Related papers: On the number of even latin squares of even order

A Latin square of order $n$ is an $n\times n$ matrix in which each row and column contains each of $n$ symbols exactly once. For $\epsilon>0$, we show that with high probability a uniformly random Latin square of order $n$ has no proper…

Two Latin squares of order $n$ are $r$-orthogonal if, when superimposed, there are exactly $r$ distinct ordered pairs. The spectrum of all values of $r$ for Latin squares of order $n$ is known. A Latin square $A$ of order $n$ is…

Every Latin square has three attributes that can be even or odd, but any two of these attributes determines the third. Hence the parity of a Latin square has an information content of 2 bits. We extend the definition of parity from Latin…

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 $k$-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 $n$…

We develop a limit theory of Latin squares, paralleling the recent limit theories of dense graphs and permutations. We introduce a notion of density, an appropriate version of the cut distance, and a space of limit objects - so-called…

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…

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.…

We report the results of a computer investigation of sets of mutually orthogonal latin squares (MOLS) of small order. For $n\le9$ we 1. Determine the number of orthogonal mates for each species of latin square of order $n$. 2. Calculate the…

The chromatic number of a cyclic Latin square of order 2n is 2n+2. The available proof for this statement includes a coloring that is rather lengthy. Here, we introduce a coloring of cyclic Latin square of even order 2n (the Latin square of…

In 1782, Euler conjectured that no Latin square of order $n\equiv 2\; \textrm{mod}\; 4$ has a decomposition into transversals. While confirmed for $n=6$ by Tarry in 1900, Bose, Parker, and Shrikhande constructed counterexamples in 1960 for…

Let $L$ be an order-$n$ Latin square. For $X, Y, Z \subseteq \{1, ... ,n\}$, let $L(X, Y. Z)$ be the number of triples $i\in X, j\in Y, k\in Z$ such that $L(i,j) = k$. We conjecture that asymptotically almost every Latin square satisfies…

We construct sequencings for many groups that are a semi-direct product of an odd-order abelian group and a cyclic group of odd prime order. It follows from these constructions that there is a group-based complete Latin square of order $n$…

Symmetries of a partial Latin square are determined by its autotopism group. Analogously to the case of Latin squares, given an isotopism $\Theta$, the cardinality of the set $\mathcal{PLS}_{\Theta}$ of partial Latin squares which are…

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 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…

A Latin square of order $n$ with symbols $a_1,\ldots,a_n$ can be considered as a multiplication table for binary operation in the set $A=\{a_1,\ldots,a_n\}$. We prove that, if this operation is associative, then $A$ is a group.

For $\mu$ given latin squares of order $n$, they have {\sf $k$ intersection} when they have $k$ identical cells and $n^2-k$ cells with mutually different entries. For each $n\geq 1$ the set of integers $k$ such that there exist $\mu$ latin…

This work discusses the Alon-Tarsi number of line graphs and total graphs. In addition, we also discuss the Alon-Tarsi number of some Erdos-Faber-Lovasz (EFL) graphs.

There exists a bijection between the set of Latin squares of order $n$ and the set of feasible solutions of the 3-dimensional planar assignment problem ($3PAP_n$). In this paper, we prove that, given a Latin square isotopism $\Theta$, we…