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(2)
Latin Square
Satisfiability
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Indivisible partitions of latin squares
Indivisible partitions of latin squares,10.1016/j.jspi.2010.06.020,Journal of Statistical Planning and Inference,Judith Egan,Ian M. Wanless
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Indivisible partitions of latin squares
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Citations: 4
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Judith Egan
,
Ian M. Wanless
In a
latin square
of order n, a kplex is a selection of kn entries in which each row, column and symbol occurs k times. A 1plex is also called a transversal. An indivisible kplex is one that contains no cplex for 0ck. For orders n∉{2,6}, existence of latin squares with a partition into 1plexes was famously shown in 1960 by Bose, Shrikhande and Parker. A main result of this paper is that, if k divides n and 1kn then there exists a
latin square
of order n with a partition into indivisible kplexes.Define κ(n) to be the largest integer k such that some
latin square
of order n contains an indivisible kplex. We report on extensive computations of indivisible plexes and partitions in latin squares of order at most 9. We determine κ(n) exactly for n≤8 and find that κ(9)∈{6,7}. Up to order 8 we count all indivisible partitions in each species.For each group table of order n≤8 we report the number of indivisible plexes and indivisible partitions. For group tables of order 9 we give the number of indivisible plexes and identify which types of indivisible partitions occur. We will also report on computations which show that the latin squares of order 9 satisfy a conjecture that every
latin square
of order n has a set of ⌊n/2⌋ disjoint 2plexes.By extending an argument used by Mann, we show that for all n≥5 there is some k∈{1,2,3,4} for which there exists a
latin square
of order n that has k disjoint transversals and a disjoint (n−k)plex that contains no cplex for any odd c.
Journal:
Journal of Statistical Planning and Inference  J STATIST PLAN INFER
, vol. 141, no. 1, pp. 402417, 2011
DOI:
10.1016/j.jspi.2010.06.020
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Citation Context
(1)
...Such conjectures and the evidence from small order examples [
5
] suggest that all latin squares are ‘highly divisible’ in the sense that they possess many different partitions...
...It would also be particularly interesting to know the asymptotic behaviour of κ(n) .I n [
5
] it is found that κ(n )> 1 n at least for 5 ≤ n ≤ 9, so there may be room to improve...
...An extension to Theorem 1.2 is shown in [
5
]...
Darryn E. Bryant
,
et al.
Indivisible plexes in latin squares
References
(17)
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(
Citations: 12
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Journal:
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Enumeration of transversals in the Cayley tables of the noncyclic groups of order 8
(
Citations: 3
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David Bedford
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Journal:
Discrete Mathematics  DM
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Indivisible plexes in latin squares
(
Citations: 6
)
Darryn E. Bryant
,
Judith Egan
,
Barbara M. Maenhaut
,
Ian M. Wanless
Journal:
Designs, Codes and Cryptography  DCC
, vol. 52, no. 1, pp. 93105, 2009
On the number of transversals in Cayley tables of cyclic groups
(
Citations: 2
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Nicholas J. Cavenagh
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Journal:
Discrete Applied Mathematics  DAM
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Latin squares with no small odd plexes
(
Citations: 9
)
Judith Egan
,
Ian M. Wanless
Journal:
Journal of Combinatorial Designs  J COMB DES
, vol. 16, no. 6, pp. 477492, 2008
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Citations
(4)
Monogamous latin squares
(
Citations: 1
)
Peter Danziger
,
Ian M. Wanless
,
Bridget S. Webb
Journal:
Journal of Combinatorial Theory  JCT
, vol. 118, no. 3, pp. 796807, 2011
A generalization of plexes of Latin squares
Kyle Pula
Journal:
Discrete Mathematics  DM
, vol. 311, no. 89, pp. 577581, 2011
A Generalization of Plexes of Latin Squares
Kyle Pula
Published in 2010.
Indivisible plexes in latin squares
(
Citations: 6
)
Darryn E. Bryant
,
Judith Egan
,
Barbara M. Maenhaut
,
Ian M. Wanless
Journal:
Designs, Codes and Cryptography  DCC
, vol. 52, no. 1, pp. 93105, 2009