On the number of even latin squares of even order
Carolin Hannusch
本文回顾了关于偶拉丁方数量的Alon-Tarsi猜想，并提出了一种在特定条件下切换拉丁方奇偶性的映射，同时给出了相关示例。
我们回顾关于偶拉丁方数量的Alon-Tarsi猜想。我们引入一个在特定要求下切换拉丁方奇偶性的映射。并包含一个示例。
We recall the Alon-Tarsi conjecture on the number of even latin squares. We introduce a map which switches the parity of a latin square under certain requirements. An example is included.
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We investigate various pursuit-evasion parameters on latin square graphs, including the cop number, metric dimension, and localization number. The cop number of latin square graphs is studied, and for $k$-MOLS$(n),$ bounds for the cop number are given. If $n>(k+1)^2,$ then the cop number is shown to be $k+2.$ Lower and upper bounds are provided for the metric dimension and localization number of latin square graphs. The metric dimension of back-circulant latin squares shows that the lower bound is close to tight. Recent results on covers and partial transversals of latin squares provide the up…
Computing random $r$-orthogonal Latin squares
本文针对$r$-正交和$r$-自正交拉丁方阵，开发了计算$n$阶$r$-正交拉丁方阵对的随机化算法，以及计算$n$阶$r$-自正交拉丁方阵的算法。
Improving SAT Solvers on Orthogonal Latin Square Problems
研究如何改进SAT求解器解决正交拉丁方问题，采用将SAT求解器与欧拉 - 帕克算法结合的混合方法，有效找到特定类型正交拉丁方，如解决了25年未解决的10×10正交拉丁方问题。
Latin squares are interesting combinatorial objects with many applications. When working with Latin squares, one is sometimes led to deal with partial Latin squares, a generalization of Latin squares. One of the problems regarding partial Latin square and with applications to Latin squares is whether a partial Latin square with a given set of conditions exists. The goal of this article is to introduce some problems of this kind and answer some existence questions regarding partial Latin squares
A (partial) Latin square is a table of multiplication of a (partial) quasigroup. Multiplication of a (partial) quasigroup may be considered as a set of triples. We give a necessary and sufficient condition when a set of triples is a quotient of a (partial) Latin square.
A multi-latin square of order $n$ and index $k$ is an $n\times n$ array of multisets, each of cardinality $k$, such that each symbol from a fixed set of size $n$ occurs $k$ times in each row and $k$ times in each column. A multi-latin square of index $k$ is also referred to as a $k$-latin square. A $1$-latin square is equivalent to a latin square, so a multi-latin square can be thought of as a generalization of a latin square. In this note we show that any partially filled-in $k$-latin square of order $m$ embeds in a $k$-latin square of order $n$, for each $n\geq 2m$, thus generalizing Evans' …
Avoiding a partial Latin square with an isotope of a given Latin square
该研究针对足够大的n阶拉丁方，通过仅置换行和列的方法，证明了可使其避开任意指定的n阶部分拉丁方，证明结合了Hall定理与随机置换的初等估计。
We show that any partial Latin square of order $n$ can be embedded in a Latin square of order at most $16n^2$ which has at least $2n$ mutually orthogonal mates. We also show that for any $t\geq 2$, a pair of orthogonal partial Latin squares of order $n$ can be embedded into a set of $t$ mutually orthogonal Latin squares (MOLS) of order a polynomial with respect to $n$. Furthermore, the constructions that we provide show that MOLS($n^2$)$\geq$MOLS($n$)+2, consequently we give a set of $9$ MOLS($576$). The maximum known size of a set of MOLS($576$) was previously given as $8$ in the literature.
The chromatic number of a Latin square is the least number of partial transversals which cover its cells. This is just the chromatic number of its associated Latin square graph. Although Latin square graphs have been widely studied as strongly regular graphs, their chromatic numbers appear to be unexplored. We determine the chromatic number of a circulant Latin square, and find bounds for some other classes of Latin squares. With a computer, we find the chromatic number for all main classes of Latin squares of order at most eight.
An $n \times n$ partial Latin square $P$ is called $α$-dense if each row and column has at most $αn$ non-empty cells and each symbol occurs at most $αn$ times in $P$. An $n \times n$ array $A$ where each cell contains a subset of $\{1,\dots, n\}$ is a $(βn, βn, βn)$-array if each symbol occurs at most $βn$ times in each row and column and each cell contains a set of size at most $βn$. Combining the notions of completing partial Latin squares and avoiding arrays, we prove that there are constants $α, β> 0$ such that, for every positive integer $n$, if $P$ is an $α$-dense $n \times n$ partial La…
Choi Seok-Jeong studied Latin squares at least 60 years earlier than Euler although this was less known. He introduced a pair of orthogonal Latin squares of order 9 in his book. Interestingly, his two orthogonal non-double-diagonal Latin squares produce a magic square of order 9, whose theoretical reason was not studied. There have been a few studies on Choi's Latin squares of order 9. The most recent one is Ko-Wei Lih's construction of Choi's Latin squares of order 9 based on the two $3 \times 3$ orthogonal Latin squares. In this paper, we give a new generalization of Choi's orthogonal Latin …
In this paper we propose an algorithm for enumerating diagonal Latin squares of small order. It relies on specific properties of diagonal Latin squares to employ symmetry breaking techniques, and on several heuristic optimizations and bit arithmetic techniques to make use of computational power of state-of-the-art CPUs. Using this approach we enumerated diagonal Latin squares of order at most 9, and vertically symmetric diagonal Latin squares of order at most 10.
A Latin square has six conjugate Latin squares obtained by uniformly permuting its (row, column, symbol) triples. We say that a Latin square has conjugate symmetry if at least two of its six conjugates are equal. We enumerate Latin squares with conjugate symmetry and classify them according to several common notions of equivalence. We also do similar enumerations under additional hypotheses, such as assuming the Latin square is reduced, diagonal, idempotent or unipotent. Our data corrected an error in earlier literature and suggested several patterns that we then found proofs for, including (1…
Symmetries of a partial Latin square are determined by its autotopism group. Analogously to the case of Latin squares, given an isotopism $Θ$, the cardinality of the set $\mathcal{PLS}_Θ$ of partial Latin squares which are invariant under $Θ$ only depends on the conjugacy class of the latter, or, equivalently, on its cycle structure. In the current paper, the cycle structures of the set of autotopisms of partial Latin squares are characterized and several related properties studied. It is also seen that the cycle structure of $Θ$ determines the possible sizes of the elements of $\mathcal{PLS}_…
We introduce a graph attached to mutually orthogonal Sudoku Latin squares. The spectra of the graphs obtained from finite fields are explicitly determined. As a corollary, we then use the eigenvalues to distinguish non-isomorphic Sudoku Latin squares.
To get another from a given latin square, we have to change at least 4 entries. We show how to find these entries and how to change them.
Consider a partial Latin square $P$ where the first two rows and first three columns are completely filled, and every other cell of $P$ is empty. It has been conjectured that all such partial Latin squares of order at least $8$ are completable. Based on a technique by Kuhl and McGinn we describe a framework for completing partial Latin squares in this class. Moreover, we use our method for proving that all partial Latin squares from this family, where the intersection of the nonempty rows and columns form a Latin rectangle with three distinct symbols, is completable.
In this paper, we first present the relation between a transversal in a Latin square with some concepts in its Latin square graph, and give an equivalent condition for a Latin square has an orthogonal mate. The most famous open problem involving Combinatorics is to find maximum number of disjoint transversals in a Latin square. So finding some family of decomposable Latin squares into disjoint transversals is our next aim. In the next section, we give an equivalent statement of a conjecture which has been attributed to Brualdi, Stein and Ryser by the concept of quasi-transversal. Finally, we p…
We investigate sets of Mutually Orthogonal Latin Squares (MOLS) generated by Cellular Automata (CA) over finite fields. After introducing how a CA defined by a bipermutive local rule of diameter $d$ over an alphabet of $q$ elements generates a Latin square of order $q^{d-1}$, we study the conditions under which two CA generate a pair of orthogonal Latin squares. In particular, we prove that the Latin squares induced by two Linear Bipermutive CA (LBCA) over the finite field $\mathbb{F}_q$ are orthogonal if and only if the polynomials associated to their local rules are relatively prime. Next, w…
A Latin square is reduced if its first row and 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 $n\rightarrow\infty$.
A classical question in combinatorics is the following: given a partial latin square P, when can we complete P to a latin square L? In this paper, we will investigate the class of \leqε-dense partial latin squares: partial latin squares in which each symbol, row, and column contains \leqεn-many nonblank cells. A conjecture of Nash-Williams on triangulations of graphs led Daykin and Häggkvist to conjecture that all \leq(1/4)-dense partial latin squares are completable. In this paper, we will discuss the proof methods and results used in previous attempts to resolve this conjecture, introduce a …
In this note we show that for each Latin square $L$ of order $n\geq 2$, there exists a Latin square $L'\neq L$ of order $n$ such that $L$ and $L'$ differ in at most $8\sqrt{n}$ cells. Equivalently, each Latin square of order $n$ contains a Latin trade of size at most $8\sqrt{n}$. We also show that the size of the smallest defining set in a Latin square is $Ω(n^{3/2})$. %That is, there are constants $c$ and $n_0$ such that for any $n>n_0$ the size of the smallest defining %set of order $n$ is at least $cn^{3/2}$.
We show that a pair of orthogonal partial latin squares of order $n$ can be embedded in a pair of orthogonal latin squares of order at most $16n^4$ and all orders greater than or equal to $48n^4$. This paper provides the first direct polynomial order embedding construction in the literature.
The logarithm of the maximum number of transversals over all latin squares of order $n$ is greater than $\frac{n}{6}(\ln n+ O(1))$.
The parity type of a Latin square is defined in terms of the numbers of even and odd rows and columns. It is related to an Alon-Tarsi-like conjecture that applies to Latin squares of odd order. Parity types are used to derive upper bounds for the size of autotopy groups. A fast algorithm for finding the autotopy group of a Latin square, based on the cycle decomposition of its rows, is presented.
An intercalate in a Latin square is a $2\times2$ Latin subsquare. Let $N$ be the number of intercalates in a uniformly random $n\times n$ Latin square. We prove that asymptotically almost surely $N\ge\left(1-o\left(1\right)\right)\,n^{2}/4$, and that $\mathbb{E}N\le\left(1+o\left(1\right)\right)\,n^{2}/2$ (therefore asymptotically almost surely $N\le fn^{2}$ for any $f\to\infty$). This significantly improves the previous best lower and upper bounds. 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 L…
Latin squares and hypercubes are combinatorial designs with several applications in statistics, cryptography and coding theory. In this paper, we generalize a construction of Latin squares based on bipermutive cellular automata (CA) to the case of Latin hypercubes of dimension $k>2$. In particular, we prove that linear bipermutive CA (LBCA) yielding Latin hypercubes of dimension $k>2$ are defined by sequences of invertible Toeplitz matrices with partially overlapping coefficients, which can be described by a specific kind of regular de Bruijn graph induced by the support of the determinant fun…
We study equitable partitions of Latin-square graphs, and give a complete classification of those whose quotient matrix does not have an eigenvalue $-3$.
In combinatorics, a latin square is a $n\times n$ matrix filled with n different symbols, each occurring exactly once in each row and exactly once in each column. Associated to each latin square, we can define a simple graph called a latin square graph. In this article, we compute lower and upper bounds for the domination number and the k-tuple total domination numbers of such graphs. Moreover, we describe a formula for the 2-tuple total domination number.
Two latin squares are orthogonal if, when they are superimposed, every ordered pair of symbols appears exactly once. This definition extends naturally to `incomplete' latin squares each having a hole on the same rows, columns, and symbols. If an incomplete latin square of order $n$ has a hole of order $m$, then it is an easy observation that $n \ge 2m$. More generally, if a set of $t$ incomplete mutually orthogonal latin squares of order $n$ have a common hole of order $m$, then $n \ge (t+1)m$. In this article, we prove such sets of incomplete squares exist for all $n,m \gg 0$ satisfying $n \g…
Difference arrays are used in applications such as software testing, authentication codes and data compression. Pseudo-orthogonal Latin squares are used in experimental designs. A special class of pseudo-orthogonal Latin squares are the mutually nearly orthogonal Latin squares (MNOLS) first discussed in 2002, with general constructions given in 2007. In this paper we develop row complete MNOLS from difference covering arrays. We will use this connection to settle the spectrum question for sets of 3 mutually pseudo-orthogonal Latin squares of even order, for all but the order 146.
Gyárfás and Sárközy conjectured that every $n\times n$ Latin square has a `cycle-free' partial transversal of size $n-2$. We confirm this conjecture in a strong sense for almost all Latin squares, by showing that as $n \rightarrow \infty$, all but a vanishing proportion of $n\times n$ Latin squares have a Hamilton transversal, i.e. a full transversal for which any proper subset is cycle-free. In fact, we prove a counting result that in almost all Latin squares, the number of Hamilton transversals is essentially that of Taranenko's upper bound on the number of full transversals. This result str…
Let $P$ be a partial latin square of prime order $p>7$ consisting of three cyclically generated transversals. Specifically, let $P$ be a partial latin square of the form: \[ P=\{(i,c+i,s+i),(i,c'+i,s'+i),(i,c''+i,s''+i)\mid 0 \leq i< p\} \] for some distinct $c,c',c''$ and some distinct $s,s',s''$. In this paper we show that any such $P$ completes to a latin square which is diagonally cyclic.
A Latin square of order $n$ is an $n\times n$ array which contains $n$ distinct symbols exactly once in each row and column. We define the adjacent distance between two adjacent cells (containing integers) to be their difference modulo $n$, and inner distance of a Latin square to be the minimum of adjacent distances in the Latin square. By first establishing upper bounds and then constructing squares with said inner distance, we found the maximum inner distance of an $n \times n$ Latin square to be $\left\lfloor\frac{n-1}{2}\right\rfloor$. We then studied special kinds of Latin squares such as…
By simulating an ergodic Markov chain whose stationary distribution is uniform over the space of nxn Latin squares, Mark T. Jacobson and Peter Matthews [4], have discussed elegant methods by which they generate Latin squares with a uniform distribution (approximately). The central issue is the construction of "moves" that connect the squares. Most of their lengthy paper is to prove that the associated graph is indeed connected. We give a short proof of this fact by using the concepts of Latin bitrades.
The problem of completing a partially specified n by n Latin square is solved by an alternative proof, based on filling the rows (or diagonals) from 1 to n, using an extended form of Hall's marriage theorem.
Symmetric Quantum Latin Squares with Maximal Cardinality
本文证明了所有$n\geq 6$阶存在最大基数的对称量子拉丁方，且所有$4k+2$阶存在最大基数的对称幂等量子拉丁方。
In recent years there has been a great deal of work done on secret sharing scehemes. Secret Sharing Schemes allow for the division of keys so that an authorised set of users may access information. In this paper we wish to present a critical comparison of two of these schemes based on Latin Squares, [Cooper et., al.] and RSA [Shoup]. These two protocols will be examined in terms of their positive and negative aspects of their secuirty.
Autoparatopisms of Quasigroups and Latin Squares
研究quasigroups和拉丁方的自paratopism性质，确定n≤17时Par(n)的结构，并分析n趋于无穷时Par(n)在所有paratopism中的比例。
Constructive and nonconstructive techniques are employed to enumerate Latin squares and related objects. It is established that there are (i) 2036029552582883134196099 main classes of Latin squares of order 11; (ii) 6108088657705958932053657 isomorphism classes of one-factorizations of $K_{11,11}$; (iii) 12216177315369229261482540 isotopy classes of Latin squares of order 11; (iv) 1478157455158044452849321016 isomorphism classes of loops of order 11; and (v) 19464657391668924966791023043937578299025 isomorphism classes of quasigroups of order 11. The enumeration is constructive for the 1151666…
Hall's Condition is a necessary condition for a partial latin square to be completable. Hilton and Johnson showed that for a partial latin square whose filled cells form a rectangle, Hall's Condition is equivalent to Ryser's Condition, which is a necessary and sufficient condition for completability. We give what could be regarded as an extension of Ryser's Theorem, by showing that for a partial latin square whose filled cells form a rectangle, where there is at most one empty cell in each column of the rectangle, Hall's Condition is a necessary and sufficient condition for completability. It …
We (1) determine the number of Latin rectangles with 11 columns and each possible number of rows, including the Latin squares of order~11, (2) answer some questions of Alter by showing that the number of reduced Latin squares of order $n$ is divisible by $f!$ where $f$ is a particular integer close to $\frac12n$, (3) provide a formula for the number of Latin squares in terms of permanents of $(+1,-1)$-matrices, (4) find the extremal values for the number of 1-factorisations of $k$-regular bipartite graphs on $2n$ vertices whenever $1\leq k\leq n\leq11$, (5) show that the proportion of Latin sq…
Latin squares have been historically used in order to create statistical designs in which, starting from a small number of experiments, it can be obtained a large experimental space. In this sense, the optimization of the selection of Latin squares can be decisive. A factor to take into account is the symmetry that the experimental space must verify and which is established by the autotopism group of each Latin square. Although the size of this group is known for Latin squares of order up to 10, a classification of the different symmetries has not yet been done. In this paper, given a cycle st…
We show that the threshold for the binomial random $3$-partite, $3$-uniform hypergraph $G^{3}((n,n,n),p)$ to contain a Latin square is $Θ(\log{n}/n)$. We also prove analogous results for Steiner triple systems and proper list edge-colorings of the complete (bipartite) graph with random lists. Our results answer several related questions of Johansson, Luria-Simkin, Casselgren-Häggkvist, Simkin, and Kang-Kelly-Kühn-Methuku-Osthus.
This is a companion note to the paper "Almost all Steiner triple systems have perfect matchings (arXiv:1611.02246). That paper contains several general lemmas about random Steiner triple systems; in this note we record analogues of these lemmas for random Latin squares, which in particular are necessary ingredients for our recent paper "Large deviations in random Latin squares" (arXiv:2106.11932). Most important is a relationship between uniformly random order-$n$ Latin squares and the triangle removal process on the complete tripartite graph $K_{n,n,n}$.
Do $K_{3,3}$-Free Latin Squares Exist?
研究是否存在不含K33子结构的拉丁方，构造了阶为16的K33自由拉丁方，并探讨了正交拉丁方的类似问题。
We prove several results about substructures in Latin squares. First, we explain how to adapt our recent work on high-girth Steiner triple systems to the setting of Latin squares, resolving a conjecture of Linial that there exist Latin squares with arbitrarily high girth. As a consequence, we see that the number of order-$n$ Latin squares with no intercalate (i.e., no $2\times2$ Latin subsquare) is at least $(e^{-9/4}n-o(n))^{n^{2}}$. Equivalently, $\mathbb{P}\left[\mathbf{N}=0\right]\ge e^{-n^{2}/4-o(n^{2})}=e^{-(1+o(1))\mathbb{E}\mathbf{N}}$, where $\mathbf{N}$ is the number of intercalates …
A classical question in combinatorics is the following:\ given a partial Latin square $P$, when can we complete $P$ to a Latin square $L$? In this paper, we investigate the class of \textbf{$ε$-dense partial Latin squares}:\ partial Latin squares in which each symbol, row, and column contains no more than $εn$-many nonblank cells. Based on a conjecture of Nash-Williams, Daykin and Häggkvist conjectured that all $\frac{1}{4}$-dense partial Latin squares are completable. In this paper, we will discuss the proof methods and results used in previous attempts to resolve this conjecture, introduce a…
We prove a conjecture by Garbe et al. [arXiv:2010.07854] by showing that a Latin square is quasirandom if and only if the density of every 2x3 pattern is 1/720+o(1). This result is the best possible in the sense that 2x3 cannot be replaced with 2x2 or 1xN for any N.
Transversals in Latin Squares
本文综述了过去十年关于Latin squares中transversals研究的最新进展。
In a latin square of order $n$, a near transversal is a collection of $n-1$ cells which intersects each row, column, and symbol class at most once. A longstanding conjecture of Brualdi, Ryser, and Stein asserts that every latin square possesses a near transversal. We show that this conjecture is true for every latin square that is main class equivalent to the Cayley table of a finite group.
Limits of Latin squares
本文发展了拉丁方的极限理论，引入了密度、割距离和极限对象Latinons，证明了极限空间的紧致性与拓扑等价性，并证明每个Latinon可由有限拉丁方近似。
In this paper we study pattern avoidance in Latin Squares, which gives us a two dimensional analogue of the well studied notion of pattern avoidance in permutations. Our main results include enumerating and characterizing the Latin Squares which avoid patterns of length three and a generalization of the Erdős-Szekeres theorem. We also discuss equivalence classes among longer patterns, and conclude by describing open questions of interest both in light of pattern avoidance and their potential to reveal information about the structure of Latin Squares. Along the way, we show that classical resul…
We generalize the notion of orthogonal latin squares to colorings of simple graphs. Two $n$-colorings of a graph are said to be \emph{orthogonal} if whenever two vertices share a color in one coloring they have distinct colors in the other coloring. We show that the usual bounds on the maximum size of a certain set of orthogonal latin structures such as latin squares, row latin squares, equi-$n$ squares, single diagonal latin squares, double diagonal latin squares, or sudoku squares are a special cases of bounds on orthogonal colorings of graphs.
It is proved that for any prescribed orientation of the triples of either a Steiner triple system or a Latin square of odd order, there exists an embedding in an orientable surface with the triples forming triangular faces and one extra large face.
The existence of non-classical orthogonal quantum Latin squares
针对量子拉丁方的特殊正交结构，本文提出幂等、自正交和带洞等概念及PBD、填洞构造，证明了多类非经典MOQLS(v)与SOQLS(v)除少数例外值外均存在。
Similar to how standard Young tableaux represent paths in the Young lattice, Latin rectangles may be use to enumerate paths in the poset of semi-magic squares with entries zero or one. The symmetries associated to determinant preserve this poset, and we completely describe the orbits, covering data, and maximal chains for squares of size 4, 5, and 6. The last item gives the number of Latin squares in these cases. To calculate efficiently for size 6, we in turn identify orbits with certain equivalence classes of hypergraphs.
The cycle structure of a Latin square autotopism $Θ=(α,β,γ)$ is the triple $(\mathbf{l}_α,\mathbf{l}_β,\mathbf{l}_γ)$, where $\mathbf{l}_δ$ is the cycle structure of $δ$, for all $δ\in\{α,β,γ\}$. In this paper we study some properties of these cycle structures and, as a consequence, we give a classification of all autotopisms of the Latin squares of order up to 11.
A critical set in an $n \times n$ array is a set $C$ of given entries, such that there exists a unique extension of $C$ to an $n\times n$ Latin square and no proper subset of $C$ has this property. For a Latin square $L$, $\scs{L}$ denotes the size of the smallest critical set of $L$, and $\scs{n}$ is the minimum of $\scs{L}$ over all Latin squares $L$ of order $n$. We find an upper bound for the number of partial Latin squares of size $k$ and prove that $$n^2-(e+o(1))n^{10/6} \le \max \scs{L} \le n^2-\frac{\sqrtπ}{2}n^{9/6}.$$ % This improves a result of N. Cavenagh (Ph.D. thesis, The Univers…
Orthogonal Quantum Latin Squares with Maximal Cardinality from Full-Rank Factorizations
本文利用有限阿贝尔群的特征与满秩分解及乘积构造，证明了两个无限阶族存在具有最大基数的正交量子拉丁方对。
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