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

Charles J. Colbourn, Petteri Kaski, Patric R.J. Östergård, David A. Pike, Olli Pottonen
2011 Discrete Mathematics  
The 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.  ... 
doi:10.1016/j.disc.2011.02.005 fatcat:qth53ipoxzd3rhhywqacdhm524

A survey of Kirkman triple systems and related designs

D.R. Stinson
1991 Discrete Mathematics  
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  ...  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  ... 
doi:10.1016/0012-365x(91)90294-c fatcat:rd2luccrfffj3f5yzb4jkgnq7m

Palettes in block colourings of designs

Charles C. Lindner, Mariusz Meszka, Alexander Rosa
2013 The Australasian Journal of Combinatorics  
We obtain bounds on the minimum possible number 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.  ... 
dblp:journals/ajc/LindnerMR13 fatcat:bkkoxo6mcrajxfbfdxj6luxzkm

An Update on the Existence of Kirkman Triple Systems with Subdesigns [article]

Peter Dukes, Esther Lamken
2021 arXiv   pre-print
A 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, .  ... 
arXiv:2110.07874v1 fatcat:5jf2o3uwjjegnp2b7sdu2mwhcq

Resolvable Group Divisible Designs with Block Size 3 [chapter]

Ahmed M. Assaf, Alan Hartman
1989 Annals of Discrete Mathematics  
Dedicated to Professor Haim 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.  ... 
doi:10.1016/s0167-5060(08)70092-x fatcat:ltco6nv355e4pexpgytxilatp4

Resolvable group divisible designs with block size 3

Ahmed M. Assaf, Alan Hartman
1989 Discrete Mathematics  
Dedicated to Professor Haim 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.  ... 
doi:10.1016/0012-365x(89)90346-4 fatcat:5nyotfmz65h7fkib2blw5gzfae

Optimal packings of K4's into a Kn

A.E. Brouwer
1979 Journal of combinatorial theory. Series A  
(3, 1, 2; 12) (also called a 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] .  ... 
doi:10.1016/0097-3165(79)90105-5 fatcat:7pczfwftsbeadhrxnwj5kbz4c4

Three-line chromatic indices of Steiner triple systems

Mike J. Grannell, Terry S. Griggs, Alexander Rosa
2000 The Australasian Journal of Combinatorics  
There are five possible structures for a set 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] .  ... 
dblp:journals/ajc/GrannellGR00 fatcat:qlje53evu5bhxdqts7kslsvrmu

The existence of restricted resolvable designs I: (1,2)-factorizations of K2n

Rolf Rees
1990 Discrete Mathematics  
Between these two extremes we define a (1, 2)factorization 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.  ... 
doi:10.1016/0012-365x(90)90179-l fatcat:wvjjpkbjj5hixc4rtbr7nvmkni

Nonsymmetric configurations with natural index

Harald Gropp
1994 Discrete Mathematics  
The existence 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.  ... 
doi:10.1016/0012-365x(94)90087-6 fatcat:j3rgfrcocfdk3b2hson7hqkwli

Zero-sum flows for Steiner triple systems [article]

S. Akbari, A.C. Burgess, P. Danziger, E. Mendelsohn
2015 arXiv   pre-print
It has been conjectured that every Steiner 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.  ... 
arXiv:1502.04096v2 fatcat:q7dq6akllvgxjk7ot6bed7ujva

Existence of resolvable group divisible designs with block size four I

Hao Shen, Jiaying Shen
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 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).  ... 
doi:10.1016/s0012-365x(01)00299-0 fatcat:7gdadgcheraofhv7dgphcovdfi

Pairwise balanced designs with block sizes three and four

Charles J. Colbourn, Alexander Rosa, Douglas R. Stinson
1991 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
Given integers v, a 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).  ... 
doi:10.4153/cjm-1991-039-9 fatcat:l5vrcmsnmndpzaybutuw4e6xa4

Index

2000 Discrete Mathematics  
Agnarsson 201 (1999) 5} 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.  ... 
doi:10.1016/s0012-365x(99)00393-3 fatcat:4m66hhtlvzd2fhtqpiqouufzlm

Generalized packing designs

Robert F. Bailey, Andrea C. Burgess
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).  ... 
doi:10.1016/j.disc.2011.11.039 fatcat:gkjrrj33onbfjcnmujtjc5xqda
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