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### Cycle Switches in Latin Squares

Ian M. Wanless
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 .  ...

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

NICHOLAS J. CAVENAGH, IAN M. WANLESS
2016 Bulletin of the Australian Mathematical Society
A 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

Nicholas J. Cavenagh, Catherine Greenhill, Ian M. Wanless
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.  ...

### Switching in One-Factorisations of Complete Graphs

Petteri Kaski, André de Souza Medeiros, Patric R.J. Östergård, Ian M. Wanless
2014 Electronic Journal of Combinatorics
For factor-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.  ...

### Completing Partial Latin Squares with Two Filled Rows and Two Filled Columns

Peter Adams, Darryn Bryant, Melinda Buchanan
2008 Electronic Journal of Combinatorics
It is shown that any partial 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  ...

### The theory and application of latin bitrades: A survey

Nicholas Cavenagh
2008 Mathematica Slovaca
This survey paper summarizes the theory of 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

M. N. Ellingham, Justin Z. Schroeder
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.  ...

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

Daniel Kotlar
2014 arXiv   pre-print
number of disjoint 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]

Omid Etesami, Willem H. Haemers
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  ...

### On triple systems and strongly regular graphs

Majid Behbahani, Clement Lam, Patric R.J. Östergård
2012 Journal of combinatorial theory. Series A
Acknowledgments This work was partly inspired by a discussion with Ted Spence 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]

Matthew Kwan, Benny Sudakov
2017 arXiv   pre-print
An intercalate 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

Roland Häggkvist, Jeannette C.M. Janssen
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.  ...

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

Y. Yang, J. Wang
2000 IEEE Transactions on Parallel and Distributed Systems
By taking advantage of fast 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

Chandra Dinavahi, Christopher A. Rodger
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.  ...

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

Darryn Bryant, C. A. Rodger
2004 Journal of the Australian Mathematical Society
A 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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