Related papers: Latin squares with no transversals

We prove that, for all even $n\geq10$, there exists a latin square of order $n$ with at least one transversal, yet all transversals coincide on $ \big\lfloor n/6 \big\rfloor$ entries. These latin squares have at least $ 19 n^2/36 + O(n)$…

A latin square of order $n$ is an $n\times n$ array of $n$ symbols in which each symbol occurs exactly once in each row and column. A transversal of such a square is a set of $n$ entries such that no two entries share the same row, column…

A $k$-plex of a latin square is a collection of cells representing each row, column, and symbol precisely $k$ times. The classic case of $k=1$ is more commonly known as a transversal. We introduce the concept of a $k$-weight, an integral…

A transversal in an $n \times n$ latin square is a collection of $n$ entries not repeating any row, column, or symbol. Kwan showed that almost every $n \times n$ latin square has $\bigl((1 + o(1)) n / e^2\bigr)^n$ transversals as $n \to…

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…

A Latin square is an $n$ by $n$ grid filled with $n$ symbols so that each symbol appears exactly once in each row and each column. A transversal in a Latin square is a collection of cells which do not share any row, column, or symbol. This…

We discuss the problem of existence of latin squares without a substructure consisting of six elements $(r_1,c_2,l_3)$, $(r_2,c_3,l_1)$, $(r_3,c_1,l_2)$, $(r_2,c_1,l_3)$, $(r_3,c_2,l_1)$, $(r_1,c_3,l_2)$. Equivalently, the corresponding…

Let m and n be integers, $2 \leq m \leq n$. An m by n array consists of mn cells, arranged in m rows and n columns, and each cell contains exactly one symbol. A transversal of an array consists of m cells, one from each row and no two from…

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…

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…

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 $k \times n$ partial Latin rectangle is \textit{$C$-sparse} if the number of nonempty entries in each row and column is at most $C$ and each symbol is used at most $C$ times. We prove that the probability a uniformly random $k \times n$…

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…

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…

A defining set of a Latin square is a partially filled-in Latin square which completes to no other Latin square of the same order. We introduce the concept of a $k$-strong defining set, in which if less than $k$ entries are deleted, the…

A Latin square of order $n$ is an $n$ by $n$ grid filled using $n$ symbols so that each symbol appears exactly once in each row and column. A transversal in a Latin square is a collection of cells which share no symbol, row or column. The…

A Latin square $L(n,k)$ is a square of order $n$ with its entries colored with $k$ colors so that all the entries in a row or column have different colors. Let $d(L(n,k))$ be the minimal number of colored entries of an $n \times n$ square…

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)…

A $d$-dimensional Latin hypercube of order $n$ is a $d$-dimensional array containing symbols from a set of cardinality $n$ with the property that every axis-parallel line contains all $n$ symbols exactly once. We show that for $(n, d)…

A $k$-plane of a $d$-dimensional array is a subarray formed by fixing $d-k$ coordinates and allowing the remaining $k$ coordinates to vary freely. A Latin hypercube of dimension $d$ and order $n$ is an $n\times n\times\cdots\times n$ array…