The importance of latin trades in the study of
completing partial latin squares.
Diane Donovan
May 13, 2013
Diane Donovan
The importance of latin trades in the study of completing parti
Definitions
Let N be a set of size n, usually {0, 1, 2, . . . , n − 1}.
Diane Donovan
The importance of latin trades in the study of completing parti
Definitions
Let N be a set of size n, usually {0, 1, 2, . . . , n − 1}.
I
A latin square (LS), of order n, is an
n × n array in which each of the
symbols of N occurs once in every row
and once in every column of the array.
Diane Donovan
The importance of latin trades in the study of completing parti
Definitions
Let N be a set of size n, usually {0, 1, 2, . . . , n − 1}.
I
A latin square (LS), of order n, is an
n × n array in which each of the
symbols of N occurs once in every row
and once in every column of the array.
Diane Donovan
0
1
2
3
4
1
2
3
4
0
2
1
4
0
3
3
4
0
2
1
4
0
1
3
2
The importance of latin trades in the study of completing parti
Definitions
Let N be a set of size n, usually {0, 1, 2, . . . , n − 1}.
I
A latin square (LS), of order n, is an
n × n array in which each of the
symbols of N occurs once in every row
and once in every column of the array.
I
A quasigroup (N, ◦) is a set N and a
binary operation ◦ satisfying the
properties, for all a, b, b 0 ∈ N,
if a ◦ b = a ◦ b 0 ,
then
b = b 0 , and
if b ◦ a = b 0 ◦ a,
then
b = b0 .
Diane Donovan
0
1
2
3
4
1
2
3
4
0
2
1
4
0
3
3
4
0
2
1
4
0
1
3
2
The importance of latin trades in the study of completing parti
Definitions
Let N be a set of size n, usually {0, 1, 2, . . . , n − 1}.
I
A latin square (LS), of order n, is an
n × n array in which each of the
symbols of N occurs once in every row
and once in every column of the array.
I
A quasigroup (N, ◦) is a set N and a
binary operation ◦ satisfying the
properties, for all a, b, b 0 ∈ N,
if a ◦ b = a ◦ b 0 ,
then
b = b 0 , and
if b ◦ a = b 0 ◦ a,
then
b = b0 .
Diane Donovan
0
1
2
3
4
◦
0
1
2
3
4
1
2
3
4
0
0
0
1
2
3
4
2
1
4
0
3
1
1
2
1
4
0
3
4
0
2
1
2
2
3
4
0
2
4
0
1
3
2
3
3
4
0
1
3
4
4
0
3
2
1
The importance of latin trades in the study of completing parti
Definitions
Let N be a set of size n, usually {0, 1, 2, . . . , n − 1}.
I
A latin square (LS), of order n, is an
n × n array in which each of the
symbols of N occurs once in every row
and once in every column of the array.
I
A decomposition of Kn,n,n into
triangles is equivalent to a latin
square of order n.
Diane Donovan
0
1
2
3
4
1
2
3
4
0
2
1
4
0
3
3
4
0
2
1
4
0
1
3
2
The importance of latin trades in the study of completing parti
Definitions
Let N be a set of size n, usually {0, 1, 2, . . . , n − 1}.
I
A latin square (LS), of order n, is an
n × n array in which each of the
symbols of N occurs once in every row
and once in every column of the array.
I
A decomposition of Kn,n,n into
triangles is equivalent to a latin
square of order n.
Diane Donovan
0
1
2
3
4
rows
•r 0
•r 1
•r 2
•r 3
•r 4
1
2
3
4
0
2
1
4
0
3
columns
•c0
•c1
•c2
•c3
•c4
3
4
0
2
1
4
0
1
3
2
symbols
•e0
•e1
•e2
•e3
•e4
The importance of latin trades in the study of completing parti
Definitions
I
In a partial latin square (PLS) each
symbol occurs at most once in every
row and at most once in every column
of the array.
Diane Donovan
The importance of latin trades in the study of completing parti
Definitions
I
In a partial latin square (PLS) each
symbol occurs at most once in every
row and at most once in every column
of the array.
Diane Donovan
1
0
0
2
0
1
2
1
The importance of latin trades in the study of completing parti
Definitions
I
I
In a partial latin square (PLS) each
symbol occurs at most once in every
row and at most once in every column
of the array.
1
0
0
2
0
1
2
1
The size is the number of filled cells.
Diane Donovan
The importance of latin trades in the study of completing parti
Definitions
I
In a partial latin square (PLS) each
symbol occurs at most once in every
row and at most once in every column
of the array.
1
0
0
2
0
1
2
1
Size 8
I
The size is the number of filled cells.
Diane Donovan
The importance of latin trades in the study of completing parti
Definitions
I
In a partial latin square (PLS) each
symbol occurs at most once in every
row and at most once in every column
of the array.
1
0
0
2
0
1
2
1
Size 8
I
The size is the number of filled cells.
I
A PLS P, of order n, is said to
complete to a LS L of order n, if P is
contained in L, written P ⊆ L.
Diane Donovan
The importance of latin trades in the study of completing parti
Definitions
I
1
In a partial latin square (PLS) each
symbol occurs at most once in every
row and at most once in every column
of the array.
0
0
2
0
1
2
1
Size 8
I
The size is the number of filled cells.
I
A PLS P, of order n, is said to
complete to a LS L of order n, if P is
contained in L, written P ⊆ L.
Diane Donovan
1
2
3
4
0
2
1
4
0
3
3
4
0
2
1
4
0
1
3
2
0
3
2
1
4
The importance of latin trades in the study of completing parti
Evans’ Conjecture, 1960
I
A PLS, of order n, with n − 1 or fewer
occupied cells can always be
completed to a LS of order n.
Diane Donovan
The importance of latin trades in the study of completing parti
Evans’ Conjecture, 1960
I
A PLS, of order n, with n − 1 or fewer
occupied cells can always be
completed to a LS of order n.
Diane Donovan
1
1
2
The importance of latin trades in the study of completing parti
Evans’ Conjecture, 1960
I
I
A PLS, of order n, with n − 1 or fewer
occupied cells can always be
completed to a LS of order n.
1
1
2
Lindner used counting arguments and
systems of distinct representatives to
solve the problem for squares where
the occupied cells occurred in the first
n/2 rows.
Diane Donovan
The importance of latin trades in the study of completing parti
Evans’ Conjecture, 1960
I
I
A PLS, of order n, with n − 1 or fewer
occupied cells can always be
completed to a LS of order n.
1
1
2
Lindner used counting arguments and
systems of distinct representatives to
solve the problem for squares where
the occupied cells occurred in the first
n/2 rows.
Diane Donovan
The importance of latin trades in the study of completing parti
Evans’ Conjecture, 1960
I
I
A PLS, of order n, with n − 1 or fewer
occupied cells can always be
completed to a LS of order n.
1
1
2
Lindner used counting arguments and
systems of distinct representatives to
solve the problem for squares where
the occupied cells occurred in the first
n/2 rows.
Diane Donovan
The importance of latin trades in the study of completing parti
Evans’ Conjecture, 1960
I
I
A PLS, of order n, with n − 1 or fewer
occupied cells can always be
completed to a LS of order n.
1
1
2
Lindner used counting arguments and
systems of distinct representatives to
solve the problem for squares where
the occupied cells occurred in the first
n/2 rows.
Diane Donovan
The importance of latin trades in the study of completing parti
Evans’ Conjecture, 1960
I
I
I
A PLS, of order n, with n − 1 or fewer
occupied cells can always be
completed to a LS of order n.
1
1
2
Lindner used counting arguments and
systems of distinct representatives to
solve the problem for squares where
the occupied cells occurred in the first
n/2 rows.
Smetaniuk went on to solve the problem in the early 80’s.
Diane Donovan
The importance of latin trades in the study of completing parti
Smetaniuk’s construction
order n + 1
order n
order n + 1
order n
order n + 1
Diane Donovan
The importance of latin trades in the study of completing parti
Completions
A PLS of order n and size n consisting of one entry from each row
and column and with no symbol repeated is called a transversal.
Diane Donovan
The importance of latin trades in the study of completing parti
Completions
A PLS of order n and size n consisting of one entry from each row
and column and with no symbol repeated is called a transversal.
I
Alspach and Heinrich posed the problem:
For each k does there exist an integer N(k)
such that every PLS, of order n ≥ N(k) and
consisting of k transversals, can be
completed?
Diane Donovan
The importance of latin trades in the study of completing parti
Completions
A PLS of order n and size n consisting of one entry from each row
and column and with no symbol repeated is called a transversal.
I
Alspach and Heinrich posed the problem:
For each k does there exist an integer N(k)
such that every PLS, of order n ≥ N(k) and
consisting of k transversals, can be
completed?
Diane Donovan
3
2
0
1
The importance of latin trades in the study of completing parti
Completions
A PLS of order n and size n consisting of one entry from each row
and column and with no symbol repeated is called a transversal.
I
Alspach and Heinrich posed the problem:
For each k does there exist an integer N(k)
such that every PLS, of order n ≥ N(k) and
consisting of k transversals, can be
completed?
3
2
0
1
3
2
Diane Donovan
1
2
3
0
1
0
The importance of latin trades in the study of completing parti
Completions
A PLS of order n and size n consisting of one entry from each row
and column and with no symbol repeated is called a transversal.
I
I
Alspach and Heinrich posed the problem:
For each k does there exist an integer N(k)
such that every PLS, of order n ≥ N(k) and
consisting of k transversals, can be
completed?
3
N(1) = 3 since there exists idempotent LS
for every order n 6= 2.
3
2
0
1
2
Diane Donovan
1
2
3
0
1
0
The importance of latin trades in the study of completing parti
Completions
A PLS of order n and size n consisting of one entry from each row
and column and with no symbol repeated is called a transversal.
I
I
I
Alspach and Heinrich posed the problem:
For each k does there exist an integer N(k)
such that every PLS, of order n ≥ N(k) and
consisting of k transversals, can be
completed?
3
N(1) = 3 since there exists idempotent LS
for every order n 6= 2.
3
Häggkvist and Daykin showed that for orders
n ≡ 0( mod 16) every PLS consisting of at
√
most n/128 transversals can be completed.
2
Diane Donovan
2
0
1
1
2
3
0
1
0
The importance of latin trades in the study of completing parti
Completions
A PLS of order n and size n consisting of one entry from each row
and column and with no symbol repeated is called a transversal.
I
I
I
I
Alspach and Heinrich posed the problem:
For each k does there exist an integer N(k)
such that every PLS, of order n ≥ N(k) and
consisting of k transversals, can be
completed?
3
N(1) = 3 since there exists idempotent LS
for every order n 6= 2.
3
2
0
1
1
2
2 0
Häggkvist and Daykin showed that for orders
0
n ≡ 0( mod 16) every PLS consisting of at
√
most n/128 transversals can be completed.
Computational results indicate N(k) = 4k − 1 and recently
Grüttmüller has shown N(k) ≥ 4k − 1.
Diane Donovan
3
1
The importance of latin trades in the study of completing parti
Completions
A PLS of order n and size n consisting of one entry from each row
and column and with no symbol repeated is called a transversal.
I
I
I
I
I
Alspach and Heinrich posed the problem:
For each k does there exist an integer N(k)
such that every PLS, of order n ≥ N(k) and
consisting of k transversals, can be
completed?
3
N(1) = 3 since there exists idempotent LS
for every order n 6= 2.
3
2
0
1
1
2
3
2 0
Häggkvist and Daykin showed that for orders
0 1
n ≡ 0( mod 16) every PLS consisting of at
√
most n/128 transversals can be completed.
Computational results indicate N(k) = 4k − 1 and recently
Grüttmüller has shown N(k) ≥ 4k − 1.
The determination of N(k) for all k is an OPEN PROBLEM.
Diane Donovan
The importance of latin trades in the study of completing parti
Restricting the problem to diagonally cyclic latin squares
Diane Donovan
The importance of latin trades in the study of completing parti
Restricting the problem to diagonally cyclic latin squares
I
A LS is diagonally cyclic if cell (i, j)
contains symbol k, implies cell
(i + 1, j + 1) contains symbol k + 1.
Diane Donovan
The importance of latin trades in the study of completing parti
Restricting the problem to diagonally cyclic latin squares
I
A LS is diagonally cyclic if cell (i, j)
contains symbol k, implies cell
(i + 1, j + 1) contains symbol k + 1.
Diane Donovan
0
2
1
2
1
0
1
0
2
The importance of latin trades in the study of completing parti
Restricting the problem to diagonally cyclic latin squares
I
A LS is diagonally cyclic if cell (i, j)
contains symbol k, implies cell
(i + 1, j + 1) contains symbol k + 1.
I
Diagonally cyclic LSs of even order do not exist.
Diane Donovan
0
2
1
2
1
0
1
0
2
The importance of latin trades in the study of completing parti
Restricting the problem to diagonally cyclic latin squares
I
A LS is diagonally cyclic if cell (i, j)
contains symbol k, implies cell
(i + 1, j + 1) contains symbol k + 1.
I
Diagonally cyclic LSs of even order do not exist.
I
Grüttmüller posed the question: For each k does there exist
an odd integer C (k) such that any PLS, of odd order
m ≥ C (k) and consisting of k cyclically generated diagonals,
can be completed?
Diane Donovan
0
2
1
2
1
0
1
0
2
The importance of latin trades in the study of completing parti
Restricting the problem to diagonally cyclic latin squares
I
A LS is diagonally cyclic if cell (i, j)
contains symbol k, implies cell
(i + 1, j + 1) contains symbol k + 1.
I
Diagonally cyclic LSs of even order do not exist.
I
Grüttmüller posed the question: For each k does there exist
an odd integer C (k) such that any PLS, of odd order
m ≥ C (k) and consisting of k cyclically generated diagonals,
can be completed?
I
Grüttmüller has shown that C (2) = 3 and C (k) ≥ 3k − 1, for
k ≥ 3.
Diane Donovan
0
2
1
2
1
0
1
0
2
The importance of latin trades in the study of completing parti
Restricting the problem to diagonally cyclic latin squares
I
A LS is diagonally cyclic if cell (i, j)
contains symbol k, implies cell
(i + 1, j + 1) contains symbol k + 1.
I
Diagonally cyclic LSs of even order do not exist.
I
Grüttmüller posed the question: For each k does there exist
an odd integer C (k) such that any PLS, of odd order
m ≥ C (k) and consisting of k cyclically generated diagonals,
can be completed?
I
Grüttmüller has shown that C (2) = 3 and C (k) ≥ 3k − 1, for
k ≥ 3.
I
Recently Cavenagh, Hämäläinen and Nelson have shown that
if the PLS has order p ≥ 7 where p is a prime and consists of
three cyclically generated diagonals then it can be completed.
Diane Donovan
0
2
1
2
1
0
1
0
2
The importance of latin trades in the study of completing parti
More on completing PLSs
I
A critical set C is a PLS, of order n, which has a unique
completion to a LS of order n, but with the additional property
that removing any entry enables at least 2 completions.
Diane Donovan
The importance of latin trades in the study of completing parti
More on completing PLSs
I
A critical set C is a PLS, of order n, which has a unique
completion to a LS of order n, but with the additional property
that removing any entry enables at least 2 completions.
I
One of the long standing open questions, is that of
establishing the spectrum of critical sets of order n.
Diane Donovan
The importance of latin trades in the study of completing parti
More on completing PLSs
I
A critical set C is a PLS, of order n, which has a unique
completion to a LS of order n, but with the additional property
that removing any entry enables at least 2 completions.
I
One of the long standing open questions, is that of
establishing the spectrum of critical sets of order n.
I
So given n, what is the set of values t, 1 ≤ t ≤ n2 , for which
there exists a critical set of order n and size t.
Diane Donovan
The importance of latin trades in the study of completing parti
More on completing PLSs
I
A critical set C is a PLS, of order n, which has a unique
completion to a LS of order n, but with the additional property
that removing any entry enables at least 2 completions.
I
One of the long standing open questions, is that of
establishing the spectrum of critical sets of order n.
I
So given n, what is the set of values t, 1 ≤ t ≤ n2 , for which
there exists a critical set of order n and size t.
I
Bate and van Rees have proven that for a certain class of
2
critical sets, t ≥ b n4 c.
Diane Donovan
The importance of latin trades in the study of completing parti
More on completing PLSs
I
A critical set C is a PLS, of order n, which has a unique
completion to a LS of order n, but with the additional property
that removing any entry enables at least 2 completions.
I
One of the long standing open questions, is that of
establishing the spectrum of critical sets of order n.
I
So given n, what is the set of values t, 1 ≤ t ≤ n2 , for which
there exists a critical set of order n and size t.
I
Bate and van Rees have proven that for a certain class of
2
critical sets, t ≥ b n4 c.
I
Cavenagh has shown that for LSs of any order n,
t ≥ nb(log n)1/3 /2c.
Diane Donovan
The importance of latin trades in the study of completing parti
More on completing PLSs
I
A critical set C is a PLS, of order n, which has a unique
completion to a LS of order n, but with the additional property
that removing any entry enables at least 2 completions.
I
One of the long standing open questions, is that of
establishing the spectrum of critical sets of order n.
I
So given n, what is the set of values t, 1 ≤ t ≤ n2 , for which
there exists a critical set of order n and size t.
I
Bate and van Rees have proven that for a certain class of
2
critical sets, t ≥ b n4 c.
I
Cavenagh has shown that for LSs of any order n,
t ≥ nb(log n)1/3 /2c.
I
An obvious upper bound is t ≤ n2 − n.
Diane Donovan
The importance of latin trades in the study of completing parti
More on completing PLSs
I
A critical set C is a PLS, of order n, which has a unique
completion to a LS of order n, but with the additional property
that removing any entry enables at least 2 completions.
I
One of the long standing open questions, is that of
establishing the spectrum of critical sets of order n.
I
So given n, what is the set of values t, 1 ≤ t ≤ n2 , for which
there exists a critical set of order n and size t.
I
Bate and van Rees have proven that for a certain class of
2
critical sets, t ≥ b n4 c.
I
Cavenagh has shown that for LSs of any order n,
t ≥ nb(log n)1/3 /2c.
I
An obvious upper bound is t ≤ n2 − n.
I
Bean and Mahmoodian have proven t ≤ n2 − 3n + 3.
Diane Donovan
The importance of latin trades in the study of completing parti
Spectrum of critical sets
Many results on critical sets are derived from studying the LS
corresponding to the cyclic group.
Diane Donovan
The importance of latin trades in the study of completing parti
Spectrum of critical sets
Many results on critical sets are derived from studying the LS
corresponding to the cyclic group.
I
Cavenagh, DD and Khodkar have shown that the cyclic group
of order n has critical sets of size t:
n2 − n
n2 − 1
≤t≤
,
4
2
n2
n2 − n
≤t≤
,
4
2
n2 − n
n2 − n
− (n − 2) ≤ t ≤
,
2
2
Diane Donovan
when n is odd
when n and t are even
when n is even.
The importance of latin trades in the study of completing parti
Example
I
LS corresponding to the cyclic group
of order 6.
Diane Donovan
0
1
2
3
4
5
1
2
3
4
5
0
2
3
4
5
0
1
3
4
5
0
1
2
4
5
0
1
2
3
5
0
1
2
3
4
The importance of latin trades in the study of completing parti
Example
I
LS corresponding to the cyclic group
of order 6.
I
A PLS of size n2 /4 = 36/4 = 9.
Diane Donovan
0
1
2
3
4
5
1
2
3
4
5
0
2
3
4
5
0
1
3
4
5
0
1
2
4
5
0
1
2
3
5
0
1
2
3
4
The importance of latin trades in the study of completing parti
Example
I
I
LS corresponding to the cyclic group
of order 6.
A PLS of size n2 /4 = 36/4 = 9.
Diane Donovan
0
1
2
3
4
5
1
2
3
4
5
0
2
3
4
5
0
1
0
1
2
1
2
2
3
4
5
0
1
2
4
5
0
1
2
3
5
0
1
2
3
4
3
3
4
The importance of latin trades in the study of completing parti
Example
I
I
LS corresponding to the cyclic group
of order 6.
A PLS of size n2 /4 = 36/4 = 9.
Diane Donovan
0
1
2
3
4
5
1
2
3
4
5
0
2
3
4
5
0
1
3
4
5
0
1
2
0
1
2
3
1
2
3
2
3
3
4
5
0
1
2
3
5
0
1
2
3
4
3
3
4
The importance of latin trades in the study of completing parti
Example
I
I
LS corresponding to the cyclic group
of order 6.
A PLS of size n2 /4 = 36/4 = 9.
0
1
2
3
4
5
1
2
3
4
5
0
2
3
4
5
0
1
3
4
5
0
1
2
4
5
0
1
2
3
0
1
2
3
4
1
2
3
4
2
3
4
3
4
4
3
Diane Donovan
5
0
1
2
3
4
3
4
The importance of latin trades in the study of completing parti
Example
I
I
LS corresponding to the cyclic group
of order 6.
A PLS of size n2 /4 = 36/4 = 9.
Diane Donovan
0
1
2
3
4
5
1
2
3
4
5
0
2
3
4
5
0
1
3
4
5
0
1
2
4
5
0
1
2
3
5
0
1
2
3
4
0
1
2
3
4
5
1
2
3
4
5
2
3
4
5
3
4
5
4
5
5
3
3
4
The importance of latin trades in the study of completing parti
Example
I
I
LS corresponding to the cyclic group
of order 6.
A PLS of size n2 /4 = 36/4 = 9.
Diane Donovan
0
1
2
3
4
5
1
2
3
4
5
0
2
3
4
5
0
1
3
4
5
0
1
2
4
5
0
1
2
3
5
0
1
2
3
4
0
1
2
3
4
5
1
2
3
4
5
0
2
3
4
5
0
3
4
5
0
4
5
0
5
0
3
3
4
The importance of latin trades in the study of completing parti
Example
I
I
LS corresponding to the cyclic group
of order 6.
A PLS of size n2 /4 = 36/4 = 9.
Diane Donovan
0
1
2
3
4
5
1
2
3
4
5
0
2
3
4
5
0
1
3
4
5
0
1
2
4
5
0
1
2
3
5
0
1
2
3
4
0
1
2
3
4
5
1
2
3
4
5
0
2
3
4
5
0
1
3
4
5
0
1
4
5
0
1
5
0
1
3
3
4
The importance of latin trades in the study of completing parti
Example
I
I
LS corresponding to the cyclic group
of order 6.
A PLS of size n2 /4 = 36/4 = 9.
Diane Donovan
0
1
2
3
4
5
1
2
3
4
5
0
2
3
4
5
0
1
3
4
5
0
1
2
4
5
0
1
2
3
5
0
1
2
3
4
0
1
2
3
4
5
1
2
3
4
5
0
2
3
4
5
0
1
3
4
5
0
1
2
4
5
0
1
2
3
5
0
1
2
3
4
The importance of latin trades in the study of completing parti
Example
0
1
2
1
2
Size
9 ×
10
11
12
13
14
15
2
3
Size 9 = n2 /4
3
4
Diane Donovan
The importance of latin trades in the study of completing parti
Example
0
1
2
1
2
2
3
Size 9 = n2 /4
0
1
2
3
3
4
1
2
3
2
3
3
4
Size 11 = n2 /4 + 2
Diane Donovan
Size
9 ×
10
11 ×
12
13
14
15
The importance of latin trades in the study of completing parti
Example
0
1
2
1
2
2
3
Size 9 = n2 /4
0
1
2
3
3
4
1
2
3
2
3
3
4
Size 11 = n2 /4 + 2
0
1
2
3
4
1
2
3
4
2
3
4
3
4
Size
9 ×
10
11 ×
12
13
14
15 ×
4
Size 15 = (n2 − n)/2
Diane Donovan
The importance of latin trades in the study of completing parti
Example
0
1
2
1
2
2
0
1
2
3
3
Size 9 = n2 /4
0
1
2
1
2
2
3
4
4
4
4
4
4
3
3
Size 13 = n2 /4 + 4
1
2
3
2
3
3
4
Size 11 = n2 /4 + 2
0
1
2
3
4
1
2
3
4
2
3
4
3
4
Size
9 ×
10
11 ×
12
13 ×
14
15 ×
4
Size 15 = (n2 − n)/2
Diane Donovan
The importance of latin trades in the study of completing parti
Spectrum of critical sets
This work raises the question of holes in the spectrum.
Diane Donovan
The importance of latin trades in the study of completing parti
Spectrum of critical sets
This work raises the question of holes in the spectrum.
I
For general latin squares of order n, DD and Howse and Bean
and DD have shown that there exists a critical set of size t:
2
n −1
n2 − n
≤t≤
.
4
2
Diane Donovan
The importance of latin trades in the study of completing parti
Spectrum of critical sets
This work raises the question of holes in the spectrum.
I
For general latin squares of order n, DD and Howse and Bean
and DD have shown that there exists a critical set of size t:
2
n −1
n2 − n
≤t≤
.
4
2
I
However, it has been conjectured that the cyclic group of even
order n does not contain a critical set of size n2 /4 + 1.
Diane Donovan
The importance of latin trades in the study of completing parti
Spectrum of critical sets
This work raises the question of holes in the spectrum.
I
For general latin squares of order n, DD and Howse and Bean
and DD have shown that there exists a critical set of size t:
2
n −1
n2 − n
≤t≤
.
4
2
I
However, it has been conjectured that the cyclic group of even
order n does not contain a critical set of size n2 /4 + 1.
I
Bate and van Rees have verified this conjecture for all n ≤ 12.
Diane Donovan
The importance of latin trades in the study of completing parti
Spectrum of critical sets
This work raises the question of holes in the spectrum.
I
For general latin squares of order n, DD and Howse and Bean
and DD have shown that there exists a critical set of size t:
2
n −1
n2 − n
≤t≤
.
4
2
I
However, it has been conjectured that the cyclic group of even
order n does not contain a critical set of size n2 /4 + 1.
I
Bate and van Rees have verified this conjecture for all n ≤ 12.
I
OPEN QUESTIONS relating to the size of the smallest and
largest critical set, and are there holes in the spectrum,
particular for specific families of LS.
Diane Donovan
The importance of latin trades in the study of completing parti
Example
0
1
2
3
1
2
3
2
3
3
4
0
1
2
3
2
3
2
3
3
4
Diane Donovan
The importance of latin trades in the study of completing parti
Example
0
1
2
3
0
1
2
3
1
2
3
2
3
2
3
2
3
3
4
0
1
2
3
4
5
1
2
3
4
5
0
2
3
4
5
0
1
3
4
5
0
1
2
4
5
0
1
2
3
5
0
1
2
3
4
4
0
1
2
3
4
5
5
2
3
4
1
0
2
3
4
5
0
1
3
4
1
0
5
2
4
5
0
1
2
3
1
0
5
2
3
4
3
Diane Donovan
The importance of latin trades in the study of completing parti
Example
0
1
2
3
0
1
2
3
1
2
3
2
3
2
3
2
3
3
1
2
3
4
5
0
2
3
4
5
0
1
3
4
5
0
1
2
4
5
0
1
2
3
5
0
1
2
3
4
1
4
0
1
2
3
4
5
5
2
3
4
1
0
2
3
4
5
0
1
3
4
1
0
5
2
4
5
0
1
2
3
1
0
5
2
3
4
5
4
0
1
2
3
4
5
3
Diane Donovan
5
1
5
5
0
1
0
1
1
1
0
1
5
1
0
5
The importance of latin trades in the study of completing parti
Critical sets and latin trades
A critical set C has a unique completion, but removing any entry
enables at least 2 completions.
Diane Donovan
The importance of latin trades in the study of completing parti
Critical sets and latin trades
A critical set C has a unique completion, but removing any entry
enables at least 2 completions.
I
For all x ∈ C , C \ {x} ⊆ M, where M
is a LS of order n, but M 6= L.
Diane Donovan
The importance of latin trades in the study of completing parti
Critical sets and latin trades
A critical set C has a unique completion, but removing any entry
enables at least 2 completions.
T
1
I
I
For all x ∈ C , C \ {x} ⊆ M, where M
is a LS of order n, but M 6= L.
Note T = L \ M and T 0 = M \ L are
two PLS satisfying the properties:
5
5
0
1
5
0
1
1
T0
5
1
Diane Donovan
1
0
1
5
1
0
5
The importance of latin trades in the study of completing parti
Critical sets and latin trades
A critical set C has a unique completion, but removing any entry
enables at least 2 completions.
T
1
I
I
For all x ∈ C , C \ {x} ⊆ M, where M
is a LS of order n, but M 6= L.
Note T = L \ M and T 0 = M \ L are
two PLS satisfying the properties:
I
5
5
0
1
5
T and T 0 have the same filled cells;
0
1
T0
5
1
Diane Donovan
1
1
0
1
5
1
0
5
The importance of latin trades in the study of completing parti
Critical sets and latin trades
A critical set C has a unique completion, but removing any entry
enables at least 2 completions.
T
1
I
I
For all x ∈ C , C \ {x} ⊆ M, where M
is a LS of order n, but M 6= L.
Note T = L \ M and T 0 = M \ L are
two PLS satisfying the properties:
I
I
T and T 0 have the same filled cells;
T and T 0 are disjoint;
5
0
1
5
0
1
1
T0
5
1
Diane Donovan
5
1
0
1
5
1
0
5
The importance of latin trades in the study of completing parti
Critical sets and latin trades
A critical set C has a unique completion, but removing any entry
enables at least 2 completions.
T
1
I
I
For all x ∈ C , C \ {x} ⊆ M, where M
is a LS of order n, but M 6= L.
Note T = L \ M and T 0 = M \ L are
two PLS satisfying the properties:
I
I
I
T and T 0 have the same filled cells;
T and T 0 are disjoint;
symbols in corresponding rows are
the same (row balanced);
5
0
1
5
0
1
1
T0
5
1
Diane Donovan
5
1
0
1
5
1
0
5
The importance of latin trades in the study of completing parti
Critical sets and latin trades
A critical set C has a unique completion, but removing any entry
enables at least 2 completions.
T
1
I
I
For all x ∈ C , C \ {x} ⊆ M, where M
is a LS of order n, but M 6= L.
Note T = L \ M and T 0 = M \ L are
two PLS satisfying the properties:
I
I
I
I
T and T 0 have the same filled cells;
T and T 0 are disjoint;
symbols in corresponding rows are
the same (row balanced);
symbols in corresponding columns
are the same (column balanced);
Diane Donovan
5
5
0
1
5
0
1
1
T0
5
1
1
0
1
5
1
0
5
The importance of latin trades in the study of completing parti
Critical sets and latin trades
A critical set C has a unique completion, but removing any entry
enables at least 2 completions.
T
1
I
I
For all x ∈ C , C \ {x} ⊆ M, where M
is a LS of order n, but M 6= L.
Note T = L \ M and T 0 = M \ L are
two PLS satisfying the properties:
I
I
I
I
I
T and T 0 have the same filled cells;
T and T 0 are disjoint;
symbols in corresponding rows are
the same (row balanced);
symbols in corresponding columns
are the same (column balanced);
5
5
0
1
5
0
1
1
T0
5
1
1
0
1
5
1
0
5
The pair of PLS {T , T 0 } are said to form a latin bitrade,
with T being a latin trade.
Diane Donovan
The importance of latin trades in the study of completing parti
Latin bitrades and Drápal triangulations
Let P be an equilateral triangle
with sides of length n, where
n ∈ Z and Q a partition of P
into be a set of integer-sided
equilateral triangles satisfying
the property that for every
vertex v of a triangle of Q
T
1
5
5
5
0
0
1
1
1
T’
5
1
Diane Donovan
1
0
1
1
0
5
5
The importance of latin trades in the study of completing parti
Latin bitrades and Drápal triangulations
Let P be an equilateral triangle
with sides of length n, where
n ∈ Z and Q a partition of P
into be a set of integer-sided
equilateral triangles satisfying
the property that for every
vertex v of a triangle of Q
I
v is either a vertex of P,
or
I
v is a vertex of exactly
three triangles of Q.
Diane Donovan
T
1
5
5
5
0
0
1
1
1
T’
5
1
1
0
1
1
0
5
5
The importance of latin trades in the study of completing parti
Latin bitrades and Drápal triangulations
Let P be an equilateral triangle
with sides of length n, where
n ∈ Z and Q a partition of P
into be a set of integer-sided
equilateral triangles satisfying
the property that for every
vertex v of a triangle of Q
I
v is either a vertex of P,
or
I
v is a vertex of exactly
three triangles of Q.
T
1
5
5
5
0
0
1
1
1
T’
5
1
1
0
1
1
0
5
5
Drápal has shown that given such a triangulation we can construct
a latin bitrade {T , T 0 } where T is contained in the cyclic group of
order n.
Diane Donovan
The importance of latin trades in the study of completing parti
Properties of Critical sets
RESULT: Let C ⊆ L be a critical set. Then C intersects every
latin trade in L and for each entry x ∈ C there exists a latin trade
T ⊆ L such that T ∩ C = {x}.
Diane Donovan
The importance of latin trades in the study of completing parti
Properties of Critical sets
RESULT: Let C ⊆ L be a critical set. Then C intersects every
latin trade in L and for each entry x ∈ C there exists a latin trade
T ⊆ L such that T ∩ C = {x}.
PROOF: If there exists a latin trade T ⊆ L, such that T ∩ C = ∅,
then C ⊆ ((L \ T ) ∪ T 0 ), where {T , T 0 } is a latin bitrade.
Diane Donovan
The importance of latin trades in the study of completing parti
Properties of Critical sets
RESULT: Let C ⊆ L be a critical set. Then C intersects every
latin trade in L and for each entry x ∈ C there exists a latin trade
T ⊆ L such that T ∩ C = {x}.
PROOF: If there exists a latin trade T ⊆ L, such that T ∩ C = ∅,
then C ⊆ ((L \ T ) ∪ T 0 ), where {T , T 0 } is a latin bitrade.
Since every entry of C is necessary for the unique completion of C ,
C \ {x} is contained in a LS L0 distinct from L. The PLS L \ L0 is
the required latin trade.
Diane Donovan
The importance of latin trades in the study of completing parti
New Critical sets from old
RESULT: Let C ⊆ L be a critical set
and {T , T 0 } is a latin bitrade with
T ⊆ L.Then (C \ T ) ∪ T 0 has a
unique completion to the LS
(L \ T ) ∪ T 0 .
Diane Donovan
The importance of latin trades in the study of completing parti
New Critical sets from old
RESULT: Let C ⊆ L be a critical set
and {T , T 0 } is a latin bitrade with
T ⊆ L.Then (C \ T ) ∪ T 0 has a
unique completion to the LS
(L \ T ) ∪ T 0 .
L
C
T
PROOF: Note (C ∪ T ) ⊆ L, so
(C \ T ) ∪ T 0 ⊆ ((L \ T ) ∪ T 0 ).
(L/T)UT’
Assume (C \ T ) ∪ T 0 completes to
distinct latin squares
M = (L \ T ) ∪ T 0 and M 0 . But this
implies M \ M 0 is a latin trade and
(M \ M 0 ) ⊆ (L \ C ) which is a
contradiction.
Diane Donovan
C
T’
The importance of latin trades in the study of completing parti
New Critical sets from old
RESULT: Let C ⊆ L be a critical set
and {T , T 0 } is a latin bitrade with
T ⊆ L.Then (C \ T ) ∪ T 0 has a
unique completion to the LS
(L \ T ) ∪ T 0 .
L
C
T
PROOF: Note (C ∪ T ) ⊆ L, so
(C \ T ) ∪ T 0 ⊆ ((L \ T ) ∪ T 0 ).
(L/T)UT’
Assume (C \ T ) ∪ T 0 completes to
distinct latin squares
M = (L \ T ) ∪ T 0 and M 0 . But this
implies M \ M 0 is a latin trade and
(M \ M 0 ) ⊆ (L \ C ) which is a
contradiction.
Diane Donovan
C
T’
The importance of latin trades in the study of completing parti
New Critical sets from old
RESULT: Let C ⊆ L be a critical set
and {T , T 0 } is a latin bitrade with
T ⊆ L.Then (C \ T ) ∪ T 0 has a
unique completion to the LS
(L \ T ) ∪ T 0 .
L
C
T
PROOF: Note (C ∪ T ) ⊆ L, so
(C \ T ) ∪ T 0 ⊆ ((L \ T ) ∪ T 0 ).
(L/T)UT’
Assume (C \ T ) ∪ T 0 completes to
distinct latin squares
M = (L \ T ) ∪ T 0 and M 0 . But this
implies M \ M 0 is a latin trade and
(M \ M 0 ) ⊆ (L \ C ) which is a
contradiction.
Diane Donovan
C
T’
The importance of latin trades in the study of completing parti
Example
0
1
2
3
Cyclic
0
1
2
3
Abelian
1 2 3
0 3 2
3 1 0
2 0 1
group order 4
1 2
0 3
3 0
2 1
2-group
3
2
1
0
order 4
Diane Donovan
The importance of latin trades in the study of completing parti
Example
0
1
2
3
Cyclic
0
1
2
3
Abelian
1 2 3
0 3 2
3 1 0
2 0 1
group order 4
1 2
0 3
3 0
2 1
2-group
0
2
2
1
Size 4 =
n2 /4
3
2
1
0
order 4
Diane Donovan
The importance of latin trades in the study of completing parti
Example
0
1
2
3
Cyclic
0
1
2
3
Abelian
1 2 3
0 3 2
3 1 0
2 0 1
group order 4
1 2
0 3
3 0
2 1
2-group
3
2
1
0
order 4
Diane Donovan
0
2
2
1
Size 4 =
0
n2 /4
2
2
0 1
1 0
Size 7 = 4m − 3n when n = 2m
The importance of latin trades in the study of completing parti
Critical sets in LS of order 23
Cyclic Group
2
4
0
2
6
3
1
4
6
5
2
3
1
0
3
2
0
1
28
29
30
6
Size
1
0
3
2
5
0
1
2
3
31
Diane Donovan
32
33
34
35
36
37
The importance of latin trades in the study of completing parti
Critical sets in LS of order 23
Cyclic Group
2
4
0
2
6
3
1
4
6
6
5
Size
28
×
2
3
1
0
3
2
0
1
29
30
1
0
3
2
5
0
1
2
3
31
Diane Donovan
32
33
34
35
36
37
The importance of latin trades in the study of completing parti
Critical sets in LS of order 23
Cyclic Group
2
4
0
2
6
3
2
1
4
6
6
5
Size
28
×
Abelian 2-group
2
4
6
0
2
3
1
0
3
2
0
1
29
30
1
0
3
2
5
0
1
2
3
31
Diane Donovan
4
6
32
33
0
1
6
1
0
4
5
5
4
34
35
6
0
1
2
3
36
1
0
3
2
4
5
2
3
0
1
5
4
3
2
1
0
37
The importance of latin trades in the study of completing parti
Critical sets in LS of order 23
Cyclic Group
2
4
0
2
6
3
2
1
4
6
6
5
Size
28
×
Abelian 2-group
2
4
6
0
2
3
1
0
3
2
0
1
29
30
1
0
3
2
5
0
1
2
3
31
Diane Donovan
4
6
32
33
0
1
6
1
0
4
5
5
4
34
35
6
0
1
2
3
36
1
0
3
2
4
5
2
3
0
1
5
4
3
2
1
0
37
×
The importance of latin trades in the study of completing parti
Critical sets in LS of order 23
Cyclic Group
2
4
0
2
6
3
2
1
4
6
6
5
Size
28
×
Abelian 2-group
2
4
6
0
2
3
1
0
3
2
0
1
29
30
1
0
3
2
5
0
1
2
3
31
Diane Donovan
4
6
32
33
0
1
6
1
0
4
5
5
4
34
35
6
0
1
2
3
36
1
0
3
2
4
5
2
3
0
1
5
4
3
2
1
0
37
×
The importance of latin trades in the study of completing parti
Critical sets in LS of order 23
Cyclic Group
2
4
0
2
6
3
2
1
4
6
6
5
Size
28
×
Abelian 2-group
2
4
6
0
0
1
2
3
1
0
3
2
29
30
2
3
0
1
5
3
2
1
0
31
Diane Donovan
4
6
32
33
0
1
6
1
0
4
5
5
4
34
35
6
0
1
2
3
36
1
0
3
2
4
5
2
3
0
1
5
4
3
2
1
0
37
×
The importance of latin trades in the study of completing parti
Critical sets in LS of order 23
Cyclic Group
2
4
0
2
6
3
2
1
4
6
6
5
Size
28
×
Abelian 2-group
2
4
6
0
0
1
2
3
1
0
3
2
29
30
2
3
0
1
5
3
2
1
0
31
Diane Donovan
4
6
32
33
0
1
6
1
0
4
5
5
4
34
35
6
0
1
2
3
36
1
0
3
2
4
5
2
3
0
1
5
4
3
2
1
0
37
×
The importance of latin trades in the study of completing parti
Critical sets in LS of order 23
0
1
2
3
4
1
0
3
2
Cyclic
2 3
3 2
1 0
0 1
6
6
5
Size
28
×
Group
4
6
0
6
0
1
2
3
29
2
1
0
3
2
30
5
3
2
1
0
2
3
0
1
31
Diane Donovan
4
6
32
33
Abelian 2-group
2
4
6
0
1
6
1
0
6
4
5
5
4
0
1
2
3
34
35
36
1
0
3
2
4
5
2
3
0
1
5
4
3
2
1
0
37
×
The importance of latin trades in the study of completing parti
Critical sets in LS of order 23
0
1
2
3
4
1
0
3
2
Cyclic
2 3
3 2
1 0
0 1
6
6
5
Size
28
×
Group
4
6
0
6
0
1
2
3
29
2
1
0
3
2
30
5
3
2
1
0
2
3
0
1
31
Diane Donovan
4
6
32
33
Abelian 2-group
2
4
6
0
1
6
1
0
6
4
5
5
4
0
1
2
3
34
35
36
1
0
3
2
4
5
2
3
0
1
5
4
3
2
1
0
37
×
The importance of latin trades in the study of completing parti
Critical sets in LS of order 23
Cyclic Group
2
4
0
2
1
0
6
4
0
1
6
5
Size
28
×
6
Abelian 2-group
2
4
6
0
6
2
0
1
2
3
1
0
3
2
29
30
2
3
0
1
5
3
2
1
0
31
Diane Donovan
4
6
32
33
0
1
6
1
0
4
5
5
4
34
35
6
0
1
2
3
36
1
0
3
2
4
5
2
3
0
1
5
4
3
2
1
0
37
×
The importance of latin trades in the study of completing parti
Critical sets in LS of order 23
0
Not Cyclic Group
2
4
6
2
0
1
6
4
1
0
6
5
Size
28
×
Abelian 2-group
2
4
6
0
6
2
0
1
2
3
1
0
3
2
29
30
2
3
0
1
5
3
2
1
0
31
Diane Donovan
4
6
32
33
0
1
6
1
0
4
5
5
4
34
35
6
0
1
2
3
36
1
0
3
2
4
5
2
3
0
1
5
4
3
2
1
0
37
×
The importance of latin trades in the study of completing parti
Critical sets in LS of order 23
0
Not Cyclic Group
2
4
6
2
0
1
6
4
1
0
6
5
Size
28
×
Abelian 2-group
2
4
6
0
6
2
0
1
2
3
1
0
3
2
29
30
2
3
0
1
5
3
2
1
0
31
Diane Donovan
4
6
32
33
0
1
6
1
0
4
5
5
4
34
35
6
0
1
2
3
36
1
0
3
2
4
5
2
3
0
1
5
4
3
2
1
0
37
×
The importance of latin trades in the study of completing parti
Critical sets in LS of order 23
0
Not Cyclic Group
2
4
6
2
0
1
6
4
1
0
6
5
Size
28
×
Abelian 2-group
2
4
6
0
6
2
0
1
2
3
1
0
3
2
29
30
2
3
0
1
5
3
2
1
0
31
×
Diane Donovan
4
6
32
33
0
1
6
1
0
4
5
5
4
34
35
6
0
1
2
3
36
1
0
3
2
4
5
2
3
0
1
5
4
3
2
1
0
37
×
The importance of latin trades in the study of completing parti
Critical sets in LS of order 23
0
2
4
6
Not Cyclic Group
2
4 5 6
5 4 7
0 1 6 7 5
1 0 7 6 4
6
0 1 2
1 0 3
2 3 0
5 3 2 1
Size
28
×
29
30
7
6
4
5
3
2
1
0
31
×
Diane Donovan
Abelian 2-group
2
4
6
0
2
4
6
32
33
0
1
6
1
0
4
5
5
4
34
35
6
0
1
2
3
36
1
0
3
2
4
5
2
3
0
1
5
4
3
2
1
0
37
×
The importance of latin trades in the study of completing parti
Critical sets in LS of order 23
0
2
4
6
Not Cyclic Group
2
4 5 6
5 4 7
0 1 6 7 5
1 0 7 6 4
6
0 1 2
1 0 3
2 3 0
5 3 2 1
Size
28
×
29
30
7
6
4
5
3
2
1
0
31
×
Diane Donovan
Abelian 2-group
2
4
6
0
2
4
6
32
33
0
1
6
1
0
4
5
5
4
34
35
6
0
1
2
3
36
1
0
3
2
4
5
2
3
0
1
5
4
3
2
1
0
37
×
The importance of latin trades in the study of completing parti
Critical sets in LS of order 23
0
Not Cyclic Group
2
4
6
2
0
1
6
4
1
0
6
5
Size
28
×
6
0
1
2
3
1
0
3
2
29
30
5
4
2
3
0
1
Abelian 2-group
2
4
6
0
4
5
3
2
1
0
31
×
Diane Donovan
2
4
6
32
33
0
1
6
1
0
4
5
5
4
34
35
6
0
1
2
3
36
1
0
3
2
4
5
2
3
0
1
5
4
3
2
1
0
37
×
The importance of latin trades in the study of completing parti
Critical sets in LS of order 23
0
Not Cyclic Group
2
4
6
2
0
1
6
4
1
0
6
5
Size
28
×
6
0
1
2
3
1
0
3
2
29
30
4
5
2
3
0
1
Abelian 2-group
2
4
6
0
5
4
3
2
1
0
31
×
Diane Donovan
2
4
6
32
33
0
1
6
1
0
4
5
5
4
34
35
6
0
1
2
3
36
1
0
3
2
4
5
2
3
0
1
5
4
3
2
1
0
37
×
The importance of latin trades in the study of completing parti
Critical sets in LS of order 23
0
Not Cyclic Group
2
4
6
2
0
1
6
4
1
0
6
5
Size
28
×
6
0
1
2
3
1
0
3
2
29
30
4
5
2
3
0
1
Abelian 2-group
2
4
6
0
5
4
3
2
1
0
31
×
Diane Donovan
2
4
6
32
33
0
1
6
1
0
4
5
5
4
34
35
6
0
1
2
3
36
1
0
3
2
4
5
2
3
0
1
5
4
3
2
1
0
37
×
The importance of latin trades in the study of completing parti
Critical sets in LS of order 23
0
Not Cyclic Group
2
4
6
2
0
1
6
4
1
0
6
5
Size
28
×
6
0
1
2
3
1
0
3
2
29
30
4
5
2
3
0
1
Abelian 2-group
2
4
6
0
5
4
3
2
1
0
31
×
Diane Donovan
2
4
6
32
33
0
1
6
1
0
4
5
5
4
34
×
35
6
0
1
2
3
36
1
0
3
2
4
5
2
3
0
1
5
4
3
2
1
0
37
×
The importance of latin trades in the study of completing parti
Critical sets in LS of order 2m
RESULT: (DD, Lefevre, van Rees) Let m ≥ 1. Then there exists a
LS of order 2m which has a critical set of size t, where
4m−1 ≤ t ≤ 4m − 3m .
Diane Donovan
The importance of latin trades in the study of completing parti
More on Latin bitrades
A latin bitrade {T , T 0 } is a pair of PLS T and T 0 of order n,
satisfying the properties:
Diane Donovan
The importance of latin trades in the study of completing parti
More on Latin bitrades
A latin bitrade {T , T 0 } is a pair of PLS T and T 0 of order n,
satisfying the properties:
I
T and T 0 have the same filled cells;
Diane Donovan
The importance of latin trades in the study of completing parti
More on Latin bitrades
A latin bitrade {T , T 0 } is a pair of PLS T and T 0 of order n,
satisfying the properties:
I
T and T 0 have the same filled cells;
I
T and T 0 are disjoint;
Diane Donovan
The importance of latin trades in the study of completing parti
More on Latin bitrades
A latin bitrade {T , T 0 } is a pair of PLS T and T 0 of order n,
satisfying the properties:
I
T and T 0 have the same filled cells;
I
T and T 0 are disjoint;
I
symbols in corresponding rows are the same (row balanced);
Diane Donovan
The importance of latin trades in the study of completing parti
More on Latin bitrades
A latin bitrade {T , T 0 } is a pair of PLS T and T 0 of order n,
satisfying the properties:
I
T and T 0 have the same filled cells;
I
T and T 0 are disjoint;
I
symbols in corresponding rows are the same (row balanced);
I
symbols in corresponding columns are the same (column
balanced);
1
0
3
0
3
3
2
2
1
1
Diane Donovan
0
3
1
3
0
2
1
1
3
2
The importance of latin trades in the study of completing parti
Drápal: Latin bitrades and permutations
1
0
3
0
3
3
2
2
1
1
0
3
1
3
0
2
1
1
3
2
•A
•E
•B
•F
•I
•C
•D
•G
•J
•H
α = (ABCD)(EF )(GH)(IJ)
β = (AIE )(BF )(CJG )(DH)
γ = (BE )(CFI )(DG )(HJ)
Diane Donovan
The importance of latin trades in the study of completing parti
Drápal: Latin bitrades and permutations
1
0
3
0
3
3
2
2
1
1
0
3
1
3
0
2
1
1
3
2
•A
•E
•I
•B
•F
•C
•D
•G
•J
•H
α = (ABCD)(EF )(GH)(IJ)
β = (AIE )(BF )(CJG )(DH)
γ = (BE )(CFI )(DG )(HJ)
Diane Donovan
The importance of latin trades in the study of completing parti
Drápal: Latin bitrades and permutations
I
Since the PLS are disjoint the permutations α, β, γ will be
fixed point free.
Diane Donovan
The importance of latin trades in the study of completing parti
Drápal: Latin bitrades and permutations
I
Since the PLS are disjoint the permutations α, β, γ will be
fixed point free.
I
Consider the occurrence of any symbol e and the mappings
α, β, γ.
c
r
r0
T
c0
c
e
r
T0
c0
e
r0
e
So
γβα = 1.
Diane Donovan
The importance of latin trades in the study of completing parti
Drápal: Latin bitrades and permutations
I
Since the PLS are disjoint the permutations α, β, γ will be
fixed point free.
I
Consider the occurrence of any symbol e and the mappings
α, β, γ.
c
r
r0
T
c0
c
e
r
T0
c0
e
r0
e
So
γβα = 1.
I
Two cycles from different permutations α, β, γ intersect in at
most one point.
Diane Donovan
The importance of latin trades in the study of completing parti
Drápal: Latin bitrades and permutations
Alternatively, let τi , 1 ≤ i ≤ 3 be permutations on a finite set X ,
and let Ai , 1 ≤ i ≤ 3, denote the set of all cycles of permutation
τi . Further assume
I
the permutations are fixed point free and τ1 τ2 τ3 = 1, and
I
two cycles τi and τj , 1 ≤ i < j ≤ 3, intersect in at most one
point.
Diane Donovan
The importance of latin trades in the study of completing parti
Drápal: Latin bitrades and permutations
Alternatively, let τi , 1 ≤ i ≤ 3 be permutations on a finite set X ,
and let Ai , 1 ≤ i ≤ 3, denote the set of all cycles of permutation
τi . Further assume
I
the permutations are fixed point free and τ1 τ2 τ3 = 1, and
I
two cycles τi and τj , 1 ≤ i < j ≤ 3, intersect in at most one
point.
Define T , T 0 to be arrays with filled cells corresponding to A1 × A2
and containing elements of A3 , in such a way that
I
cell (ρ1 , ρ2 ) of T contains ρ3 if and only if ρ1 , ρ2 , ρ3 all act
on some x ∈ X , and
I
cell (ρ1 , ρ2 ) of T 0 contains ρ3 if and only if there exists x ∈ X
such that ρ3 acts on x, ρ2 acts on ρ3 (x), and ρ1 acts on
ρ2 (ρ3 (x)).
Diane Donovan
The importance of latin trades in the study of completing parti
Drápal: Latin bitrades and permutations
Alternatively, let τi , 1 ≤ i ≤ 3 be permutations on a finite set X ,
and let Ai , 1 ≤ i ≤ 3, denote the set of all cycles of permutation
τi . Further assume
I
the permutations are fixed point free and τ1 τ2 τ3 = 1, and
I
two cycles τi and τj , 1 ≤ i < j ≤ 3, intersect in at most one
point.
Define T , T 0 to be arrays with filled cells corresponding to A1 × A2
and containing elements of A3 , in such a way that
I
cell (ρ1 , ρ2 ) of T contains ρ3 if and only if ρ1 , ρ2 , ρ3 all act
on some x ∈ X , and
I
cell (ρ1 , ρ2 ) of T 0 contains ρ3 if and only if there exists x ∈ X
such that ρ3 acts on x, ρ2 acts on ρ3 (x), and ρ1 acts on
ρ2 (ρ3 (x)).
Then {T , T 0 } is a latin bitrade.
Diane Donovan
The importance of latin trades in the study of completing parti
Biembeddings of Latin squares
Consider a latin bitrade {T , T 0 } where T and T 0 are both latin
squares of order n and the permutations α, β and γ can be written
as compositions of cycles of length n.
0
1
1
0
1
0
0
1
row 0
row 1
Diane Donovan
c0
c1
c0
c1
The importance of latin trades in the study of completing parti
Biembeddings of Latin squares
Consider a latin bitrade {T , T 0 } where T and T 0 are both latin
squares of order n and the permutations α, β and γ can be written
as compositions of cycles of length n.
0
1
1
0
1
0
0
1
By adding vertices for each
element we obtain:
e1
row 0
row 1
c0
e0
c1
c0
e1
c1
e0
Diane Donovan
The importance of latin trades in the study of completing parti
Biembeddings of Latin squares
Consider a latin bitrade {T , T 0 } where T and T 0 are both latin
squares of order n and the permutations α, β and γ can be written
as compositions of cycles of length n.
0
1
1
0
1
0
0
1
By adding vertices for each
element and each row we
obtain:
e1
r0
c0
e0
c1
c0
e1
c1
r1
e0
Diane Donovan
The importance of latin trades in the study of completing parti
Biembeddings of Latin squares
Consider a latin bitrade {T , T 0 } where T and T 0 are both latin
squares of order n and the permutations α, β and γ can be written
as compositions of cycles of length n.
0
1
1
0
1
0
0
1
By adding vertices for each
element, each row and joining
these vertices we obtain:
e1
r0
c0
e0
c1
c0
e1
c1
r1
e0
Diane Donovan
The importance of latin trades in the study of completing parti
An Example of a Biembedding of Latin squares
Consider a latin bitrade {T , T 0 } where T and T 0 are both latin
squares of order n and the permutations α, β and γ can be written
as compositions of cycles of length n.
0
1
1
0
1
0
0
1
By adding vertices for each
element, each row and joining
these vertices we obtain:
Finally 2-colouring the
triangles gives:
e1
r0
c0
e0
c1
c0
e1
c1
r1
e0
Diane Donovan
The importance of latin trades in the study of completing parti
An Example of a Biembedding of Latin squares
By glueing common edges we obtain a 2-colouring of the complete
tripartite graph
K2,2,2 .
e
A decomposition of Kn,n,n ,
r
V = V1 ∪ V2 ∪ V3 , into triangles
c
c
e
corresponds to a LS. (Triangle
{v1 , v2 , v3 } gives rise to entry v3 in cell
c
e
c
(v1 , v2 ) of the array.
r
If M be a face 2-colourable triangular
e
embedding of Kn,n,n then the faces in
c0
e0
c0
each colour class (blue and white)
determine a LS of order n and the face
r1
r0
2-colourable triangular embedding of
Kn,n,n may be regarded as a
e1
c1
e1
biembedding of two LSs of order n.
1
0
0
0
1
0
1
1
1
0
Diane Donovan
The importance of latin trades in the study of completing parti
An Example of a Biembedding of Latin squares
By glueing common edges we obtain a 2-colouring of the complete
tripartite graph
K2,2,2 .
e
Grannell and Griggs state
r
1
0
c0
e0
c1
c0
e1
c1
r1
e0
c0
e0
c0
r1
r0
e1
c1
e1
Diane Donovan
The importance of latin trades in the study of completing parti
An Example of a Biembedding of Latin squares
By glueing common edges we obtain a 2-colouring of the complete
tripartite graph
K2,2,2 .
e
Grannell and Griggs state
r
1
0
c0
e0
c1
c0
e1
c1
I
r1
e0
c0
e0
c0
We may reasonably enquire about
existence of these biembeddings for
each n, the number of
biembeddings for each n, whether
every LS is biembeddable, and
whether every pair of LSs of the
same size is biembeddable.
r1
r0
e1
c1
e1
Diane Donovan
The importance of latin trades in the study of completing parti
Number of Latin squares of a given order
McKay, Wanless has shown that there are
776966836171770144107444346734230682311065600000 LS of
order 11.
Focusing on reduced LS (first row and column are in natural order)
then:
Order
2
3
4
5
6
7
8
9
10
11
Number of reduced LS
1
1
22
23.7
26.3.72
210.3.5.1103
217.3.1361291
221.32.5231.3824477
228.32.5.31.37.547135293937
235.34.5.2801.2206499.62368028479
Diane Donovan
The importance of latin trades in the study of completing parti
Number of Latin squares of a given order
McKay, Wanless has shown that there are
776966836171770144107444346734230682311065600000 LS of
order 11.
Focusing on reduced LS (first row and column are in natural order)
then:
Order
2
3
4
5
6
7
8
9
10
11
Number of reduced LS
1
1
22
23.7
26.3.72
210.3.5.1103
217.3.1361291
221.32.5231.3824477
228.32.5.31.37.547135293937
235.34.5.2801.2206499.62368028479
OPEN PROBLEM: count number of LSs of order 12 or more!!
Diane Donovan
The importance of latin trades in the study of completing parti