n} satisfying conditions (1) (2) and the condition: (3) s+z=o. We will refer to near-Skolem and hooked near-Skolem sequences of order n and defect m as m-near-Skolem and hooked m-near-Skolem sequences, respectively. Skolem (1957) [9] showed that the set (1,. . . , 2n) can be partitioned into distinct pairs (a,, b,) such that b,-a, =I, r = 1, . . _, n. He proved that the necessary and sufficient condition for the existence of such partition is n = 0,l (mod 4). O’Keefe (1961) [7] proved that the necessary and sufficient condition for partitioning the set (1, . . . ,2n 1,2n + 1) into distinct pairs (a,, b,) such that b,a,=r, r= 1, . . ., n, is n=2,3 (mod4). The results of Skolem and O’Keefe led to the construction of cyclic Steiner triple systems of order v E 1 (mod 6). Note that if we did not skip the difference m in the above definitions, we would obtain the definitions of Skolem and hooked Skolem sequences, respectively. The pairs of subscripts (i, j) whenever si = sj= k give the required partitioning in both Skolem and O’Keefe cases.

Anti-mitre steiner triple systems

A mitre in a Steiner triple system is a set of five triples on seven points, in which two are disjoint. Recursive constructions for Steiner triple systems containing no mitre are developed, leading to such anti-mitre systems for at least 9/16 of the admissible orders. Computational results for small cyclic Steiner triple systems are also included.

Concerning multiplier automorphisms of cyclic Steiner triple systems

A cyclic Steiner triple system, presented additively over Zv as a set B of starter blocks, has a non-trivial multiplier automorphism λ ≠ 1 when λB is a set of starter blocks for the same Steiner triple system. When does a cyclic Steiner triple system of order v having a nontrivial multiplier automorphism exist? Constructions are developed for such systems; of most interest, a novel extension of Netto's classical construction for prime orders congruent to 1 (mod 6) to prime powers is proved. Nonexistence results are then established, particularly in the cases when v = (2β + 1)α, when v = 9p with p ≡ 5 (mod 6), and in certain cases when all prime divisors are congruent to 5 (mod 6). Finally, a complete solution is given for all v < 1000, in which the remaining cases are produced by simple computations.

Triple systems in PG(2, q)

Let G be a cyclic Singer group for the Desarguesian projective plane P = PG(2, q) . Then there exists a cyclic Steiner triple system on the point set of P which is invariant under G and the blocks of which are triangles of P .

Partitioning Steiner triple systems into complete arcs

For a Steiner triple system of order v to have a complete s-arc one must have s( s + 1)/2⩾ v with equality only if s = 1 or 2 mod 4. To partition a Steiner triplesystem of order s(s + 1) 2 into complete s-arcs, one must have s = 1 mod 4. In this paper wegive constructions of Steiner triple systems of order s(s + 1) 2 which can be partitioned into complete s-arcs for all s = 1 mod 4. For s = 1 or 5 mod 12, we construct cyclic Steiner triple systems having this property. For s = 9 mod 12 we use Kirkman triple systems of order s having one additional property to construct these Steiner triple systems. We further establish that Kirkman triple systems having this additional property exist at least for s = 9 mod 24 and s = 21 mod 120.

Cyclic Steiner Triple Systems with Cyclic Subsystems

A Steiner triple system of order v (briefly STS(v)) is a pair (V, B), where V is a v-set and B is a set of 3-subsets of V called triples such that each 2-subset of V is contained in exactly one triple. A cyclic Steiner triple system of order v (briefly CTS(v)), i.e. an STS with a regular cyclic automorphism group) can be thought of as a pair (Zv, B), where Zv is the additive group of integers modulo v fixing the set of triples B. An STS(v) exists iff v == lor 3 (mod 6), and a CTS(v) exists iff v == lor 3 (mod 6), v * 9 (see [5]). For a general reference on CTSs, see [2). When v = nu, n > 1, u > 1, then the group of integers modulo v will have a (non-trivial) subgroup of order u and index n for each divisor u of v. In a similar fashion, one can define a cyclic subsystem of a CTS(v) as having order u and index n, where (H, Bu) is the subsystem, and H is the subgroup of order u and index n which fixes Bu' A natural question asked, amongst others, by Peter Tannenbaum [6], is the following: What are the necessary and sufficient conditions for a CTS(v) to contain a cyclic subsystem of order u and index n? The problem becomes more interesting when one realizes that the obvious necessary conditions--namely that u divides v, and that u == lor 3 (mod 6), u * 9, v * 9--are not sufficient. First we consider construction of such systems. The methods we use are variants of known composition methods for cyclic designs. Recent references on such methods include Colbourn and Colbourn [1] and limbo and Kuriki [3]. Some cases of the main results of this paper are probable consequences of previously published work and, to an extent, may be considered folklore. However, for the purposes of clarity and simplicity, our presentation will be self-contained.

A construction of cyclic Steiner triple systems of order pn

On presente des constructions de systemes cycliques de triplets de Steiner d'ordre p n , pour p premier ≡1(mod 6)

Skolem method of constructing cyclic Steiner triple systems

Skolem method of constructing cyclic Steiner triple systems