Cyclic Triple Systems Research Articles

Decomposition of Complete Multigraph into Wheel Graphs for Cyclic Triple System

Let 𝑛 be positive integer and 𝑊𝑛 be a wheel graph of order 𝑛. In this paper, we construct a new decomposition of 8𝐾𝑣 into wheel graphs for 𝑣 ≡ 2 (mod 12). Then, new cyclic triple system will be defined to arrange 𝑣 × 2(𝑣 − 1) triples satisfying certain criteria. In this development, the decomposition of 8𝐾𝑣 will be used to construct a cyclic 12-fold triple system of order 𝑣 ≡ 2 (mod 12).

On Cyclic Triple System and Factorization

On Cyclic Triple System and Factorization

Skolem and Rosa rectangles and related designs

A Skolem sequence of ordern is a sequence Sn=(s1,s2,…,s2n) of 2n integers such that (1) for all k∈{1,2,…,n} there are exactly two terms si, sj such that si=sj=k, and (2) if si=sj=k and i<j, then j−i=k.Two Skolem sequences Sn, Sn′ are disjoint if si=sj=k=st′=su′ implies that {i,j}≠{t,u}, for all k=1,2,…,n. For example, the two Skolem sequences of order four 1,1,4,2,3,2,4,3 and 2,3,2,4,3,1,1,4 are disjoint. A set of m pairwise disjoint Skolem sequences forms a Skolem rectangle of strength m. The above sequences then form a Skolem rectangle of strength two:1,1,4,2,3,2,4,32,3,2,4,3,1,1,4.We introduce several new constructions for Skolem and Rosa rectangles then we apply them to generate simple cyclic triple systems and disjoint cyclic triple systems. For example, for certain constants C1, C2 we obtain ⌊log3(2n+C1)⌋+C2 disjoint Skolem sequences of order n. Finally, we obtain the analogous results for hooked Skolem and Rosa sequences.

Decomposing triples of $\mathbb{Z}_{p^n } $ and $\mathbb{Z}_{3p^n } $ into cyclic designs

In this paper, we give some decompositions of triples of \(\mathbb{Z}_{p^n } \) or \(\mathbb{Z}_{3p^n } \) into cyclic triple systems. New constructions of difference families are given. Some infinite classes of simple cyclic triple systems are obtained from these decompositions.

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.

Decomposing triples into cyclic designs

Motivated by constructing cyclic simple designs, we consider how to decomposing all the triples of Z v into cyclic triple systems. Furthermore, we define a large set of cyclic triple systems to be a decomposition of triples of Z v into indecomposable cyclic designs. Constructions of decompositions and large sets are given. Some infinite classes of decompositions and large sets are obtained. Large sets of small v with odd v < 97 are also given. As an application, the results are used to construct cyclic simple triple systems.

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.

Simple and indecomposable twofold cyclic triple systems from Skolem sequences

Simple and indecomposable twofold cyclic triple systems from Skolem sequences

On the application of a pure cyclic triple system for plant breeding experiments

Given three distinct lines (varieties or strains), say A, B and C, we can form three distinct three-way crosses (AB)C, (AC) B and (BC)A. The set of all possible three-way crosses VT = v(v — 1 )(v — 2)/2 among the given set of v lines is termed triallel crosses or a triallel mating design. As the number of lines v increases, the triallel crosses become too many to be accommodated within the resources of the breeder. One of the ways to overcome this situation is to take a sample of three-way crosses, called partial triallel crosses. Estimators for design and genetic components of variance have been developed based on Hinkelmann's model.

Triple systems with a fixed number of repeated triples

AbstractIn this paper, linear embeddings of partial designs into designs are found where no repeated blocks are introduced in the embedding process. Triple systems, pure cyclic triple systems, cyclic and directed triple systems are considered. In particular, a partial triple system with no repeated triples is embedded linearly in a triple system with no repeated triples.

Graph factorization, general triple systems, and cyclic triple systems

In this self-contained exposition, results are developed concerning one-factorizations of complete graphs, and incidence matrices are used to turn these factorization results into embedding theorems on Steiner triple systems. The result is a constructive graphical proof that a Steiner triple system exists for any order congruent to 1 or 3 modulo 6. A pairing construction is then introduced to show that one can also obtain triple systems which are cyclically generated.

Extended cyclic triple systems

An extended cyclic triple system is a pair ( S, W) where S is a finite set and W is a collection of cyclic triples from S, where each triple may have repeated elements, such that every ordered pair of elements of S, not necessarily distinct, is contained in exactly one triple on W. The triples of W are of three types: (1) { a, a, a}, (2) { b, b, c}, (3) { x, y, z}, where { b, b, c} contains the pairs bc, bc and cb, while { x, y, z} contains xy, yz and zx but not xz, yx or zy. The element z is called an idempotent and b a non-idempotent of ( S, W). We denote by CTS ( n; α) be class of all extended cyclic triple systems on n elements which have α idempotents. We say CTS ( n; α) exists if there is a system with parameters n and α. We investigate conditions for the existence of CTS ( n; α) and show that if the necessary conditions are satisfied then CTS( n; α) exists.

Finite partial cyclic triple systems can be finitely embedded

By a cyclic triple system is meant a pair (S, T) where S is a finite set and T is a collection of three element subsets of S arranged cyclically and such that each ordered pair of elements of S appears in exactly one cyclic triplet. By a partial cyclic triple system (PCTS) is meant a pair (S, T) where S is a finite set and T is a collection of three element subsets of S arranged cyclically and such that each ordered pair of elements of S appears in at most one cyclic triplet. Algebraically a CTS is a quasigroup satisfying the identities x 2 =x and x (yx)=y [8], so that algebraically a PCTS is a partial quasigroup Q such that a 2 =a for all a in Q and whenever ab is defined in a then so are b(ab) and (ab)a and further b(ab)=a, (ab)a=b. In what follows we will consider CTS's and PCTS's algebraically. If we omit the words cyclic and ordered in the definition of a CTS and a PCTS we obtain the definition of a Steiner triple system (STS) and a partial Steiner triple system (PSTS) respectively.