scholarly journals Bicoloring Steiner Triple Systems

10.37236/1457 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Charles J. Colbourn ◽  
Jeffrey H. Dinitz ◽  
Alexander Rosa
Keyword(s):  
Triple System ◽  
Necessary Conditions ◽  
Steiner Triple System ◽  
Steiner Triple Systems ◽  
Triple Systems ◽  
Multiplication Theorem

A Steiner triple system has a bicoloring with $m$ color classes if the points are partitioned into $m$ subsets and the three points in every block are contained in exactly two of the color classes. In this paper we give necessary conditions for the existence of a bicoloring with 3 color classes and give a multiplication theorem for Steiner triple systems with 3 color classes. We also examine bicolorings with more than 3 color classes.

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.

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.

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].

The configuration polytope of ℓ-line configurations in Steiner triple systems

2009 ◽  
Vol 59 (1) ◽  
Author(s):  
Charles Colbourn
Keyword(s):  
Linear Combination ◽  
Triple System ◽  
Steiner Triple System ◽  
Effective Procedure ◽  
Steiner Triple Systems ◽  
Triple Systems ◽  
Linear Basis ◽  
Line Configuration

AbstractIt has been shown that the number of occurrences of any ℓ-line configuration in a Steiner triple system can be written as a linear combination of the numbers of full m-line configurations for 1 ≤ m ≤ ℓ; full means that every point has degree at least two. More precisely, the coefficients of the linear combination are ratios of polynomials in v, the order of the Steiner triple system. Moreover, the counts of full configurations, together with v, form a linear basis for all of the configuration counts when ℓ ≤ 7. By relaxing the linear integer equalities to fractional inequalities, a configuration polytope is defined that captures all feasible assignments of counts to the full configurations. An effective procedure for determining this polytope is developed and applied when ℓ = 6. Using this, minimum and maximum counts of each configuration are examined, and consequences for the simultaneous avoidance of sets of configurations explored.

Construction of Large Sets of Almost Disjoint Steiner Triple Systems

1975 ◽  
Vol 27 (2) ◽  
pp. 256-260 ◽  
Author(s):  
C. C. Lindner ◽  
A. Rosa
Keyword(s):  
Triple System ◽  
Steiner Triple System ◽  
Steiner Triple Systems ◽  
Triple Systems ◽  
Large Sets ◽  
Almost Disjoint

A Steiner triple system (briefly STS) is a pair (S, t) where S is a set and t is a collection of 3-subsets of S (called triples) such that every 2-subset of S is contained in exactly one triple of t. The number |S| is called the order of the STS (S, t). It is well-known that there is an STS of order v if and only if v = 1 or 3 (mod 6). Therefore in saying that a certain property concerning STS is true for all v it is understood that v = 1 or 3 (mod 6).

Varieties Of Steiner Loops and Steiner Quasigroups

1976 ◽  
Vol 28 (6) ◽  
pp. 1187-1198 ◽  
Author(s):  
Robert W. Quackenbush
Keyword(s):  
Triple System ◽  
Steiner Triple System ◽  
Variety Of Algebras ◽  
Steiner Triple Systems ◽  
Triple Systems ◽  
Set Of Points

A Steiner Triple System (STS) is a pair (P, B) where P is a set of points and B is a set of 3-elenient subsets of P called blocks (or triples) such that for distinct p, q ∈ P there is a unique block b ∈ B with ﹛p, q) ⊂ b. There are two well-known methods for turning Steiner Triple Systems into algebras; both methods are due to R. H. Bruck [1]. Each method gives rise to a variety of algebras; in this paper we will study these varieties.

scholarly journals On 2-ranks of Steiner triple systems

10.37236/1203 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
E. F. Assmus Jr.
Keyword(s):  
Uniqueness Theorem ◽  
Triple System ◽  
Existence And Uniqueness ◽  
Binary Code ◽  
Steiner Triple System ◽  
Direct Proof ◽  
Steiner Triple Systems ◽  
Existence And Uniqueness Theorem ◽  
Triple Systems ◽  
Quadruple System

Our main result is an existence and uniqueness theorem for Steiner triple systems which associates to every such system a binary code — called the "carrier" — which depends only on the order of the system and its 2-rank. When the Steiner triple system is of 2-rank less than the number of points of the system, the carrier organizes all the information necessary to construct directly all systems of the given order and $2$-rank from Steiner triple systems of a specified smaller order. The carriers are an easily understood, two-parameter family of binary codes related to the Hamming codes. We also discuss Steiner quadruple systems and prove an analogous existence and uniqueness theorem; in this case the binary code (corresponding to the carrier in the triple system case) is the dual of the code obtained from a first-order Reed-Muller code by repeating it a certain specified number of times. Some particularly intriguing possible enumerations and some general open problems are discussed. We also present applications of this coding-theoretic classification to the theory of triple and quadruple systems giving, for example, a direct proof of the fact that all triple systems are derived provided those of full 2-rank are and showing that whenever there are resolvable quadruple systems on $u$ and on $v$ points there is a resolvable quadruple system on $uv$ points. The methods used in both the classification and the applications make it abundantly clear why the number of triple and quadruple systems grows in such a staggering way and why a triple system that extends to a quadruple system has, generally, many such extensions.

scholarly journals Model theory of Steiner triple systems

2019 ◽  
Vol 20 (02) ◽  
pp. 2050010
Author(s):  
Silvia Barbina ◽  
Enrique Casanovas
Keyword(s):  
Model Theory ◽  
Triple System ◽  
Steiner Triple System ◽  
Quantifier Elimination ◽  
Steiner Triple Systems ◽  
Triple Systems ◽  
Limit Formula ◽  
Model Completion ◽  
Theory Formula

A Steiner triple system (STS) is a set [Formula: see text] together with a collection [Formula: see text] of subsets of [Formula: see text] of size 3 such that any two elements of [Formula: see text] belong to exactly one element of [Formula: see text]. It is well known that the class of finite STS has a Fraïssé limit [Formula: see text]. Here, we show that the theory [Formula: see text] of [Formula: see text] is the model completion of the theory of STSs. We also prove that [Formula: see text] is not small and it has quantifier elimination, [Formula: see text], [Formula: see text], elimination of hyperimaginaries and weak elimination of imaginaries.

scholarly journals Combinatorial Game Distributions of Steiner Systems

10.37236/9252 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Yuki Irie
Keyword(s):  
Triple System ◽  
Steiner Triple System ◽  
Steiner System ◽  
Steiner Triple Systems ◽  
Steiner Systems ◽  
Combinatorial Game ◽  
Combinatorial Structures ◽  
Triple Systems ◽  
Minimum Number ◽  
Point Set

The $P$-position sets of some combinatorial games have special combinatorial structures. For example, the $P$-position set of the hexad game, first investigated by Conway and Ryba, is the block set of the Steiner system $S(5, 6, 12)$ in the shuffle numbering, denoted by $D_{\text{sh}}$. However, few games were known to be related to Steiner systems in this way. For a given Steiner system, we construct a game whose $P$-position set is its block set. By using constructed games, we obtain the following two results. First, we characterize $D_{\text{sh}}$ among the 5040 isomorphic $S(5, 6, 12)$ with point set $\{0, 1, ..., 11\}$. For each $S(5, 6, 12)$, our construction produces a game whose $P$-position set is its block set. From $D_{\text{sh}}$, we obtain the hexad game, and this game is characterized as the unique game with the minimum number of positions among the obtained 5040 games. Second, we characterize projective Steiner triple systems by using game distributions. Here, the game distribution of a Steiner system $D$ is the frequency distribution of the numbers of positions in games obtained from Steiner systems isomorphic to $D$. We find that the game distribution of an $S(t, t + 1, v)$ can be decomposed into symmetric components and that a Steiner triple system is projective if and only if its game distribution has a unique symmetric component.

A Method of Constructing Steiner Triple System of Order 2t+1

2014 ◽  
Vol 513-517 ◽  
pp. 2995-2998
Author(s):  
Zhao Di Xu ◽  
Xiao Yi Li ◽  
Wan Xi Chou
Keyword(s):  
Complete Graph ◽  
Basic Concept ◽  
Triple System ◽  
Steiner Triple System ◽  
Steiner Triple Systems ◽  
Triple Systems ◽  
Enumeration Problem

The basic concept of constructing the Steiner triple systems of order is described. The detinition of edges matrix of the complete graph is given. A method of constructing Steiner triple system of order is proposed. The errtire procedure of constructing Steiner triple systems of orderand is presented respectively, The enumeration problem of Steiner triple systems of order is discussed [1-1.

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