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Cycle Switches in Latin Squares

2004
*
Graphs and Combinatorics
*

*Cycle*

*switches*are the simplest changes which can be used to alter

*latin*

*squares*, and as such have found many applications

*in*the generation of

*latin*

*squares*. ... They also provide the simplest examples of

*latin*interchanges or trades

*in*

*latin*

*square*designs.

*In*this paper we construct graphs

*in*which the vertices are classes of

*latin*

*squares*. ...

*Cycle*

*Switching*Any

*cycle*

*in*a

*latin*

*square*L can be

*switched*to create a slightly different

*Latin*

*square*L 0 . ...

##
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THERE ARE ASYMPTOTICALLY THE SAME NUMBER OF LATIN SQUARES OF EACH PARITY

2016
*
Bulletin of the Australian Mathematical Society
*

A

doi:10.1017/s0004972716000174
fatcat:qidlnorw5ng7rnpyiht3lyy6ve
*Latin**square*is reduced if its first row and first column are*in*natural order. For*Latin**squares*of a particular order$n$, there are four possible different parities. ... We confirm a conjecture of Stones and Wanless by showing asymptotic equality between the numbers of reduced*Latin**squares*of each possible parity as the order$n\rightarrow \infty$. ...*Switching*the first switchable*cycle*that we find*in*this way would give us an involution, because*switching**cycles**in*rows x and y never affects the*cycle*lengths between rows other than x and y. ...##
###
The cycle structure of two rows in a random Latin square

2008
*
Random structures & algorithms (Print)
*

*In*this paper we avoid this problem by doing the

*switchings*within

*latin*

*squares*themselves. This requires the use of more complicated

*switchings*. ... Finally, we give computational data on the

*cycle*structure of

*latin*

*squares*of orders n ≤ 11. ... We expect that this also holds for the number of

*cycles*

*in*σ 1,2 for a random n × n

*latin*

*square*. ...

##
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Switching in One-Factorisations of Complete Graphs

2014
*
Electronic Journal of Combinatorics
*

For factor-

doi:10.37236/3606
fatcat:63raxxv7wjeirlk4dh5rgk2kue
*switching*the isolated vertices are perfect one-factorisations, while for vertex-*switching*the isolated vertices are closely related to atomic*Latin**squares*. ... We define two types of*switchings*between one-factorisations of complete graphs, called factor-*switching*and vertex-*switching*. ... There are corresponding notions of column*cycle**switching*and symbol*cycle**switching*, which are related to row*cycle**switching*by the conjugacy operation on*Latin**squares*. ...##
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Completing Partial Latin Squares with Two Filled Rows and Two Filled Columns

2008
*
Electronic Journal of Combinatorics
*

It is shown that any partial

doi:10.37236/780
fatcat:uite3zttwnbplomviqgbgmjoem
*Latin**square*of order at least six which consists of two filled rows and two filled columns can be completed. ... The partial*Latin**square*shown on the left below has column*cycle*type {2 * , 2, 3 * } and its transpose shown on the right has column*cycle*type {5 * * , 2}. ...*Switching**in*these created intercalates*in*the completion of S will result*in*a*Latin**square**in*which symbols x and y occur*in*the same cells as those*in*which they occur*in*P , and hence we obtain a completion ...##
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The theory and application of latin bitrades: A survey

2008
*
Mathematica Slovaca
*

This survey paper summarizes the theory of

doi:10.2478/s12175-008-0103-2
fatcat:a4xbmub3kbbbdi4xfgfjikxlgu
*latin*bitrades, detailing their applications to critical sets, random*latin**squares*and existence constructions for*latin**squares*. ... AbstractA*latin*bitrade is a pair of partial*latin**squares*which are disjoint, occupy the same set of non-empty cells, and whose corresponding rows and columns contain the same sets of symbols. ... However,*cycle**switches*are insufficient as it is not always possible to "walk" between any two*latin**squares*via*cycle**switches*alone, as shown*in*[61] . ...##
###
Orientable Hamilton Cycle Embeddings of Complete Tripartite Graphs I: Latin Square Constructions

2013
*
Journal of combinatorial designs (Print)
*

*In*this two-part series, we extend those results to orientable surfaces for all n = 2.

*In*part I, we explore a connection between orthogonal

*latin*

*squares*and embeddings. ... Moreover, it is shown that the

*latin*

*square*construction utilized to get hamilton

*cycle*embeddings of Kn,n,n can also be used to obtain triangulations of Kn,n,n. ... Acknowledgements The first author would like to thank Wendy Myrvold for helpful discussions regarding

*latin*

*squares*. ...

##
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Computing the autotopy group of a Latin square by cycle structure
[article]

2014
*
arXiv
*
pre-print

number of disjoint

arXiv:1305.1406v5
fatcat:homu6w4mwbcx5poan72bre5fiu
*cycles*, is polynomial*in*the order n. ... An algorithm that uses the*cycle*structure of the rows, or the columns, of a*Latin**square*to compute its autotopy group is introduced. ... Acknowledgment I thank Ian Wanless for providing the programs for generating*Latin**squares*for the experiments mentioned*in*Remark 4. ...##
###
On NP-hard graph properties characterized by the spectrum
[article]

2020
*
arXiv
*
pre-print

*In*addition, we discuss a possible spectral characterization of some well-known NP-hard problems. ...

*In*particular, for every integer k≥ 6 we construct a pair of k-regular cospectral graphs, where one graph is Hamiltonian and the other one not. ...

*Latin*

*square*graphs A good source for such pairs of graphs are the

*Latin*

*square*graphs defined as follows: Given an m × m

*Latin*

*square*L, the vertices of the

*Latin*

*square*graph G(L) are the m 2 entries ...

##
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On triple systems and strongly regular graphs

2012
*
Journal of combinatorial theory. Series A
*

Acknowledgments This work was partly inspired by a discussion with Ted Spence

doi:10.1016/j.jcta.2012.03.013
fatcat:be3jw4h5lvbnxdfbtmj7albmcq
*in*2003; Willem Haemers is acknowledged for more recent discussions. ...*Cycle**switching*has been applied extensively to the catalogues of the*Latin**squares*up to order 8 and the Steiner triple systems up to order 19; see [32] and [13, 14] , respectively. ... A*Latin**square*of order n can be viewed as a 3-GDD of type n 3 . ...##
###
Intercalates and Discrepancy in Random Latin Squares
[article]

2017
*
arXiv
*
pre-print

An intercalate

arXiv:1607.04981v2
fatcat:e2fsicvtwfgrzctn6v7tzsglrq
*in*a*Latin**square*is a 2×2*Latin*subsquare. Let N be the number of intercalates*in*a uniformly random n× n*Latin**square*. ... We also give an upper tail bound for the number of intercalates*in*two fixed rows of a random*Latin**square*.*In*addition, we discuss a problem of Linial and Luria on low-discrepancy*Latin**squares*. ... This situation is analogous to the use of 6-*cycle**switchings*rather than 4-*cycle**switchings**in*the analysis of random regular graphs (see for example [17] ). Lemma 13. ...##
###
All-even latin squares

1996
*
Discrete Mathematics
*

*In*this paper it is proved that the proportion of

*latin*

*squares*of order n which are all-even is at most c", where fi < c c 1. ... All-even

*latin*

*squares*are

*latin*

*squares*of which all rows are even permutations. All-even

*latin*rectangles are defined accordingly. ... This work was done while the second author was a visitor at Ume& University

*in*Sweden. She wishes to thank this U~versity, especially the Mathema~~ Department, for the support and hospitality. ...

##
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Optimal all-to-all personalized exchange in self-routable multistage networks

2000
*
IEEE Transactions on Parallel and Distributed Systems
*

By taking advantage of fast

doi:10.1109/71.841742
fatcat:3boeh55hu5ckzpqyyeftzbk5yi
*switch*setting of self-routable*switches*and the property of a single input/output port per processor*in*a multistage network, we believe that a multistage network could be ... AbstractÐAll-to-all personalized exchange is one of the most dense collective communication patterns and occurs*in*many important applications*in*parallel computing. ...*In*Fig. 8 , we list all possible*switch*settings*in*an V Â V omega network, and the corresponding*Latin**Square*is v P*in*(15). ...##
###
Diagonally switchable 4-cycle systems revisited

2009
*
The Australasian Journal of Combinatorics
*

*In*this paper we give an alternative proof of this result and use the method to prove a new result for K v − I, where I is any one factor of K v . ... the transformed set of 4-

*cycles*forms another 4-

*cycle*system. ... A self-orthogonal

*latin*

*square*of order v, or SOLS(v), is a

*latin*

*square*of order v which is orthogonal to its transpose. ...

##
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On the completion of latin rectangles to symmetric latin squares

2004
*
Journal of the Australian Mathematical Society
*

A

doi:10.1017/s1446788700008739
fatcat:znjmtjqebrbalp4zm4spxvqm54
*latin**square*is symmetric if for i ^ j , the symbol*in*cell (i, j) is also*in*cell (j, i). Consider the more difficult problem of completing*latin*rectangles to symmetric*latin**squares*. ... A*latin**square*is idempotent if symbol i is*in*cell (i, i) for all i. ... A partial*latin**square*is said to be completed to (or embedded*in*) a*latin**square*if its empty cells are filled to produce a*latin**square*. ...
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