scholarly journals Pairwise Balanced Design of Order 6n+4 and 2-Fold System of Order 3n+2

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Manjusri Basu ◽  
Debabrata Kumar Ghosh ◽  
Satya Bagchi
Keyword(s):  
Triple System ◽  
Steiner Triple System ◽  
Pairwise Balanced Design ◽  
Balanced Design ◽  
Fold System

In the Steiner triple system, Bose (1939) constructed the 2−(v,3,1) design for v=6n+3 and later on Skolem (1958) constructed the same for v=6n+1. In the literature we found a pairwise balanced design (PBD) for v=6n+5. We also found the 2-fold triple system of the orders 3n and 3n+1. In this paper, we construct a PBD for v=6n+4 and a 2-fold system of the order 3n+2. The second construction completes the 2-fold system for all n∈N.

scholarly journals Characterisation Results for Steiner Triple Systems and Their Application to Edge-Colourings of Cubic Graphs

2010 ◽  
Vol 62 (2) ◽  
pp. 355-381 ◽  
Author(s):  
Daniel Král’ ◽  
Edita Máčajov´ ◽  
Attila Pór ◽  
Jean-Sébastien Sereni
Keyword(s):  
Triple System ◽  
Steiner Triple System ◽  
Cubic Graphs ◽  
Steiner Triple Systems ◽  
Cubic Graph ◽  
Triple Systems ◽  
Forbidden Configurations ◽  
Edge Colourings

AbstractIt is known that a Steiner triple system is projective if and only if it does not contain the four-triple configuration C14. We find three configurations such that a Steiner triple system is affine if and only if it does not contain one of these configurations. Similarly, we characterise Hall triple systems using two forbidden configurations.Our characterisations have several interesting corollaries in the area of edge-colourings of graphs. A cubic graph G is S-edge-colourable for a Steiner triple system S if its edges can be coloured with points of S in such a way that the points assigned to three edges sharing a vertex form a triple in S. Among others, we show that all cubic graphs are S-edge-colourable for every non-projective nonaffine point-transitive Steiner triple system S.

scholarly journals Some Pairwise Balanced Designs

10.37236/1491 ◽  
1999 ◽  
Vol 7 (1) ◽  
Author(s):  
Malcolm Greig
Keyword(s):  
Block Design ◽  
Basis Sets ◽  
Combinatorial Designs ◽  
Pairwise Balanced Design ◽  
Balanced Design ◽  
Pairwise Balanced Designs ◽  
Balanced Designs ◽  
Block Sizes

A pairwise balanced design, $B(K;v)$, is a block design on $v$ points, with block sizes taken from $K$, and with every pair of points occurring in a unique block; for a fixed $K$, $B(K)$ is the set of all $v$ for which a $B(K;v)$ exists. A set, $S$, is a PBD-basis for the set, $T$, if $T=B(S)$. Let $N_{a(m)}=\{n:n\equiv a\bmod m\}$, and $N_{\geq m}=\{n:n\geq m\}$; with $Q$ the corresponding restriction of $N$ to prime powers. This paper addresses the existence of three PBD-basis sets. 1. It is shown that $Q_{1(8)}$ is a basis for $N_{1(8)}\setminus E$, where $E$ is a set of 5 definite and 117 possible exceptions. 2. We construct a 78 element basis for $N_{1(8)}$ with, at most, 64 inessential elements. 3. Bennett and Zhu have shown that $Q_{\geq8}$ is a basis for $N_{\geq8}\setminus E'$, where $E'$ is a set of 43 definite and 606 possible exceptions. Their result is improved to 48 definite and 470 possible exceptions. (Constructions for 35 of these possible exceptions are known.) Finally, we provide brief details of some improvements and corrections to the generating/exception sets published in The CRC Handbook of Combinatorial Designs.

Direct Product of Derived Steiner Systems Using Inversive Planes

1981 ◽  
Vol 33 (6) ◽  
pp. 1365-1369 ◽  
Author(s):  
K. T. Phelps
Keyword(s):  
Direct Product ◽  
Triple System ◽  
Steiner Triple System ◽  
Steiner System ◽  
Steiner Systems

A Steiner system S(t, k, v) is a pair (P, B) where P is a v-set and B is a collection of k-subsets of P (usually called blocks) such that every t-subset of P is contained in exactly one block of B. As is well known, associated with each point x ∈ P is a S(t � 1, k � 1, v � 1) defined on the set Px = P\{x} with blocksB(x) = {b\{x}|x ∈ b and b ∈ B}.The Steiner system (Px, B(x)) is said to be derived from (P, B) and is called (obviously) a derived Steiner (t – 1, k – 1)-system. Very little is known about derived Steiner systems despite much effort (cf. [11]). It is not even known whether every Steiner triple system is derived.Steiner systems are closely connected to equational classes of algebras (see [7]) for certain values of k.

scholarly journals C-LOOPS: EXTENSIONS AND CONSTRUCTIONS

2007 ◽  
Vol 06 (01) ◽  
pp. 1-20 ◽  
Author(s):  
MICHAEL K. KINYON ◽  
J. D. PHILLIPS ◽  
PETR VOJTĚCHOVSKÝ
Keyword(s):  
Abelian Group ◽  
Triple System ◽  
Abelian Groups ◽  
Steiner Triple System

C-loops are loops satisfying the identity x(y · yz) = (xy · y)z. We develop the theory of extensions of C-loops, and characterize all nuclear extensions provided the nucleus is an abelian group. C-loops with central squares have very transparent extensions; they can be built from small blocks arising from the underlying Steiner triple system. Using these extensions, we decide for which abelian groups K and Steiner loops Q there is a nonflexible C-loop C with center K such that C/K is isomorphic to Q. We discuss possible orders of associators in C-loops. Finally, we show that the loops of signed basis elements in the standard real Cayley–Dickson algebras are C-loops.

Steiner triple systems of order 19 constructed from the Steiner triple system of order 9

1983 ◽  
Vol 94 (1-2) ◽  
pp. 89-92 ◽  
Author(s):  
Alan R. Prince
Keyword(s):  
Standard Method ◽  
Triple System ◽  
Isomorphism Class ◽  
Automorphism Groups ◽  
Steiner Triple System ◽  
Steiner Triple Systems ◽  
Triple Systems ◽  
Isomorphism Classes

SynopsisA standard method of constructing Steiner triple systems of order 19 from the Steiner triple system of order 9 gives rise to 212 different such systems. It is shown that there are just three isomorphism classes amongst these systems. Representatives of each isomorphism class are described and the orders of their automorphism groups are determined.

Methods of Constructing and Enumerating the Steiner Triple System with Order 31

2014 ◽  
Vol 513-517 ◽  
pp. 3061-3064
Author(s):  
Xiao Yi Li ◽  
Zhao Di Xu ◽  
Wan Xi Chou
Keyword(s):  
Complete Graph ◽  
Basic Concept ◽  
Triple System ◽  
Steiner Triple System ◽  
Enumeration Problem ◽  
Entire Procedure

This paper describes the basic concept of constructing Steiner triple system of order and gives the definition edge matrix of a complete graph. It proposes a method of constructing Steiner triple system of order. The entire procedure of constructing Steiner triple system of order .It discusses the enumeration problem of Steiner triple system.

Matrix and Steiner-triple-system smart pooling assays for high-performance transcription regulatory network mapping

2007 ◽  
Vol 4 (8) ◽  
pp. 659-664 ◽  
Author(s):  
Vanessa Vermeirssen ◽  
Bart Deplancke ◽  
M Inmaculada Barrasa ◽  
John S Reece-Hoyes ◽  
H Efsun Arda ◽  
...  
Keyword(s):  
Regulatory Network ◽  
Triple System ◽  
High Performance ◽  
Steiner Triple System ◽  
Transcription Regulatory Network ◽  
Network Mapping

Construction of Steiner Triple Systems Having Exactly One Triple in Common

1974 ◽  
Vol 26 (1) ◽  
pp. 225-232 ◽  
Author(s):  
Charles C. Lindner
Keyword(s):  
Triple System ◽  
Steiner Triple System ◽  
Steiner Triple Systems ◽  
Triple Systems

A Steiner triple system is a pair (Q, t) where Q is a set and t a collection of three element subsets of Q such that each pair of elements of Q belong to exactly one triple of t. The number |Q| is called the order of the Steiner triple system (Q, t). It is well-known that there is a Steiner triple system of order n if and only if n ≡ 1 or 3 (mod 6). Therefore in saying that a certain property concerning Steiner triple systems is true for all n it is understood that n ≡ 1 or 3 (mod 6). Two Steiner triple systems (Q, t1) and (Q, t2) are said to be disjoint provided that t1 ∩ t2 = Ø. Recently, Jean Doyen has shown the existence of a pair of disjoint Steiner triple systems of order n for every n ≧ 7 [1].

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