Related papers: Computing random $r$-orthogonal Latin squares

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…

Given two integers $m$ and $n$ with $m\leq n$, a Latin rectangle of size $m\times n$ is a bi-dimensional array with $m$ rows and $n$ columns filled with symbols from an alphabet with $n$ symbols, such that each row contains a permutation of…

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…

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…

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…

Let $L(n)$ be the number of Latin squares of order $n$, and let $L^{\textrm{even}}(n)$ and $L^{\textrm{odd}}(n)$ be the number of even and odd such squares, so that $L(n) = L^{\textrm{even}}(n) + L^{\textrm{odd}}(n)$. The Alon-Tarsi…

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…

We use a greedy probabilistic method to prove that for every $\epsilon > 0$, every $m\times n$ Latin rectangle on $n$ symbols has an orthogonal mate, where $m=(1-\epsilon)n$. That is, we show the existence of a second Latin rectangle such…

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)…

The logarithm of the maximum number of transversals over all latin squares of order $n$ is greater than $\frac{n}{6}(\ln n+ O(1))$.

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.

There exist pairs of orthogonal Latin squares of any order n except if n=2 or n=6 [Bose, Shrikhande and Parker, 1960]. In particular, the problem of Euler's thirty-six officers does not have a solution. However, it has a "quantum solution":…

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…

Ever since E. T. Parker constructed an orthogonal pair of $10\times10$ Latin squares in 1959, an orthogonal triple of $10\times10$ Latin squares has been one of the most sought-after combinatorial designs. Despite extensive work, the…

It is known that no-boundary Cellular Automata (CA) defined by bipermutive local rules give rise to Latin squares. In this paper, we study under which conditions the Latin square generated by a bipermutive CA is self-orthogonal, i.e.…

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…

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…

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…

Latin squares are $n\times n$ matrices containing $n$ symbols, where each symbol appears exactly once in each row and column. They were studied by Euler, later popularized through Sudoku, and remain a rich source of difficult combinatorial…

Latin squares are well studied combinatorial objects. In this paper we generalize the concept and propose new objects like Latin triangles, free Latin squares, Latin tetrahedra, free Latin cubes, etc. We start with a classic definition of…