Existence of weakly pandiagonal orthogonal Latin squares

A weakly pandiagonal Latin square of order n over the number set {0, 1, . . . , n-1} is a Latin square having the property that the sum of the n numbers in each of 2n diagonals is the same. In this paper, we shall prove that a pair of orthogonal weakly pandiagonal Latin squares of order n exists if and only if n ≡ 0, 1, 3 (mod 4) and n≠3.     

A generalization of sum composition: Self orthogonal Latin square design with sub self orthogonal Latin square designs

A generalization of the theory of sum composition of Latin square designs is given. Via this generalized theory it is shown that a self orthogonal Latin square design of order $\text{(3p}{}^{\mathrm{\alpha }}\text{? 1)}\text{2}$ with a subself orthogonal Latin square design of order $\text{(p}{}^{\mathrm{\alpha }}\text{? 1)}\text{2}$ can be constructed for any prime p > 2 and any positive integer α as long as p ≠ 3, 5, 7 and 13 if α = 1. Additional results concerning sets of orthogonal Latin square designs are also provided.     

Existence of conjugate orthogonal diagonal Latin squares

We shall refer to a diagonal Latin square which is orthogonal to its (3, 2, 1)-conjugate and having its (3, 2, 1)-conjugate also a diagonal Latin square as a (3, 2, 1)-conjugate orthogonal diagonal Latin square, briefly CODLS. This article investigates the spectrum of CODLS and it is found that it contains all positive integers v except 2, 3, 6, and possibly 10. © 1997 John Wiley & Sons, Inc. J Combin Designs 5: 449–461, 1997     

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     

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.     

Existence of r-self-orthogonal Latin squares

Two Latin squares of order v are r-orthogonal if their superposition produces exactly r distinct ordered pairs. If the second square is the transpose of the first one, we say that the first square is r-self-orthogonal, denoted by r-SOLS(v). It has been proved that for any integer v?28, there exists an r-SOLS(v) if and only if v?r?v² and r∉{v+1,v²-1}. In this paper, we give an almost complete solution for the existence of r-self-orthogonal Latin squares.     

Latin squares with pairwise orthogonal conjugates

Let L¹ denote the set of integers n such that there exists an idempotent Latin square of order n with all of its conjugates distinct and pairwise orthogonal. It is known that L¹ contains all sufficiently large integers. That is, there is a smallest integer n_o such that L¹ contains all integers greater than n_o. However, no upper bound for n_o has been given and the term “sufficiently large” is unspecified. The main purpose of this paper is to establish a concrete upper bound for n_o. In particular it is shown that L¹ contain all integers n>5594, with the possible exception of n=6810.     

用正交对角拉丁方及幻矩与和阵构作mn阶标准二次幻方

当m和n为同奇或同偶的正整数且m,n≠1,2,3,6时,用m和n阶正交对角拉丁方及{0,1,…,mn-1）上的m×n幻矩与和阵,构作了mn阶标准二次幻方．     

Further results on the existence of nested orthogonal arrays

Nested orthogonal arrays provide an option for designing an experimental setup consisting of two experiments, the expensive one of higher accuracy being nested in a larger and relatively less expensive one of lower accuracy. We denote by OA_(λ, μ)(t, k, (v, w)) (or OA(t, k, (v, w)) if λ = μ = 1) a (symmetric) orthogonal array OA _λ (t, k, v) with a nested OA _μ (t, k, w) (as a subarray). It is proved in this article that an OA(t, t + 1,(v, w)) exists if and only if v ≥ 2w for any positive integers v, w and any strength t ≥ 2. Some constructions of OA_(λ, μ)(t, k, (v, w))′s with λ ≠ μ and k ? t > 1 are also presented.     

6-Valent analogues of Eberhard’s theorem

It is shown that for every sequence of non-negative integers (p _n|1≦n≠3) satisfying the equation {ie19-1} (respectively, =0) there exists a 6-valent, planar (toroidal, respectively) multi-graph that has preciselyp _n n gonal faces for alln, 1≦n≠3. This extends Eberhard’s theorem that deals, in a similar fashion, with 3-valent, 3-connected planar graphs; the equation involved follows from the famous Euler’s equation.     

Mutually orthogonal Latin squares based on general linear groups

Given a finite group G, how many squares are possible in a set of mutually orthogonal Latin squares based on G? This is a question that has been answered for a few classes of groups only, and for no nonsoluble group. For a nonsoluble group G, we know that there exists a pair of orthogonal Latin squares based on G. We can improve on this lower bound when G is one of GL(2, q) or SL(2, q), q a power of 2, q ≠ 2, or is obtained from these groups using quotient group constructions. For nonsoluble groups, that is the extent of our knowledge. We will extend these results by deriving new lower bounds for the number of squares in a set of mutually orthogonal Latin squares based on the group GL(n, q), q a power of 2, q ≠ 2.     

The Existence and Application of Strongly Idempotent Self-orthogonal Row Latin Magic Arrays

Let N = {0, 1, · · ·, n ? 1}. A strongly idempotent self-orthogonal row Latin magic array of order n (SISORLMA(n) for short) based on N is an n × n array M satisfying the following properties: (1) each row of M is a permutation of N, and at least one column is not a permutation of N; (2) the sums of the n numbers in every row and every column are the same; (3) M is orthogonal to its transpose; (4) the main diagonal and the back diagonal of M are 0, 1, · · ·, n ? 1 from left to right. In this paper, it is proved that an SISORLMA(n) exists if and only if n ? {2, 3}. As an application, it is proved that a nonelementary rational diagonally ordered magic square exists if and only if n ? {2, 3}, and a rational diagonally ordered magic square exists if and only if n ≠ 2.     

The use of hill-climbing to construct orthogonal steiner triple systems

In a related article, Colbourn, Gibbons, Mathon, Mullin, and Rosa [7] have shown that a pair of orthogonal Steiner triple systems exists for all v ≡ 1, 3 (mod 6), v ≥ 7 and v ≠ 9. This result is based on the construction of a finite set of pairs of orthogonal Steiner triple systems followed by the application of recursive constructions to settle the remaining undecided cases. In this article we report on the computational aspects of that investigation, and in particular the remarkable success of the hill-climbing method. © 1993 John Wiley & Sons, Inc.     

Atomic Latin squares of order eleven

A Latin square is pan‐Hamiltonian if the permutation which defines row i relative to row j consists of a single cycle for every i ≠ j. A Latin square is atomic if all of its conjugates are pan‐Hamiltonian. We give a complete enumeration of atomic squares for order 11, the smallest order for which there are examples distinct from the cyclic group. We find that there are seven main classes, including the three that were previously known. A perfect 1‐factorization of a graph is a decomposition of that graph into matchings such that the union of any two matchings is a Hamiltonian cycle. Each pan‐Hamiltonian Latin square of order n describes a perfect 1‐factorization of K_n,n, and vice versa. Perfect 1‐factorizations of K_n,n can be constructed from a perfect 1‐factorization of K_n+1. Six of the seven main classes of atomic squares of order 11 can be obtained in this way. For each atomic square of order 11, we find the largest set of Mutually Orthogonal Latin Squares (MOLS) involving that square. We discuss algorithms for counting orthogonal mates, and discover the number of orthogonal mates possessed by the cyclic squares of orders up to 11 and by Parker's famous turn‐square. We find that the number of atomic orthogonal mates possessed by a Latin square is not a main class invariant. We also define a new sort of Latin square, called a pairing square, which is mapped to its transpose by an involution acting on the symbols. We show that pairing squares are often orthogonal mates for symmetric Latin squares. Finally, we discover connections between our atomic squares and Franklin's diagonally cyclic self‐orthogonal squares, and we correct a theorem of Longyear which uses tactical representations to identify self‐orthogonal Latin squares in the same main class as a given Latin square. © 2003 Wiley Periodicals, Inc.     

Existence of generalized Latin squares which are not embeddable in any group

Let n be a positive integer. A generalized Latin square of order n is an \(n\times n\) matrix such that the elements in each row and each column are distinct. In this paper, we show that for any integer \(n\ge 6\) and any integer m where \(m\in \left\{ n, n+1, \dots , \frac{n(n+1)}{2}-2\right\} \), there exists a commutative generalized Latin square of order n with m distinct elements which is not embeddable in any group. In addition, we show that for any integer \(r\ge 3\) and any integer s where \(s\in \{ r, r+1, \dots , r^2-2\}\), there exists a non-commutative generalized Latin square of order r with s distinct elements which is not embeddable in any group.     

Totally conjugate orthogonal quasigroups and complete graphs

In this article, we study the spectrum of quasigroups all conjugates of which are distinct and pairwise orthogonal. We call such quasigroups totally conjugate orthogonal quasigroups (for brevity, totCO-quasigroups). Every totCO-quasigroup defines the complete conjugate orthogonal Latin square graph K ₆. Examples of totCO-quasigroups of different orders are given.     

An existence theory for pairwise balanced designs,III: Proof of the existence conjectures

Given positive integers k and λ, balanced incomplete block designs on v points with block size k and index λ exist for all sufficiently large integers v satisfying the congruences λ(v ? 1) ≡ 0 (mod k ? 1) and λv(v ? 1) ≡ 0 (mod k(k ? 1)). Analogous results hold for pairwise balanced designs where the block sizes come from a given set K of positive integers. We also observe that the number of nonisomorphic designs on v points with given block size k > 2 and index λ tends to infinity as v increases (subject to the above congruences).     

On the decomposition of λK into regular tournaments of order 5

An RTD[5,λ; v] is a decomposition of the complete symmetric directed multigraph, denoted by λK, into regular tournaments of order 5. In this article we show that an RTD[5,λ; v] exists if and only if (v?1)λ ≡ 0 (mod 2) and v(v?1)λ ≡ 0 (mod 10), except for the impossible case (v,λ) = (15,1). Furthermore, we show that for each v ≡ 1,5 (mod 20), v ≠ 5, there exists a B[5,2; v] which is not RT₅-directable. © 1994 John Wiley & Sons, Inc.     

Generalized latin square

LetX be a n-set and letA = [a_ij] be an xn matrix for whichaij ?X, for 1 ≤i, j ≤n. A is called a generalized Latin square onX, if the following conditions is satisfied: $ \cup _{i = 1}^n a_{ij} = X = \cup _{j = 1}^n a_{ij} $ . In this paper, we prove that every generalized Latin square has an orthogonal mate and introduce a H_v -structure on a set of generalized Latin squares. Finally, we prove that every generalized Latin square of ordern, has a transversal set.     

On large sets of disjoint Steiner triple systems II

Let D(v) denote the maximum number of pairwise disjoint Steiner triple systems of order v. In this paper, we prove that if n is an odd number, there exist 12 mutually orthogonal Latin squares of order n and D(1 + 2n) = 2n ? 1, then D(1 + 12n) = 12n ? 1.