We define a cover of a Latin square to be a set of entries that includes at least one representative of each row, column and symbol. A cover is minimal if it does not contain any smaller cover. A partial transversal is a set of entries that includes at most one representative of each row, column and symbol. A partial transversal is maximal if it is not contained in any larger partial transversal. We explore the relationship between covers and partial transversals.
We prove the following: (1) The minimum size of a cover in a Latin square of order $n$ is $n+a$ if and only if the maximum size of a partial transversal is either $n-2a$ or $n-2a+1$. (2) A minimal cover in a Latin square of order $n$ has size at most $μ_n=3(n+1/2-\sqrt{n+1/4})$. (3) There are infinitely many orders $n$ for which there exists a Latin square having a minimal cover of every size from $n$ to $μ_n$. (4) Every Latin square of order $n$ has a minimal cover of a size which is asymptotically equal to $μ_n$. (5) If $1\le k\le n/2$ and $n\ge5$ then there is a Latin square of order $n$ with a maximal partial transversal of size $n-k$. (6) For any $ε>0$, asymptotically almost all Latin squares have no maximal partial transversal of size less than $n-n^{2/3+ε}$.
An array is row-Latin if no symbol is repeated within any row. An array is Latin if it and its transpose are both row-Latin. A transversal in an $n\times n$ array is a selection of $n$ different symbols from different rows and different columns. We prove that every $n \times n$ Latin array containing at least $(2-\sqrt{2}) n^2$ distinct symbols has a transversal. Also, every $n \times n$ row-Latin array containing at least $\frac14(5-\sqrt{5})n^2$ distinct symbols has a transversal. Finally, we show by computation that every Latin array of order $7$ has a transversal, and we describe all small…
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…
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…
A $k$-plex in a latin square of order $n$ is a selection of $kn$ entries that includes $k$ representatives from each row and column and $k$ occurrences of each symbol. A $1$-plex is also known as a transversal. It is well known that if $n$ is even then $B_n$, the addition table for the integers modulo $n$, possesses no transversals. We show that there are a great many latin squares that are similar to $B_n$ and have no transversal. As a consequence, the number of species of transversal-free latin squares is shown to be at least $n^{n^{3/2}(1/2-o(1))}$ for even $n\rightarrow\infty$. We also pro…
A Latin array is a matrix of symbols in which no symbol occurs more than once within a row or within a column. A diagonal of an $n\times n$ array is a selection of $n$ cells taken from different rows and columns of the array. The weight of a diagonal is the number of different symbols on it. We show via computation that every Latin array of order $n\le11$ has a diagonal of weight at least $n-1$. A corollary is the existence of near transversals in Latin squares of these orders. More generally, for all $k\le20$ we compute a lower bound on the order of any Latin array that does not have a diagon…
We are seeking a sufficient condition that forces a transversal in a generalized Latin square. A generalized Latin square of order $n$ is equivalent to a proper edge-coloring of $K_{n,n}$. A transversal corresponds to a multicolored perfect matching. Akbari and Alipour defined $l(n)$ as the least integer such that every properly edge-colored $K_{n,n}$, which contains at least $l(n)$ different colors, admits a multicolored perfect matching. They conjectured that $l(n)\leq n^2/2$ if $n$ is large enough. In this note we prove that $l(n)$ is bounded from above by $0.75n^2$ if $n>1$. We point out a…
A partial transversal $T$ of a Latin square $L$ is a set of entries of $L$ in which each row, column and symbol is represented at most once. A partial transversal is maximal if it is not contained in a larger partial transversal. Any maximal partial transversal of a Latin square of order $n$ has size at least $\lceil\frac{n}{2}\rceil$ and at most $n$. We say that a Latin square is omniversal if it possesses a maximal partial transversal of all feasible sizes and is near-omniversal if it possesses a maximal partial transversal of all feasible sizes except one. Evans showed that omniversal Latin…
Embedding a Latin square with transversal into a projective space
一个n边的拉丁方自然地定义了一个3 n个点上的有限几何，其中有三条长度为n的直线和n^2条长度为3的直线。一个n边的拉丁方有一个截线，类似地在3 n +1个点上定义了一个有限几何，其中有三条长度为n的线，n^2-n条长度为3的线，n条长度为4的共点线。由k个相互正交拉丁方组成的集合定义了一个在kn个点上的几何，其中k条线的长度为n，n^2条线的长度为k。Bruen和Colbourn（J. Combin.日Ser.A92（2000），88-94），我们将这些有限几何嵌入到除环上的射影空间中。
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.
We introduce a notion of parity for transversals, and use it to show that in Latin squares of order $2 \bmod 4$, the number of transversals is a multiple of 4. We also demonstrate a number of relationships (mostly congruences modulo 4) involving $E_1,\dots, E_n$, where $E_i$ is the number of diagonals of a given Latin square that contain exactly $i$ different symbols. Let $A(i\mid j)$ denote the matrix obtained by deleting row $i$ and column $j$ from a parent matrix $A$. Define $t_{ij}$ to be the number of transversals in $L(i\mid j)$, for some fixed Latin square $L$. We show that $t_{ab}\equi…
An $n$-ary quasigroup $f$ of order $q$ is an $n$-ary operation over a set of cardinality $q$ such that the Cayley table of the operation is an $n$-dimensional latin hypercube of order $q$. A transversal in a quasigroup $f$ (or in the corresponding latin hypercube) is a collection of $q$ $(n+1)$-tuples from the Cayley table of $f$, each pair of tuples differing at each position. The problem of transversals in latin hypercubes was posed by Wanless in 2011. An $n$-ary quasigroup $f$ is called reducible if it can be obtained as a composition of two quasigroups whose arity is at least 2, and it is …
Given a binary quasigroup $G$ of order $n$, a $d$-iterated quasigroup $G[d]$ is the $(d+1)$-ary quasigroup equal to the $d$-times composition of $G$ with itself. The Cayley table of every $d$-ary quasigroup is a $d$-dimensional latin hypercube. Transversals and diagonals in multiary quasigroups are defined so as to coincide with those in the corresponding latin hypercube. We prove that if a group $G$ of order $n$ satisfies the Hall--Paige condition, then the number of transversals in $G[d]$ is equal to $ \frac{n!}{ |G'| n^{n-1}} \cdot n!^{d} (1 + o(1))$ for large $d$, where $G'$ is the commuta…
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.
A $d$-ary quasigroup of order $n$ is a $d$-ary operation over a set of cardinality $n$ such that the Cayley table of the operation is a $d$-dimensional latin hypercube of the same order. Given a binary quasigroup $G$, the $d$-iterated quasigroup $G^{\left[d\right]}$ is a $d$-ary quasigroup that is a $d$-time composition of $G$ with itself. A $k$-multiplex (a $k$-plex) $K$ in a $d$-dimensional latin hypercube $Q$ of order $n$ or in the corresponding $d$-ary quasigroup is a multiset (a set) of $kn$ entries such that each hyperplane and each symbol of $Q$ is covered by exactly $k$ elements of $K$…
A Latin square of order $n$ is an $n \times n$ array filled with $n$ symbols such that each symbol appears only once in every row or column and a transversal is a collection of cells which do not share the same row, column or symbol. The study of Latin squares goes back more than 200 years to the work of Euler. One of the most famous open problems in this area is a conjecture of Ryser-Brualdi-Stein from 60s which says that every Latin square of order $n\times n$ contains a transversal of order $n-1$. In this paper we prove the existence of a transversal of order $n-O(\log{n}/\log{\log{n}})$, i…
Akbari and Alipour conjectured that any Latin array of order $n$ with at least $n^2/2$ symbols contains a transversal. We confirm this conjecture for large $n$, and moreover, we show that $n^{399/200}$ symbols suffice.
The chromatic number of a latin square $L$, denoted $χ(L)$, is the minimum number of partial transversals needed to cover all of its cells. It has been conjectured that every latin square satisfies $χ(L) \leq |L|+2$. If true, this would resolve a longstanding conjecture---commonly attributed to Brualdi---that every latin square has a partial transversal of size $|L|-1$. Restricting our attention to Cayley tables of finite groups, we prove two main results. First, we resolve the chromatic number question for Cayley tables of finite Abelian groups: the Cayley table of an Abelian group $G$ has ch…
We prove several structural properties of Steiner triple systems (STS) of order 3w+3 that include one or more transversal subdesigns TD(3,w). Using an exhaustive search, we find that there are 2004720 isomorphism classes of STS(21) including a subdesign TD(3,6), or, equivalently, a 6-by-6 latin square.
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…
Let $B_p$ be the Latin square given by the addition table for the integers modulo an odd prime $p$. Here we consider the properties of Latin trades in $B_p$ which preserve orthogonality with one of the $p-1$ MOLS given by the finite field construction. We show that for certain choices of the orthogonal mate, there is a lower bound logarithmic in $p$ for the number of times each symbol occurs in such a trade, with an overall lower bound of $(\log{p})^2/\log\log{p}$ for the size of such a trade. Such trades imply the existence of orthomorphisms of the cyclic group which differ from a linear orth…
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.
Myrvold's Results on Orthogonal Triples of $10 \times 10$ Latin Squares: A SAT Investigation
School of Computer Science, University of Waterloo（滑铁卢大学计算机学院）；Department of Mathematics and Statistics, University of Windsor（温莎大学数学与统计系）；School of Mathematics and Statistics, Carleton University（卡尔顿大学数学与统计学院）
We explore the interplay between algebraic combinatorics and algorithmic problems in graph theory by defining a polynomial with connections to correspondence colouring (also known as DP-colouring), a recent generalization of list-colouring, and the Unique Games Conjecture. Like the chromatic polynomial of a graph, we are able to evaluate this polynomial at a point, despite the complexity of computing this polynomial. We construct a cover of a graph $X$ by blowing up each vertex to a set of $r$ vertices and joining each pair of sets corresponding to adjacent vertices by a matching with $r$ edge…
Mathematics Institute, University of Warwick（华威大学数学研究所）；Department of Computer Science, ETH Zürich（苏黎世联邦理工学院计算机科学系）；Department of Mathematics, ETH Zürich（苏黎世联邦理工学院数学系）
本文肯定回答等 n 方阵是否总有近似完全横截的问题，同时改进存在性反例界与普适的近乎完全互不相交横截界。
In a hypergraph with vertex set $V$ and edge set $E$, a real-valued function $f: V \to [0, 1]$ is a fractional transversal if $\sum_{v\in e} f(v) \ge 1$ for every edge $e \in E$. Its size is $|f| := \sum_{v \in V} f(v)$, and the fractional transversal number is the smallest possible $|f|$. We consider a game scenario where two players with opposite goals construct a fractional transversal incrementally, trying to minimize and maximize $|f|$, respectively. We prove that both players have strategies to achieve their common optimum, and they can reach their goals using rational weights.
A Bruhat order for Latin squares and alternating sign hypermatrices
School of Mathematical and Statistical Sciences, University of Galway（高威大学数学与统计科学学院）；Department of Mathematics and Computer Studies, Mary Immaculate College（玛丽不可瑕疵学院数学与计算机研究系）
Let $vc(G)$, $fvs(G)$ and $oct(G)$, respectively, denote the size of a minimum vertex cover, minimum feedback vertex set and minimum odd cycle transversal in a graph $G$. One can ask, when looking for these sets in a graph, how much bigger might they be if we require that they are independent; that is, what is the price of independence? If $G$ has a vertex cover, feedback vertex set or odd cycle transversal that is an independent set, then we let $ivc(G)$, $ifvs(G)$ or $ioct(G)$, respectively, denote the minimum size of such a set. Similar to a recent study on the price of connectivity (Hartin…
