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On large sets of disjoint Kirkman triple systems

2002
*
Discrete Mathematics
*

In this paper, we introduce LR(u) designs and use these designs together with

doi:10.1016/s0012-365x(01)00470-8
fatcat:g7rbpllc3bag7lgwedznvtu5i4
*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*...##
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On large sets of Kirkman triple systems and 3-wise balanced design

2004
*
Discrete Mathematics
*

In this paper, the existence

doi:10.1016/s0012-365x(03)00279-6
fatcat:qvr6zf3cpjckfoibeuqrhvu4g4
*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 ...##
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Forthcoming papers

1991
*
Discrete Mathematics
*

Simpson,

doi:10.1016/0012-365x(91)90052-4
fatcat:3g4xmr656vaczdgn4xzyl6d6ra
*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. ...##
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On intersections of pairs of steiner triple systems

1977
*
Indagationes Mathematicae (Proceedings)
*

We have already defined a

doi:10.1016/1385-7258(77)90034-8
fatcat:2blgad3b4fcylf4i4ybhaprq2a
*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. ...##
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A construction for large sets of disjoint Kirkman triple systems

2007
*
Designs, Codes and Cryptography
*

A

doi:10.1007/s10623-007-9069-2
fatcat:mdly2gicvrhqxpgen755irzpjm
*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). ...##
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A survey of Kirkman triple systems and related designs

1991
*
Discrete Mathematics
*

., A survey

doi:10.1016/0012-365x(91)90294-c
fatcat:rd2luccrfffj3f5yzb4jkgnq7m
*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*...##
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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 ...##
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An improved product construction for large sets of Kirkman triple systems

2003
*
Discrete Mathematics
*

It has been shown by Lei, in his recent paper, that there exists a

doi:10.1016/s0012-365x(02)00766-5
fatcat:rlyplkabtjc2nmqj7zgyub3j7e
*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). ...##
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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. ...##
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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 ...##
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Transitive Steiner and Kirkman triple systems of order $27$

1992
*
Mathematics of Computation
*

There are 71 Steiner

doi:10.1090/s0025-5718-1992-1106962-5
fatcat:lc5xo7fjd5anlif4xb72s34kwy
*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." ...##
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Transitive Steiner and Kirkman Triple Systems of Order 27

1992
*
Mathematics of Computation
*

There are 71 Steiner

doi:10.2307/2153046
fatcat:krgzxytx2vbstjpmedgzlnud6e
*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." ...##
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Construction of large sets of almost disjoint Steiner triple systems

1975
*
Canadian Journal of Mathematics - Journal Canadien de Mathematiques
*

Construction

doi:10.4153/cjm-1975-031-3
fatcat:uk5mr4yrp5ahvfpystqfvqsfxu
*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*. ...##
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Kirkman triple systems of order 21 with nontrivial automorphism group

2001
*
Mathematics of Computation
*

There are 50,024

doi:10.1090/s0025-5718-01-01372-2
fatcat:i2fb7wry2zhwpebcdjgomfwckm
*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). ...##
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Large sets of Kirkman triple systems with orderqn+2

2017
*
Discrete Mathematics
*

The existence

doi:10.1016/j.disc.2016.11.008
fatcat:e2hqm7tarzg3pmhvdwrqwa2gcy
*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*...
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