Latin bitrades derived from groups

A Latin bitrade is a pair of partial Latin squares which are disjoint, occupy the same set of non-empty cells, and whose corresponding rows and columns contain the same set of entries. In [A. Drápal, On geometrical structure and construction of Latin trades, Advances in Geometry (in press)] it is shown that a Latin bitrade may be thought of as three derangements of the same set, whose product is the identity and whose cycles pairwise have at most one point in common. By letting a group act on itself by right translation, we show how some Latin bitrades may be derived directly from groups. Properties of Latin bitrades such as homogeneity, minimality (via thinness) and orthogonality may also be encoded succinctly within the group structure. We apply the construction to some well-known groups, constructing previously unknown Latin bitrades. In particular, we show the existence of minimal, k-homogeneous Latin bitrades for each odd k≥3. In some cases these are the smallest known such examples.     

The theory and application of latin bitrades: A survey

A latin bitrade is a pair of partial latin squares which are disjoint, occupy the same set of non-empty cells, and whose corresponding rows and columns contain the same sets of symbols. This survey paper summarizes the theory of latin bitrades, detailing their applications to critical sets, random latin squares and existence constructions for latin squares.      

3-Homogeneous latin trades

Let T be a partial latin square and L be a latin square with TL. We say that T is a latin trade if there exists a partial latin square T^′ with T^′∩T= such that (LT)T^′ is a latin square. A k-homogeneous latin trade is one which intersects each row, each column and each entry either 0 or k times. In this paper, we construct 3-homogeneous latin trades from hexagonal packings of the plane with circles. We show that 3-homogeneous latin trades of size 3 m exist for each m3. This paper discusses existence results for latin trades and provides a glueing construction which is subsequently used to construct all latin trades of finite order greater than three.     

Planar Eulerian triangulations are equivalent to spherical Latin bitrades

Given a pair of Latin squares, we may remove from both squares those cells that contain the same symbol in corresponding positions. The resulting pair T={P₁,P₂} of partial Latin squares is called a Latin bitrade. The number of filled cells in P₁ is called the size of T. There are at least two natural ways to define the genus of a Latin bitrade; the bitrades of genus 0 are called spherical. We construct a simple bijection between the isomorphism classes of planar Eulerian triangulations on v vertices and the main classes of spherical Latin bitrades of size v−2. Since there exists a fast algorithm (due to Batagelj, Brinkmann and McKay) for generating planar Eulerian triangulations up to isomorphism, our result implies that also spherical Latin bitrades can be generated very efficiently.     

CONSTRUCTING SELF-CONJUGATE SELF-ORTHOGONAL DIAGONAL LATIN SQUARES

1.IntroductionALatinsquareofordernisannxnarraysuchthateveryrowandeverycolumnisapermutationofann-setN.AtransversalinaLatinsquareisasetofpositions,oneperrowandonepercolumn,amongwhichthesymbolsoccurpreciselyonceeach.AdiagonalLatinsquareisaLatinsquarewhosemaindiagonalandbackdiagonalarebothtransversals.TwoLatinsquaresofordernareorthogonalifeachsymbolinthefirstsquaremeetseachsymbolinthesecondsquareexactlyoncewhentheyaresuperposed.ALatinsquareisself-orthogonalifitisorthogonaltoitstranspose.Inanea…     

Latin bitrades,dissections of equilateral triangles,and abelian groups

Let T=(T^*, T^?) be a spherical latin bitrade. With each a=(a₁, a₂, a₃)∈T^* associate a set of linear equations Eq(T, a) of the form b₁+b₂=b₃, where b=(b₁, b₂, b₃) runs through T^*\{a}. Assume a₁=0=a₂ and a₃=1. Then Eq(T,a) has in rational numbers a unique solution $b_{i}=\bar{b}_{i}Let T=(T^*, T^?) be a spherical latin bitrade. With each a=(a₁, a₂, a₃)∈T^* associate a set of linear equations Eq(T, a) of the form b₁+b₂=b₃, where b=(b₁, b₂, b₃) runs through T^*\{a}. Assume a₁=0=a₂ and a₃=1. Then Eq(T,a) has in rational numbers a unique solution $b_{i}=\bar{b}_{i}$. Suppose that $\bar{b}_{i}\not= \bar{c}_{i}$ for all b, c∈T^* such that $\bar{b}_{i}\not= \bar{c}_{i}$ and i∈{1, 2, 3}. We prove that then T^? can be interpreted as a dissection of an equilateral triangle. We also consider group modifications of latin bitrades and show that the methods for generating the dissections can be used for a proof that T^* can be embedded into the operational table of a finite abelian group, for every spherical latin bitrade T. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 1–24, 2010     

Monogamous latin squares

We show for all n∉{1,2,4} that there exists a latin square of order n that contains two entries γ₁ and γ₂ such that there are some transversals through γ₁ but they all include γ₂ as well. We use this result to show that if n>6 and n is not of the form 2p for a prime p?11 then there exists a latin square of order n that possesses an orthogonal mate but is not in any triple of MOLS. Such examples provide pairs of 2-maxMOLS.     

Minimal homogeneous latin trades

A latin trade is a subset of a latin square which may be replaced with a disjoint mate to obtain a new latin square. A d-homogeneous latin trade is one which intersects each row, each column and each entry of the latin square either 0 or d times. In this paper we give a construction for minimal d-homogeneous latin trades of size dm, for every integer d?3, and m?1.75d²+3. We also improve this bound for small values of d. Our proof relies on the construction of cyclic sequences whose adjacent sums are distinct.     

Existence of self-conjugate self-orthogonal diagonal latin squares

In this article we give some new constructions of self-conjugate self-orthogonal diagonal Latin squares (SCSODLS). As an application of such constructions, we give a conclusive result regarding the existence of SCSODLS and show that there exists an SCSODLS of order n if and only if n ≡ 0, 1 (mod 4), except for n = 5. This result completely disproves a conjecture of Danhof, Phillips, and Wallis about SCSODLS in Danhof, Philips, and Wallis, JCMCC, 8 (1990), 3–8. © 1998 John Wiley & Sons, Inc. J Combin Designs 6:51–62, 1998     

Cycle structure of autotopisms of quasigroups and latin squares

An autotopism of a Latin square is a triple (α, β, γ) of permutations such that the Latin square is mapped to itself by permuting its rows by α, columns by β, and symbols by γ. Let Atp(n) be the set of all autotopisms of Latin squares of order n. Whether a triple (α, β, γ) of permutations belongs to Atp(n) depends only on the cycle structures of α, β, and γ. We establish a number of necessary conditions for (α, β, γ) to be in Atp(n), and use them to determine Atp(n) for n?17. For general n, we determine if (α, α, α)∈Atp(n) (that is, if αis an automorphism of some quasigroup of order n), provided that either αhas at most three cycles other than fixed points or that the non‐fixed points of α are in cycles of the same length. © 2011 Wiley Periodicals, Inc. J Combin Designs     

Existence of (3, 1, 2)‐conjugate orthogonal diagonal latin squares

We shall refer to a diagonal Latin square which is orthogonal to its (3,1,2)‐conjugate, and the latter is also a diagonal Latin square, as a (3,1, 2)‐conjugate orthogonal diagonal Latin square, briefly CODLS. This article investigates the spectrum of CODLS and it is found that it contains all positive integers υ except 2, 3, 6, and possibly 10. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 297–308, 2001     

Rank 3 Latin square designs

A Latin square design whose automorphism group is transitive of rank at most 3 on points must come from the multiplication table of an elementary abelian p-group, for some prime p.     

Small latin squares,quasigroups, and loops

We present the numbers of isotopy classes and main classes of Latin squares, and the numbers of isomorphism classes of quasigroups and loops, up to order 10. The best previous results were for Latin squares of order 8 (Kolesova, Lam, and Thiel, 28 ), quasigroups of order 6 (Bower, 7 ), and loops of order 7 (Brant and Mullen, 8 ). The loops of order 8 have been independently found by “QSCGZ” and Guérin (unpublished, 25 ). We also report on the most extensive search so far for a triple of mutually orthogonal Latin squares (MOLS) of order 10. Our computations show that any such triple must have only squares with trivial symmetry groups. © 2006 Wiley Periodicals, Inc. J Combin Designs     

Distances of group tables and latin squares via equilateral triangle dissections

Denote by gdist(p)$\mathrm{gdist}(p)$ the least non-zero number of cells that have to be changed to get a latin square from the table of addition modulo p  . A conjecture of Drápal, Cavenagh and Wanless states that there exists c>0$c>0$ such that gdist(p)?clog(p)$\mathrm{gdist}(p)?c\log (p)$. In this paper the conjecture is proved for c≈7.21$c\approx 7.21$, and as an intermediate result it is shown that an equilateral triangle of side n   can be non-trivially dissected into at most 5_log2(n)$5{\log }_2(n)$ integer-sided equilateral triangles. The paper also presents some evidence which suggests that gdist(p)/log(p)≈3.56$\mathrm{gdist}(p)/\log (p)\approx 3.56$ for large values of p.     

Existence of three HMOLS of type 2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript><Emphasis Type="Italic">u</Emphasis><Superscript>1</Superscript>

A Latin squares of order v with ni missing sub-Latin squares （holes） of order hi （1 〈= i 〈 k）, which are disjoint and spanning （i.e. ∑k i=l1 nihi = v）, is called a partitioned incomplete Latin squares and denoted by PILS. The type of PILS is defined by （h1n1 h2n2…hknk ）. If any two PILS inaset of t PILS of type T are orthogonal, then we denote the set by t-HMOLS（T）. It has been proved that 3-HMOLS（2n31） exist for n ≥6 with 11 possible exceptions. In this paper, we investigate the existence of 3-HMOLS（2nu1） with u ≥ 4, and prove that 3-HMOLS（2~u1） exist if n ≥ 54 and n ≥7/4u ＋ 7.     

Generating spherical Eulerian triangulations

We show that all spherical Eulerian triangulations can be inductively generated from the set of all even double wheels using just one of the two local transformations used in the algorithm by Brinkmann and McKay and originally proposed by Batagelj.     

14阶循环拉丁方型均匀设计

本文利用电子计算机求得全部１４阶循环拉丁方型均匀设计。这些设计的均匀性比现有的均匀设计Ｕ１４（１４＾ｓ）有显著改进，且参数ｓ的范围从ｓ≤８扩大到ｓ≤１４。