Latin Squares with a Unique Intercalate

The paper is devoted to discovering new features of diagonal Latin squares of small order. We present an algorithm, based on a special kind of transformations, that constructs a canonical form of a given diagonal Latin square. Each canonical form corresponds to one isotopy class of diagonal Latin squares. The algorithm was implemented and used to enumerate the isotopy classes of diagonal Latin squares of order at most 8. For order 8 the computational experiment was conducted in a volunteer computing project. The algorithm was also used to estimate how long it would take to enumerate the isotopy classes of diagonal Latin squares of order 9 in the same volunteer computing project.

Embedding partial Latin squares in Latin squares with many mutually orthogonal mates

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

Indivisible partitions of latin squares

The maximum number of mutually orthogonal Sudoku Latin squares (MOSLS) of order $$n=m^2$$n=m2 is $$n-m$$n-m. In this paper, we construct for $$n=q^2$$n=q2, q a prime power, a set of $$q^2-q-1$$q2-q-1 MOSLS of order $$q^2$$q2 that cannot be extended to a set of $$q^2-q$$q2-q MOSLS. This contrasts to the theory of ordinary Latin squares of order n, where each set of $$n-2$$n-2 mutually orthogonal Latin Squares (MOLS) can be extended to a set of $$n-1$$n-1 MOLS (which is best possible). For this proof, we construct a particular maximal partial spread of size $$q^2-q+1$$q2-q+1 in $$\mathrm {PG}(3,q)$$PG(3,q) and use a connection between Sudoku Latin squares and projective geometry, established by Bailey, Cameron and Connelly.

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 nthat 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 Latin square has six conjugate Latin squares obtained by uniformly permuting its (row, column, symbol) triples. We say that a Latin square has conjugate symmetry if at least two of its six conjugates are equal. We enumerate Latin squares with conjugate symmetry and classify them according to several common notions of equivalence. We also do similar enumerations under additional hypotheses, such as assuming the Latin square is reduced, diagonal, idempotent or unipotent. Our data corrected an error in earlier literature and suggested several patterns that we then found proofs for, including (1) the number of isomorphism classes of semisymmetric idempotent Latin squares of order equals the number of isomorphism classes of semisymmetric unipotent Latin squares of order , and (2) suppose and are totally symmetric Latin squares of order . If and are paratopic then and are isomorphic.

Diagonally cyclic latin squares

An orthogonal latin square graph (OLSG) is one in which the vertices are latin squares of the same order and on the same symbols, and two vertices are adjacent if and only if the latin squares are orthogonal. If G is an arbitrary finite graph, we say that G is realizable as an OLSG if there is an OLSG isomorphic to G. The spectrum of G [Spec(G)] is defined as the set of all integers n that there is a realization of G by latin squares of order n. The two basic theorems proved here are (1) every graph is realizable and (2) for any graph G, Spec G contains all but a finite set of integers. A number of examples are given that point to a number of wide open questions. An example of such a question is how to classify the graphs for which a given n lies in the spectrum.

ABSTRACTA Latin square of order is an matrix in which each row and column contains each of symbols exactly once. For , we show that with high probability a uniformly random Latin square of order has no proper subsquare of order larger than . Using this fact, we present a canonical labeling algorithm for Latin squares of order that runs in average time bounded by a polynomial in . The algorithm can be used to solve isomorphism problems for many combinatorial objects that can be encoded using Latin squares, including quasigroups, Steiner triple systems, Mendelsohn triple systems, 1‐factorizations, nets, affine planes, and projective planes.

In this paper we consider the approach to search for systems of orthogonal Latin Squares, that is based on reducing the corresponding problems to Boolean Satisfiability problem. We constructed the SAT encoding for finding orthogonal diagonal Latin Squares of order 10. Using this encoding and the resources provided by the volunteer computing project SAT@home we managed to find 17 previously unknown pairs. Based on these 17 pairs and on 3 previously published pairs we constructed the pseudotriples of diagonal Latin Squares of order 10. To construct pseudotriples we employed the computing cluster. The last step required us to make parallel implementation of the algorithm for generating diagonal Latin Squares of order 10.

Paratopism is a well‐known action of the wreath product on Latin squares of order n. A paratopism that maps a Latin square to itself is an autoparatopism of that Latin square. Let Par(n) denote the set of paratopisms that are an autoparatopism of at least one Latin square of order n. We prove a number of general properties of autoparatopisms. Applying these results, we determine Par(n) for . We also study the proportion of all paratopisms that are in Par(n) as .

This paper deals with the problem of finding sets of mutually orthogonal Latin squares of order 4t (where 4t – 1 is a prime power) based on orthogonal mappings of a group. For the group G we take the module G(2, 2t) whose elements are vectors (a 1, a 2,) where a 1 is a residue class (mod 2) and a 2 is a residue class (mod 2t), the addition being defined by (a 1, a 2) + (b 1, b 2) = (c 1, c 2) where a 1 + b 1 = C 1 (mod 2) and a 2 + b 2 = c 2 (mod 2t). Then the search for orthogonal mappings is materially simplified by using a configuration based on the balanced incomplete block design with parameters v = b = 4t – 1, r = k = 2t – 1, λ = t – 1. Using this method, two sets of five mutually orthogonal Latin squares of order 12 were obtained.

Large sets of resolvable idempotent Latin squares

This paper deals with the search for triples of partially orthogonal Latin squares of order 10. For every known pair of orthogonal Latin squares of order 10 we add a third diagonal Latin square in such a way that the orthogonality condition between it and squares from a considered pair coincides in the maximum possible number of cells. Two approaches are used: the first one is based on reducing an original problem to Boolean satisfiability problem; the second one is based on Brute force method. Several triples of the aforementioned kind with high characteristics were constructed. The experiments were held in the volunteer computing project SAT@home and on a computing cluster.