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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. ...*An**improved**construction**for*LKTS We ÿrst recall the deÿnition*of*LR-design [7] . Let X be a*set**of*size u. ...##
###
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). ... A*large**set**of**Kirkman**triple**systems**of*order v, denoted by LKTS(v), is*an*LSTS(v) where each STS(v) is resolvable. ...##
###
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 ...##
###
Further results on large sets of Kirkman triple systems

2008
*
Discrete Mathematics
*

*An*LR design is introduced by the second author in his recent paper and it plays a very important role in the

*construction*

*of*LKTS (a

*large*

*set*

*of*disjoint

*Kirkman*

*triple*

*system*). ... Some

*constructions*

*for*these two designs are also presented. With the relationship between them and LKTS, we obtain some new LKTSs. ... Zhu

*for*helpful suggestions on this topic and the referees

*for*many helpful comments. ...

##
###
On colorful edge triples in edge-colored complete graphs
[article]

2018
*
arXiv
*
pre-print

*For*the first two meaningful cases, when F=K_1,3 and F=P_4 we determine

*for*infinitely many n the minimum number

*of*colors needed

*for*

*an*F-caring edge-coloring

*of*K_n. ...

*An*explicit family

*of*2_2 n 3-edge-colorings

*of*K_n so that every quadruple

*of*its vertices contains a totally multicolored P_4 in at least one

*of*them is also presented. ... Resolvable Steiner

*triple*

*systems*are called

*Kirkman*

*triple*

*systems*(KTS

*for*short) referring to Kirkman's puzzle in the middle

*of*the 19th century asking

*for*the existence

*of*such a

*system*

*of*

*triples*...

##
###
On Colorful Edge Triples in Edge-Colored Complete Graphs

2020
*
Graphs and Combinatorics
*

*For*the first two meaningful cases, when $$F=K_{1,3}$$ F = K 1 , 3 and $$F=P_4$$ F = P 4 we determine

*for*infinitely many n the minimum number

*of*colors needed

*for*

*an*F-caring edge-coloring

*of*$$K_n$$ ...

*An*edge-coloring

*of*the complete graph $$K_n$$ K n we call F-caring if it leaves no F-subgraph

*of*$$K_n$$ K n monochromatic and at the same time every subset

*of*|V(F)| vertices contains in it at least ... Resolvable Steiner

*triple*

*systems*are called

*Kirkman*

*triple*

*systems*(KTS

*for*short) referring to Kirkman's puzzle in the middle

*of*the 19th century asking

*for*the existence

*of*such a

*system*

*of*

*triples*...

##
###
Transitive resolvable idempotent quasigroups and large sets of resolvable Mendelsohn triple systems

2009
*
Discrete Mathematics
*

Then we use TRIQ to present a

doi:10.1016/j.disc.2009.04.012
fatcat:v4gak2dpobbunbxys66tbffioa
*tripling**construction**for**large**sets**of*resolvable Mendelsohn*triple**systems*LRMTS(v)s, which*improves**an*earlier version*of**tripling**construction*by Kang. ... As*an*application we obtain*an*LRMTS(4 · 3 n )*for*any integer n ≥ 1, which provides*an*infinite family*of*even orders. ... Zhu*for*his helpful discussions on this topic. Research supported in part by NSFC Grant 10771013 and 10831002. ...##
###
Efficient Two-Stage Group Testing Algorithms for Genetic Screening

2013
*
Algorithmica
*

The main focus is on novel combinatorial

doi:10.1007/s00453-013-9791-2
fatcat:xgleqxnr7fgwhkshrwdvnnhl7y
*constructions*in order to minimize the number*of*individual tests at the second stage*of*a two-stage disjunctive testing procedure. ... Efficient two-stage group testing algorithms that are particularly suited*for*rapid and less-expensive DNA library screening and other*large*scale biological group testing efforts are investigated in this ... Acknowledgments The author thanks the anonymous referees*for*their careful reading and suggestions that helped*improving*the presentation*of*the paper. ...##
###
Linear spaces with many small lines

1994
*
Discrete Mathematics
*

In this paper some

doi:10.1016/0012-365x(92)00511-o
fatcat:6rzza6mlfnhrto72aw2q2hyhz4
*of*the work in linear spaces in which most*of*the lines have few points is surveyed. This includes existence results, blocking*sets*and embeddings. ... Also, it is shown that any linear space*of*order c can be embedded in a linear space*of*order about 13v in which there are no lines*of*size 2. ... Acknowledgment The author would like to thank Kevin Phelps and Rolf Rees*for*their useful comments during the preparation*of*this survey. ...##
###
Steiner quadruple systems - a survey

1978
*
Discrete Mathematics
*

However, several partial answers were given only 3 years later in 1847 by

doi:10.1016/0012-365x(78)90122-x
fatcat:hdr3ujbwrjetnclzfgmifwxzmy
*Kirkman*[28] who showed that a*system*S(2,3, v) i(i.e.*an*STS) exists if and only if v = 1 or 3 (mod 6) and*constructed**systems*... A Steiner*system*S( t, k, O) is a pair (S, B) where S is a u-*set*and B is a collection*of*k-subsets*of*S such that every t-subset*of*S is contained in exactly one member*of*B. ... Any collection*of**triple**systems*(S, Tkj, (~5, Tzj, . . . , (S, Tkj SUC~I that T1 U 'If2 U l l . U Tk = T(S) is called a*large**set**of*STS. ! ...##
###
Resolvable 2-designs for regular low-density parity-check codes

2003
*
IEEE Transactions on Communications
*

The resulting codes are ¡ ¢ -regular or £ ¢ -regular

doi:10.1109/tcomm.2003.816946
fatcat:kgwy4b3pxrdkvaa2zg6yq54obi
*for*any value*of*and*for*a flexible choice*of*code lengths. ... This paper extends the class*of*low-density paritycheck (LDPC) codes that can be algebraically*constructed*. ... Neal*for*his online repository*of*LDPCrelated software. ...##
###
Steiner systems $S(2,4,v)$ - a survey

2010
*
Electronic Journal of Combinatorics
*

We survey the basic properties and results on Steiner

doi:10.37236/39
fatcat:pnaqutsoz5ebhoppf6cm35hnaq
*systems*$S(2, 4, v)$, as well as open problems. ... OGP007268 (AR). the electronic journal*of*combinatorics (2010), #DS18 ... The difficulty*of**constructing**large**sets**of*S(2, 4, v)s is partly due to the fact that the number*of**systems*in a*large**set*LS(2, 4, v) is quadratic in v. ...##
###
R.A.Fisher, design theory, and the Indian connection
[article]

2009
*
arXiv
*
pre-print

This essay will present these structures and ideas as well as how the field came together, in itself

arXiv:0910.4163v1
fatcat:n52w6keffvhfpp6aov7oltvw4u
*an*interesting story. ... Design Theory, a branch*of*mathematics, was born out*of*the experimental statistics research*of*the population geneticist R. A. Fisher and*of*Indian mathematical statisticians in the 1930s. ...*Kirkman**constructed*the solution with b = 35, these being the number*of*rows*of*three, 5*for*each*of*the 7 days. ...##
###
The triangle chromatic index of Steiner triple systems

2001
*
The Australasian Journal of Combinatorics
*

The minimum number

dblp:journals/ajc/GrannellGH01
fatcat:y6knmyk76nebjgkcywc5zhqclu
*of*triangle-free*sets*required to partition the lines*of*a Steiner*triple**system*S, is called the triangle chromatic index*of*S. ... In a Steiner*triple**system**of*order v, STS(v), a*set**of*three lines intersecting pairwise in three distinct points is called a triangle. A*set**of*lines containing no triangle is called triangle-free. ... Acknowledgement The authors thank Professor Alex Rosa*of*McMaster University*for*his helpful comments and advice concerning the preparation*of*this paper. ...##
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Almost resolvable decompositions of 2Kn into cycles of odd length

1988
*
Journal of combinatorial theory. Series A
*

We prove that there exists

doi:10.1016/0097-3165(88)90053-2
fatcat:avscrzbknfcjppke5o7hfjf6y4
*an*almost resolvable m-cycle decomposition*of*2K" + 1*for*all odd ma3 and all S> 1. SC ... That is, they*constructed*what they called*an*almost resolvable 2-fold*triple**system**of*order n*for*all n = 1 (mod 3). ... When m = 3 and n is even the solution*of*the Oberwolfach problem is equivalent to finding a nearly*Kirkman**triple**system**for*all n = 0 (mod 6). ...
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