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On large sets of disjoint Kirkman triple systems

Jian-guo Lei
2002 Discrete Mathematics  
In this paper, we introduce LR(u) designs and use these designs together with large sets of Kirkman triple systems (LKTS) and transitive KTS (TKTS) of order v to construct an LKTS(uv).  ...  For example, a large set of disjoint Steiner triple systems, a large set of disjoint Mendelsohn triple systems, a large set of disjoint groupdivisible designs, etc.  ...  It is easy to see that the maximum number of pairwise disjoint KTS(v)s on the same point set is v − 2, and a set of v − 2 pairwise disjoint KTS(v)s is called a large set of disjoint Kirkman triple systems  ... 
doi:10.1016/s0012-365x(01)00470-8 fatcat:g7rbpllc3bag7lgwedznvtu5i4

On large sets of Kirkman triple systems and 3-wise balanced design

Jian-guo Lei
2004 Discrete Mathematics  
In this paper, the existence of large sets of Kirkman triple system is transformed to the existence of ÿnite OLKFs and LGKSs.  ...  As well, we present a construction of 3-wise balanced design.  ...  For example, a large set of disjoint Steiner triple systems, a large set of disjoint transitive triple systems, a large set of disjoint Mendelsohn triple systems, a large set of disjoint group-divisible  ... 
doi:10.1016/s0012-365x(03)00279-6 fatcat:qvr6zf3cpjckfoibeuqrhvu4g4

Forthcoming papers

1991 Discrete Mathematics  
Simpson, Disjoint covering systems of rational Beatty sequences A rational Beatty sequence is a sequence {[on + /I]}, where (Y is rational, n runs through the integers and square brackets denote integer  ...  It is shown that the collection of all ('4) quadruples chosen from a set of eleven points can be partitioned into eleven mutually disioint 3-(10.4. 11 desiens in oreciselv 21 non-isomomhic wavn. R.J.  ...  Stinson, A survey of Kirkman triple systems and related designs The purpose of this paper is to survey results on Kirkman triple systems and generalizations.  ... 
doi:10.1016/0012-365x(91)90052-4 fatcat:3g4xmr656vaczdgn4xzyl6d6ra

On intersections of pairs of steiner triple systems

J.I Hall, J.T Udding
1977 Indagationes Mathematicae (Proceedings)  
We have already defined a Kirkman system of order 6t + 3 as a Steiner triple system, (X, A), whose set of triples admits a partition A =A1 u . .  ...  For most orders v = 3 (mod 6), Theorem 4 seems to provide the largest set of disjoint Kirkman systems known to date. (iv) C. C.  ... 
doi:10.1016/1385-7258(77)90034-8 fatcat:2blgad3b4fcylf4i4ybhaprq2a

A construction for large sets of disjoint Kirkman triple systems

Lijun Ji
2007 Designs, Codes and Cryptography  
A set of v − 2 pairwise disjoint KTS(v) is called a large set of disjoint Kirkman triple systems of order v and briefly denoted by LKTS(v).  ...  also called a Kirkman triple system of order v and shortly denoted by KTS(v). The KTS(v) is known to exist for any non-negative integer v ≡ 3 (mod 6).  ... 
doi:10.1007/s10623-007-9069-2 fatcat:mdly2gicvrhqxpgen755irzpjm

A survey of Kirkman triple systems and related designs

D.R. Stinson
1991 Discrete Mathematics  
., A survey of Kirkman triple systems and related designs, Discrete Mathematics 92 (1991) 371-393. The purpose of this paper is lo survey results on Kirkman triple systems and generalizations.  ...  These generalizations include nearly Kirkman triple systems and resolvable group-divisible designs with block size three, Kirkman frames, Kirkman triple systems containing Kirkman and/or Steiner triple  ...  These generalizations include nearly Kirkman triple systems and resolvable group-divisible designs with block size three, Kirkman frames, Kirkman triple systems containing Kirkman and/or Steiner triple  ... 
doi:10.1016/0012-365x(91)90294-c fatcat:rd2luccrfffj3f5yzb4jkgnq7m

Page 1808 of Mathematical Reviews Vol. , Issue 85e [page]

1985 Mathematical Reviews  
If it is possible to partition this set into 4(v—2) subsets T} , Tz, - -- ,T4»-2) 80 that each (S,T;) is an OTS, we have a large set of pairwise disjoint OTSs.  ...  A Kirkman triple system (KTS) is a Steiner triple system (V, B) together with a partition R = (R,,---,R,) of the set of triples B into parallel classes: B = R,UR2U---UR, and each R; is a parallel class  ... 

An improved product construction for large sets of Kirkman triple systems

S. Zhang, L. Zhu
2003 Discrete Mathematics  
It has been shown by Lei, in his recent paper, that there exists a large set of Kirkman triple systems of order uv (LKTS(uv)) if there exist an LKTS(v), a TKTS(v) and an LR(u), where a TKTS(v) is a transitive  ...  Kirkman triple system of order v, and an LR(u) is a new kind of design introduced by Lei.  ...  A large set of KTS(v) (denoted LKTS(v)) is a collection of v − 2 pairwise disjoint KTS(v) on the same point set. Note that the necessary condition for the existence of LKTS(v) is v ≡ 3 (mod 6).  ... 
doi:10.1016/s0012-365x(02)00766-5 fatcat:rlyplkabtjc2nmqj7zgyub3j7e

Page 3199 of Mathematical Reviews Vol. , Issue 2003e [page]

2003 Mathematical Reviews  
A large set of disjoint Kirkman triple systems of order v, or LKTS(wv), isa partition of the set of all 3-subsets of a v-set into v — 2 Kirkman triple systems of order v.  ...  Ryoh Fuji-Hara (J-TSUKS-IPS; Tsukuba) 2003e:05023 05B07 Lei, Jian-guo (PRC-HNU-IM; Shijiazhuang) On large sets of Kirkman systems with holes.  ... 

Page 8081 of Mathematical Reviews Vol. , Issue 2003k [page]

2003 Mathematical Reviews  
[Zhu, Lie'] (PRC-SOO; Suzhou) An improved product construction for large sets of Kirkman triple systems. (English summary) Discrete Math. 260 (2003), no. 1-3, 307-313.  ...  Lei [Discrete Math. 257 (2002), no. 1, 63-81; MR 2003h:05041] that there exists a large set of Kirkman triple systems of order uv (LKTS(uv)) if there exist an LKTS(v), a TKTS(v) and an LR(u), where a TKTS  ... 

Transitive Steiner and Kirkman triple systems of order $27$

Charles J. Colbourn, Spyros S. Magliveras, Rudolf A. Mathon
1992 Mathematics of Computation  
There are 71 Steiner triple systems of order 27 whose automorphism groups are point-transitive, and there are 248 transitive Kirkman triple systems of order 27.  ...  The designs and some of their properties are presented.  ...  Steiner triple systems with large automorphism groups, and in particular transitive STS, are studied in large part because they yield examples of Steiner triple systems with interesting "regularity."  ... 
doi:10.1090/s0025-5718-1992-1106962-5 fatcat:lc5xo7fjd5anlif4xb72s34kwy

Transitive Steiner and Kirkman Triple Systems of Order 27

Charles J. Colbourn, Spyros S. Magliveras, Rudolf A. Mathon
1992 Mathematics of Computation  
There are 71 Steiner triple systems of order 27 whose automorphism groups are point-transitive, and there are 248 transitive Kirkman triple systems of order 27.  ...  The designs and some of their properties are presented.  ...  Steiner triple systems with large automorphism groups, and in particular transitive STS, are studied in large part because they yield examples of Steiner triple systems with interesting "regularity."  ... 
doi:10.2307/2153046 fatcat:krgzxytx2vbstjpmedgzlnud6e

Construction of large sets of almost disjoint Steiner triple systems

C. C. Lindner, A. Rosa
1975 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
Construction of large sets of MAD Steiner triple systems.  ...  A Steiner triple system (briefly STS) is a pair (5, t) where 5 is a set and t is a collection of 3-subsets of 5 (called triples) such that every 2-subset of 5 is contained in exactly one triple of t.  ...  In order to show that a large set of MAD STS of order v must contain at least v -1 triple systems we show that no set of v -2 MAD STS can form a large set.  ... 
doi:10.4153/cjm-1975-031-3 fatcat:uk5mr4yrp5ahvfpystqfvqsfxu

Kirkman triple systems of order 21 with nontrivial automorphism group

Myra B. Cohen, Charles J. Colbourn, Lee A. Ives, Alan C. H. Ling
2001 Mathematics of Computation  
There are 50,024 Kirkman triple systems of order 21 admitting an automorphism of order 2. There are 13,280 Kirkman triple systems of order 21 admitting an automorphism of order 3.  ...  Together with the 192 known systems and some simple exchange operations, this leads to a collection of 63,745 nonisomorphic Kirkman triple systems of order 21.  ...  Research of the authors is supported by ARO grant DAAG55-98-1-0272 (Colbourn).  ... 
doi:10.1090/s0025-5718-01-01372-2 fatcat:i2fb7wry2zhwpebcdjgomfwckm

Large sets of Kirkman triple systems with orderqn+2

C. Wang, C. Shi
2017 Discrete Mathematics  
The existence of Large sets of Kirkman Triple Systems (LKTS) is an old problem in combinatorics.  ...  The only known recursive constructions are an tripling construction by Denniston MR535159and a product construction by Lei MR1931492, both constructs an LKTS(uv) on the basis of an LKTS(v).  ...  By a simple counting argument, there can be at most v −2 mutually disjoint STS(v)s on the same set X; such a set of v −2 disjoint systems must contain every possible triple in X, and is called a large  ... 
doi:10.1016/j.disc.2016.11.008 fatcat:e2hqm7tarzg3pmhvdwrqwa2gcy
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