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Nearly Kirkman triple systems of order 18 and Hanani triple systems of order 19

2011
*
Discrete Mathematics
*

The

doi:10.1016/j.disc.2011.02.005
fatcat:qth53ipoxzd3rhhywqacdhm524
*Hanani**triple**systems**of**order*6n + 1*and*the*nearly**Kirkman**triple**systems**of**order*6n can be classified using the classification*of*the Steiner*triple**systems**of**order*6n +1. ... A*nearly**Kirkman**triple**system**of**order*6n, NKTS(6n), is a decomposition*of*K 6n − F into 3n − 1 sets*of*2n disjoint triangles; here F is a one-factor*of*K 6n . ... The fifth author was supported by the Graduate School in Electronics, Telecommunication*and*Automation, by the Nokia Foundation,*and*by the Academy*of*Finland, Grant No. 110196. ...##
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A survey of Kirkman triple systems and related designs

1991
*
Discrete Mathematics
*

These generalizations include

doi:10.1016/0012-365x(91)90294-c
fatcat:rd2luccrfffj3f5yzb4jkgnq7m
*nearly**Kirkman**triple**systems**and*resolvable group-divisible designs with block size three,*Kirkman*frames,*Kirkman**triple**systems*containing*Kirkman**and*/or Steiner*triple*... A*Kirkman**triple**system**of**order*9. 123 147 159 168 456 258 267 249 789 369 348 357 A suruey*of**Kirkman*tr+ ~ys.2m.s*and*related designs 373 ... 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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Palettes in block colourings of designs

2013
*
The Australasian Journal of Combinatorics
*

We obtain bounds on the minimum possible number

dblp:journals/ajc/LindnerMR13
fatcat:bkkoxo6mcrajxfbfdxj6luxzkm
*of*distinct palettes in proper block colourings*of*Steiner*triple**systems**and*Steiner*systems*S(2, 4, v). ... For any proper block colouring*of*a Steiner*system*, a palette*of*an element is the set*of*colours on blocks incident with it. ... Acknowledgements The research*of*the second author was supported by the NCN Grant No. 2011/01/B/ ST1/04056. The research*of*the third author was supported by NSERC*of*Canada Grant No. A7268. ...##
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An Update on the Existence of Kirkman Triple Systems with Subdesigns
[article]

2021
*
arXiv
*
pre-print

A

arXiv:2110.07874v1
fatcat:5jf2o3uwjjegnp2b7sdu2mwhcq
*Kirkman**triple**system**of**order*v, KTS(v), is a resolvable Steiner*triple**system*on v elements. ... In this paper, we investigate an open problem posed by Doug Stinson, namely the existence*of*KTS(v) which contain as a subdesign a Steiner*triple**system**of**order*u, an STS(u). ... Assuming v ≥ 7, the assumption on chromatic index can be achieved using a KTS(w) if w ≡ 3 (mod 6), a '*Hanani**triple**system*' if w ≡ 1 (mod 6), w ≥*19*, or directly for w ∈ {7, 13}; see [4, . ...##
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Resolvable Group Divisible Designs with Block Size 3
[chapter]

1989
*
Annals of Discrete Mathematics
*

Dedicated to Professor Haim

doi:10.1016/s0167-5060(08)70092-x
fatcat:ltco6nv355e4pexpgytxilatp4
*Hanani*on the occasion*of*his 75th birthday. Let Y be a non negative integer, let I be a positive integer,*and*let K*and*M be sets*of*positive integers. ... A group divisible design, denoted by GD[K, 1, M, v], is a*triple*(X, r, b) where X is a set*of*points, r = {G" Gz, .} is a partition*of*X,*and*/! ... When m = 2*and*u > 9 the result follows from the existence*of**nearly**Kirkman**triple**systems*(Theorem 1.6)*and*the Addition Lemma. ...##
###
Resolvable group divisible designs with block size 3

1989
*
Discrete Mathematics
*

Dedicated to Professor Haim

doi:10.1016/0012-365x(89)90346-4
fatcat:5nyotfmz65h7fkib2blw5gzfae
*Hanani*on the occasion*of*his 75th birthday. Let Y be a non negative integer, let I be a positive integer,*and*let K*and*M be sets*of*positive integers. ... A group divisible design, denoted by GD[K, 1, M, v], is a*triple*(X, r, b) where X is a set*of*points, r = {G" Gz, .} is a partition*of*X,*and*/! ... When m = 2*and*u > 9 the result follows from the existence*of**nearly**Kirkman**triple**systems*(Theorem 1.6)*and*the Addition Lemma. ...##
###
Optimal packings of K4's into a Kn

1979
*
Journal of combinatorial theory. Series A
*

(3, 1, 2; 12) (also called a

doi:10.1016/0097-3165(79)90105-5
fatcat:7pczfwftsbeadhrxnwj5kbz4c4
*Nearly**Kirkman**Triple**System*NKTS(l2)). ... under Z, x Z,, . j&r fact, using a similar solution for n = 89 (also found by PDPIl), the case n = 5 (mod 12) can be solved completely without recourse to*Nearly**Kirkman**Triple**systems*.] the*triples*form ...*Triple**Systems*is now settled completely. It follows that the condition t # 14, 17,29 may be dropped in Lemma 8. (iii) Reference [2] can be found in [22] . Reference [5] has appeared as [24] . ...##
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Three-line chromatic indices of Steiner triple systems

2000
*
The Australasian Journal of Combinatorics
*

There are five possible structures for a set

dblp:journals/ajc/GrannellGR00
fatcat:qlje53evu5bhxdqts7kslsvrmu
*of*three lines*of*a Steiner*triple**system*. ... Aspects which have received significant attention have been the counting*of*configurations in Steiner*triple**systems*[12, 8, 15] , the decomposition*of*Steiner*triple**systems*into various n-line configurations ... If v == 1 (mod 12)*and*-:;; =I-13 thenthere exists a*nearly*-*Kirkman**triple**system**of**order*v-I, i.e. a resolvable 3-GDD*of*type 2(v-l)!2 [21] . ...##
###
The existence of restricted resolvable designs I: (1,2)-factorizations of K2n

1990
*
Discrete Mathematics
*

Between these two extremes we define a (1, 2)factorization

doi:10.1016/0012-365x(90)90179-l
fatcat:wvjjpkbjj5hixc4rtbr7nvmkni
*of*K, with cardinal@ k to be a pair (T, P) where T is a decomposition*of*K, into edges*and*triangles (K,'s*and*K,'s)*and*P is a partition*of*T ... is a decomposition*of*K" into triangles (K3's)*and*P is a partition*of*T into subsets T" . , Tc3n_,jn so that each T consists*of*n vertex-disjoint triangles. ... There exists a*Kirkman**Triple**System**of**order*v if*and*only if v = 3 module 6. ...##
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Nonsymmetric configurations with natural index

1994
*
Discrete Mathematics
*

The existence

doi:10.1016/0012-365x(94)90087-6
fatcat:j3rgfrcocfdk3b2hson7hqkwli
*of*a cfz. (*18*1 s0, 2715,) which is a Steiner*system*S(2,4,18 1) has been known already as well as the existence*of*S (2, 4, 193) or cfz. (19364, 30884). ... By using results about resolvable*and*near-resolvable Steiner*systems*as well as difference triangle sets, the existence*of*all configurations with k = 3 is proved. ... There is a*nearly**Kirkman**triple**system*NK TS(u) iff v = 0 mad (6) , v 3*18*. This implies the existence*of*a resolvable configuration (ur, b3), v=Omod (6) , r=(u-2)/2, b=u(u-2)/6 for all ~318. ...##
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Zero-sum flows for Steiner triple systems
[article]

2015
*
arXiv
*
pre-print

It has been conjectured that every Steiner

arXiv:1502.04096v2
fatcat:q7dq6akllvgxjk7ot6bed7ujva
*triple**system*, STS(v), on v points (v>7) admits a zero-sum 3-flow. ... We show that for every pair (v,λ), for which a*triple**system*, TS(v,λ) exists, there exists one which has a zero-sum 3-flow, except when (v,λ)∈{(3,1), (4,2), (6,2), (7,1)}*and*except possibly when v ≡ 1012 ... Acknowledgments This work was done while S.Akbari was visiting the University*of*Toronto*and*supported by NSERC Discovery Grant 455994. ...##
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Existence of resolvable group divisible designs with block size four I

2002
*
Discrete Mathematics
*

It is proved in this paper that for m ≡ 0; 2; 6; 10 (mod 12) there exists a resolvable group divisible design

doi:10.1016/s0012-365x(01)00299-0
fatcat:7gdadgcheraofhv7dgphcovdfi
*of**order*v, block size 4*and*group size m if*and*only v ≡ 0 (mod 4), v ≡ 0 (mod m), v − m ≡ ... 0 (mod 3), except when (3; 12)*and*except possibly when (3; 264); (3; 372); ... An RGD(k; 1; v) is known as a*Kirkman**system**and*denoted KS(2; k; v), An RGD(k; k − 1; v) is called a*nearly**Kirkman**system**and*denoted NKS(2; k; v). ...##
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Pairwise balanced designs with block sizes three and four

1991
*
Canadian Journal of Mathematics - Journal Canadien de Mathematiques
*

Given integers v, a

doi:10.4153/cjm-1991-039-9
fatcat:l5vrcmsnmndpzaybutuw4e6xa4
*and*b, when does a pairwise balanced design on v elements with a*triples**and*b quadruples exist? Necessary conditions are developed,*and*shown to be sufficient for all v > 96. ... An extensive set*of*constructions for pairwise balanced designs is used to obtain the result. ... We thank Rolf Rees*and*Yeow Meng Chee for pointing out relevant literature. Research*of*the authors is supported by NSERC Canada under grants numbered A0579 (CJC), A7268 (AR)*and*A9287 (DRS). ...##
###
Index

2000
*
Discrete Mathematics
*

Agnarsson
201
(1999)
5}

doi:10.1016/s0012-365x(99)00393-3
fatcat:4m66hhtlvzd2fhtqpiqouufzlm
*19*three dimensional partial*orders*which are not sphere*orders*201 (1999) 101}132 Ferenczi, S., Complexity*of*sequences*and*dynamical*systems*206 (1999) 145}154 FereticH ... Gionfriddo, On the edge- coloring property for*Hanani**triple**systems*208/209 (1999) 205}209 Laskar*and*L.R. Markus, Restrained domination in graphs 203 (1999) 61} 69 Dong, F.M.*and*K.M. ...##
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Generalized packing designs

2013
*
Discrete Mathematics
*

*of*

*triple*

*systems*,

*and*also concepts such as resolvability

*and*block colouring

*of*ordinary designs

*and*packings,

*and*orthogonal resolutions

*and*colourings. ... In this paper, we define a related class

*of*combinatorial designs which simultaneously generalize packing designs

*and*packing arrays. ... with v = (6, 6, 6, 7)

*and*k = (1, 1, 1, 1),

*and*Doug Stinson for the argument in the case v = (6, 5, 5)

*and*k = (2, 1, 1). ...

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