Latin squares with restricted transversals

We prove that for all odd m ≥ 3 there exists a latin square of order 3 m that contains an ( m ? 1 ) × m latin subrectangle consisting of entries not in any transversal. We prove that for all even n ≥ 10 there exists a latin square of order n in which there is at least one transversal, but all transversals coincide on a single entry. A corollary is a new proof of the existence of a latin square without an orthogonal mate, for all odd orders n ≥ 11 . Finally, we report on an extensive computational study of transversal‐free entries and sets of disjoint transversals in the latin squares of order n ? 9 . In particular, we count the number of species of each order that possess an orthogonal mate. © 2011 Wiley Periodicals, Inc. J Combin Designs 20:124‐141, 2012     

Latin Squares with Restricted Transversals

The original article to which this erratum refers was correctly published online on 1 December 2011. Due to an error at the publisher, it was then published in Journal of Combinatorial Designs 20: 124–141, 2012 without the required shading in several examples. To correct this, the article is here reprinted in full. The publisher regrets this error. We prove that for all odd there exists a latin square of order 3m that contains an latin subrectangle consisting of entries not in any transversal. We prove that for all even there exists a latin square of order n in which there is at least one transversal, but all transversals coincide on a single entry. A corollary is a new proof of the existence of a latin square without an orthogonal mate, for all odd orders . Finally, we report on an extensive computational study of transversal‐free entries and sets of disjoint transversals in the latin squares of order . In particular, we count the number of species of each order that possess an orthogonal mate. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 344–361, 2012     

On the number of transversals in Cayley tables of cyclic groups

It is well known that if n is even, the addition table for the integers modulo n (which we denote by B_n) possesses no transversals. We show that if n is odd, then the number of transversals in B_n is at least exponential in n. Equivalently, for odd n, the number of diagonally cyclic latin squares of order n, the number of complete mappings or orthomorphisms of the cyclic group of order n, the number of magic juggling sequences of period n and the number of placements of n non-attacking semi-queens on an n×n toroidal chessboard are at least exponential in n. For all large n we show that there is a latin square of order n with at least (3.246)ⁿ transversals.We diagnose all possible sizes for the intersection of two transversals in B_n and use this result to complete the spectrum of possible sizes of homogeneous latin bitrades.We also briefly explore potential applications of our results in constructing random mutually orthogonal latin squares.     

Bachelor latin squares with large indivisible plexes

In a latin square of order n , a k ‐plex is a selection of kn entries in which each row, column, and symbol occurs k times. A 1 ‐plex is also called a transversal. A k ‐plex is indivisible if it contains no c ‐plex for 0 < c < k . We prove that, for all n ≥ 4 , there exists a latin square of order n that can be partitioned into an indivisible ? n / 2 ?‐plex and a disjoint indivisible ? n / 2 ?‐plex. For all n ≥ 3 , we prove that there exists a latin square of order n with two disjoint indivisible ? n / 2 ?‐plexes. We also give a short new proof that, for all odd n ≥ 5 , there exists a latin square of order n with at least one entry not in any transversal. Such latin squares have no orthogonal mate. Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 19:304‐312, 2011     

A superlinear lower bound for the size of a critical set in a latin square

A critical set is a partial latin square that has a unique completion to a latin square, and is minimal with respect to this property. Let scs(n) denote the smallest possible size of a critical set in a latin square of order n. We show that for all n, . Thus scs(n) is superlinear with respect to n. We also show that scs(n) ≥ 2n?32 and if n ≥ 25, . © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 269–282, 2007     

On the spectrum of critical sets in latin squares of order 2n

Suppose that L is a latin square of order m and P ? L is a partial latin square. If L is the only latin square of order m which contains P, and no proper subset of P has this property, then P is a critical set of L. The critical set spectrum problem is to determine, for a given m, the set of integers t for which there exists a latin square of order m with a critical set of size t. We outline a partial solution to the critical set spectrum problem for latin squares of order 2ⁿ. The back circulant latin square of even order m has a well‐known critical set of size m²/4, and this is the smallest known critical set for a latin square of order m. The abelian 2‐group of order 2ⁿ has a critical set of size 4ⁿ‐3ⁿ, and this is the largest known critical set for a latin square of order 2ⁿ. We construct a set of latin squares with associated critical sets which are intermediate between the back circulant latin square of order 2ⁿ and the abelian 2‐group of order 2ⁿ. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 25–43, 2008     

Transversals in completely reducible multiary quasigroups and in multiary quasigroups of order 4

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 completely reducible if it can be decomposed into binary quasigroups.In this paper we investigate transversals in reducible quasigroups and in quasigroups of order 4. We find a lower bound on the number of transversals for a vast class of completely reducible quasigroups. Next we prove that, except for the iterated group ${\mathbb{Z}}_4$ of even arity, every $n$-ary quasigroup of order 4 has a transversal. Also we obtain a lower bound on the number of transversals in quasigroups of order 4 and odd arity and count transversals in the iterated group ${\mathbb{Z}}_4$ of odd arity and in the iterated group ${\mathbb{Z}}_2^2.$All results of this paper can be regarded as those concerning latin hypercubes.     

Row-complete latin squares of every composite order exist

A construction for a row-complete latin square of order n, where n is any odd composite number other than 9, is given in this article. Since row-complete latin squares of order 9 and of even order have previously been constructed, this proves that row-complete latin squares of every composite order exist. © 1998 John Wiley & Sons, Inc. J Combin Designs 6:63–77, 1998     

Multidimensional Permanents and an Upper Bound on the Number of Transversals in Latin Squares

The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. A nonnegative matrix whose every 1‐dimensional plane sums to 1 is called polystochastic. A latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and each column. A transversal of such a square is a set of n entries such that no two entries share the same row, column, or symbol. Let T(n) be the maximum number of transversals over all latin squares of order n. Here, we prove that over the set of multidimensional polystochastic matrices of order n the permanent has a local extremum at the uniform matrix for whose every entry is equal to . Also, we obtain an asymptotic value of the maximal permanent for a certain set of nonnegative multidimensional matrices. In particular, we get that the maximal permanent of polystochastic matrices is asymptotically equal to the permanent of the uniform matrix, whence as a corollary we have an upper bound on the number of transversals in latin squares      

Symmetries that latin squares inherit from 1‐factorizations

A 1‐factorization of a graph is a decomposition of the graph into edge disjoint perfect matchings. There is a well‐known method, which we call the ??‐construction, for building a 1‐factorization of K_n,n from a 1‐factorization of K_{n + 1}. The 1‐factorization of K_n,n can be written as a latin square of order n. The ??‐construction has been used, among other things, to make perfect 1‐factorizations, subsquare‐free latin squares, and atomic latin squares. This paper studies the relationship between the factorizations involved in the ??‐construction. In particular, we show how symmetries (automorphisms) of the starting factorization are inherited as symmetries by the end product, either as automorphisms of the factorization or as autotopies of the latin square. Suppose that the ??‐construction produces a latin square L from a 1‐factorization F of K_{n + 1}. We show that the main class of L determines the isomorphism class of F, although the converse is false. We also prove a number of restrictions on the symmetries (autotopies and paratopies) which L may possess, many of which are simple consequences of the fact that L must be symmetric (in the usual matrix sense) and idempotent. In some circumstances, these restrictions are tight enough to ensure that L has trivial autotopy group. Finally, we give a cubic time algorithm for deciding whether a main class of latin squares contains any square derived from the ??‐construction. The algorithm also detects symmetric squares and totally symmetric squares (latin squares that equal their six conjugates). © 2005 Wiley Periodicals, Inc. J Combin Designs 13: 157–172, 2005.     

The cycle structure of two rows in a random Latin square

Let L be chosen uniformly at random from among the latin squares of order n ≥ 4 and let r,s be arbitrary distinct rows of L. We study the distribution of σ_r,s, the permutation of the symbols of L which maps r to s. We show that for any constant c > 0, the following events hold with probability 1 ‐ o(1) as n → ∞: (i) σ_r,s has more than (log n)^1?c cycles, (ii) σ_r,s has fewer than 9 cycles, (iii) L has fewer than n^5/2 intercalates (latin subsquares of order 2). We also show that the probability that σ_r,s is an even permutation lies in an interval and the probability that it has a single cycle lies in [2n^‐2,2n^‐2/3]. Indeed, we show that almost all derangements have similar probability (within a factor of n^3/2) of occurring as σ_r,s as they do if chosen uniformly at random from among all derangements of {1,2,…,n}. We conjecture that σ_r,s shares the asymptotic distribution of a random derangement. Finally, we give computational data on the cycle structure of latin squares of orders n ≤ 11. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Construction of doubly diagonalized orthogonal latin squares

A transversal T of a latin square is a collection of n cells no two in the same row or column and such that each of the integers 1, 2, …, n appears in exactly one of the cells of T. A latin square is doubly diagonalized provided that both its main diagonal and off-diagonal are transversals. Although it is known that a doubly diagonalized latin square of every order n ≥ 4 exists and that a pair of orthogonal latin squares of order n exists for every n ≠ 2 or 6, it is still an open question as to what the spectrum is for pairs of doubly diagonalized orthogonal latin squares. The best general result seems to be that pairs of orthogonal doubly diagonalized latin squares of order n exist whenever n is odd or a multiple of 4, except possibly when n is a multiple of 3 but not of 9. In this paper we give a new construction for doubly diagonalized latin squares which is used to enlarge the known class for doubly diagonalized orthogonal squares. The construction is based on Sade's singular direct product of quasigroups.     

The number of transversals in a Latin square

A Latin Square of order n is an n × n array of n symbols, in which each symbol occurs exactly once in each row and column. A transversal is a set of n entries, one selected from each row and each column of a Latin Square of order n such that no two entries contain the same symbol. Define T(n) to be the maximum number of transversals over all Latin squares of order n. We show that for n ≥ 5, where b ≈ 1.719 and c ≈ 0.614. A corollary of this result is an upper bound on the number of placements of n non-attacking queens on an n × n toroidal chess board. Some divisibility properties of the number of transversals in Latin squares based on finite groups are established. We also provide data from a computer enumeration of transversals in all Latin Squares of order at most 9, all groups of order at most 23 and all possible turn-squares of order 14.     

The Order of Automorphisms of Quasigroups

We prove quadratic upper bounds on the order of any autotopism of a quasigroup or Latin square, and hence also on the order of any automorphism of a Steiner triple system or 1‐factorization of a complete graph. A corollary is that a permutation σ chosen uniformly at random from the symmetric group will almost surely not be an automorphism of a Steiner triple system of order n, a quasigroup of order n or a 1‐factorization of the complete graph . Nor will σ be one component of an autotopism for any Latin square of order n. For groups of order n it is known that automorphisms must have order less than n, but we show that quasigroups of order n can have automorphisms of order greater than n. The smallest such quasigroup has order 7034. We also show that quasigroups of prime order can possess autotopisms that consist of three permutations with different cycle structures. Our results answer three questions originally posed by D.  Stones.     

Latin Squares without Orthogonal Mates

In 1779 Euler proved that for every even n there exists a latin square of order n that has no orthogonal mate, and in 1944 Mann proved that for every n of the form 4k + 1, k ≥ 1, there exists a latin square of order n that has no orthogonal mate. Except for the two smallest cases, n = 3 and n = 7, it is not known whether a latin square of order n = 4k + 3 with no orthogonal mate exists or not. We complete the determination of all n for which there exists a mate-less latin square of order n by proving that, with the exception of n = 3, for all n = 4k + 3 there exists a latin square of order n with no orthogonal mate. We will also show how the methods used in this paper can be applied more generally by deriving several earlier non-orthogonality results.     

The recognition of symmetric latin squares

A latin square S is isotopic to another latin square S′ if S′ can be obtained from S by permuting the row indices, the column indices and the symbols in S. Because the three permutations used above may all be different, a latin square which is isotopic to a symmetric latin square need not be symmetric. We call the problem of determining whether a latin square is isotopic to a symmetric latin square the symmetry recognition problem. It is the purpose of this article to give a solution to this problem. For this purpose we will introduce a cocycle corresponding to a latin square which transforms very simply under isotopy, and we show this cocycle contains all the information needed to determine whether a latin square is isotopic to a symmetric latin square. Our results relate to 1‐factorizations of the complete graph on n + 1 vertices, K_{n + 1}. There is a well known construction which can be used to make an n × n latin square from a 1‐factorization on n + 1 vertices. The symmetric idempotent latin squares are exactly the latin squares that result from this construction. The idempotent recognition problem is simple for symmetric latin squares, so our results enable us to recognize exactly which latin squares arise from 1‐factorizations of K_{n + 1}. As an example we show that the patterned starter 1‐factorization for the group G gives rise to a latin square which is in the main class of the Cayley latin square for G if and only if G is abelian. Hence, every non‐abelian group gives rise to two latin squares in different main classes. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 291–300, 2008     

The Existence of Latin Squares without Orthogonal Mates

A latin square is a bachelor square if it does not possess an orthogonal mate; equivalently, it does not have a decomposition into disjoint transversals. We define a latin square to be a confirmed bachelor square if it contains an entry through which there is no transversal. We prove the existence of confirmed bachelor squares for all orders greater than three. This resolves the existence question for bachelor squares.     

Line Transversals to Disjoint Balls

We prove that the set of directions of lines intersecting three disjoint balls in ℝ³ in a given order is a strictly convex subset of . We then generalize this result to n disjoint balls in ℝ ^d . As a consequence, we can improve upon several old and new results on line transversals to disjoint balls in arbitrary dimension, such as bounds on the number of connected components and Helly-type theorems.     

Latin squares with no small odd plexes

A k‐plex in a Latin square of order n is a selection of kn entries in which each row, column, and symbol is represented precisely k times. A transversal of a Latin square corresponds to the case k = 1. We show that for all even n > 2 there exists a Latin square of order n which has no k‐plex for any odd but does have a k‐plex for every other . © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 477–492, 2008     

Symmetric latin square and complete graph analogues of the evans conjecture

With the proof of the Evans conjecture, it was established that any partial latin square of side n with a most n ? 1 nonempty cells can be completed to a latin square of side n. In this article we prove an analogous result for symmetric latin squares: a partial symmetric latin square of side n with an admissible diagonal and at most n ? 1 nonempty cells can be completed to a symmetric latin square of side n. We also characterize those partial symmetric latin squares of side n with exactly n or n + 1 nonempty cells which cannot be completed. From these results we deduce theorems about completing edge-colorings of complete graphs K_2m and K_{2m ? 1} with 2m ? 1 colors, with m + 1 or fewer edges getting prescribed colors. © 1994 John Wiley & Sons, Inc.