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.     

Crisscross Latin squares

Let A be a Latin square of order n. Then the jth right diagonal of A is the set of n cells of A: {(i,j+i):i=0,1…,n?1(modn); and the jth left diagonal of A is the set {(i,j?i):i=0,1…,n?1(modn); A diagonal is said to be complete if every element appears in it exactly once. For n = 2m even, we introduce the concept of a crisscross Latin square which is something in between a diagonal Latin square and a Knut Vik design. A crisscross Latin square is a Latin square such that all the jth right diagonals for even j and all the jth left diagonals for odd j are complete. We show that a necessary and sufficient condition for the existence of a crisscross Latin square of order 2m is that m is even.     

An inequality for pseudo-subplanes of sets of orthogonal latin squares

Let L₁, L₂,…, L_t be a given set of t mutually orthogonal order-n latin squares defined on a symbol set S, |S| = n. The squares are equivalent to a (t + 2)-netN of order n which has n² points corresponding to the n² cells of the squares. A line of the net N defined by the latin square L_i comprises the n points of the net which are specified by a set of n cells of L_i all of which contain the same symbol x of S. If we pick out a particular r × r block B of cells, a line which contains points corresponding to r of the cells of B will be called an r-cell line. If there exist r(r ? 1) such lines among the tn lines of N, we shall say that they form a pseudo-subplane of order r-the “pseudo” means that these lines need not belong to only r ? 1 of the latin squares. The purpose of the present note is to prove that the hypothesis that such a pseudo-plane exists in N implies that $\text{r}{}^3\text{? (t + 2)r}{}^2\text{+ r + nt ?}{}^1\text{0}$.     

Row complete squares and a problem of A. Kotzig concerning P-quasigroups and eulerian circuits

An n × n square L on n symbols is called row (column) complete if every ordered pair of the symbols of L occurs just once as an adjacent pair of elements in some row (column) of L. It is called row (column) latin if each symbol occurs exactly once in each row (column) of the square. A square which is both row latin and column latin is called a latin square. All known examples of row complete latin squares can be made column complete as well by suitable reordering of their rows and in the present paper we provide a sufficient condition that a given row complete latin square should have this property.Using row complete and column latin squares as a tool we follow this by showing how to construct code words on n symbols of the maximum possible length $\text{l =}\text{1}\text{2}\text{n(n ? 1) + 1}$ with the two properties that (i) no unordered pair of consecutive symbols is repeated more than once and (ii) no unordered pair of nearly consecutive symbols is repeated more than once. (Two symbols are said to be nearly consecutive if they are separated by a single symbol.) We prove that such code words exist whenever n = 4r + 3 with r ? 1 mod 6 and r ? 2 mod 5. We show that the existence of such a code word for a given value of n guarantees the existence of an Eulerian circuit in the complete undirected n-graph which corresponds to a P-quasigroup, thus answering a question raised by A. Kotzig in the affirmative. (Kotzig has defined a P-groupoid as a groupoid (G, ·) having the following three properties: (i) a . a = a for all a?G; (ii) a ≠ b implies a ≠ a . b and b ≠ a. b for all a, b?G; and (iii) a . b = c implies c. b = a for all a, b, c?G. Every decomposition of the complete undirected n-graph into disjoint closed circuits defines such a P-groupoid, as is easily seen by defining a . b = c if and only if a, b, c are consecutive edges of one such closed circuit. A P-groupoid whose multiplication table is a latin square is called a P-quasigroup.)     

Subsquares in orthogonal latin squares as subspaces in affine geometries: A generalization of an equivalence of bose

The combinatorial properties of subsquares in orthogonal latin squares are examined. Using these properties it is shown that in appropriate orthogonal latin squares of orderm ^h blocks of subsquares of orderm ^h(i–1)/i , wherei dividesh, form the hyperplanes of the affine geometryAG (2i, m ^h/i ). This means that a given set of mutually orthogonal latin squares may be equivalent simultaneously to a number of different geometries depending on the order of the subsquares used to form the hyperplanes. In the case thati=1, the subsquares become points, the hyperplanes become lines, and the equivalence reduces to the well known result of Bose relating orthogonal latin squares and affine planes.The author would like to thank the Natural Sciences and Engineering Research Council of Canada for partial support under grant no. OGP0014645.     

The Orthogonality Spectrum for Latin Squares of Different Orders

Two orthogonal latin squares of order n have the property that when they are superimposed, each of the n ² ordered pairs of symbols occurs exactly once. In a series of papers, Colbourn, Zhu, and Zhang completely determine the integers r for which there exist a pair of latin squares of order n having exactly r different ordered pairs between them. Here, the same problem is considered for latin squares of different orders n and m. A nontrivial lower bound on r is obtained, and some embedding-based constructions are shown to realize many values of r.     

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.     

On the separation power and the completion of partial latin squares

Let n and m be natural numbers, n ? m. The separation power of order n and degree m is the largest integer k = k(n, m) such that for every (0, 1)-matrix A of order n with constant linesums equal to m and any set of k 1's in A there exist (disjoint) permutation matrices P₁,…, P_m such that A = P₁ + … + P_m and each of the k 1's lies in a different P_i. Almost immediately we have 1 ? k(n, m) ? m ? 1, yet in all cases where the value of k(n, m) is actually known it equals m ? 1 (except under the somewhat trivial circumstances of k(n, m) = 1). This leads to a conjecture about the separation power, namely that k(n, m) = m ? 1 if $\text{m ? [}\text{n}\text{2}\text{] + 1}$. We obtain the bound $\text{k(n, m) ? m ? [}\text{n}\text{2}\text{] + 2}$, so that this conjecture holds for n ? 7. We then move on to latin squares, describing several equivalent formulations of the concept. After establishing a sufficient condition for the completion of a partial latin square in terms of the separation power, we can show that the Evans conjecture follows from this conjecture about the separation power. Finally the lower bound on k(n, m) allows us to show, after some calculations, that the Evans conjecture is true for orders n ? 11.     

Latin squares and superqueens

Let L be a Latin square of order n with entries from {0, 1,…, n ? 1}. In addition, L is said to have the (n, k) property if, in each right or left wrap around diagonal, the number of cells with entries smaller than k is exactly k. It is established that a necessary and sufficient condition for the existence of Latin squares having the (n, k) property is that of (2|n ? 2| k) and (3|n ? 3| k). Also, these Latin squares are related to a problem of placing nonattacking queens on a toroidal chessboard.     

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.     

Permutation Polynomials Modulo 2w

We give an exact characterization of permutation polynomials modulo n=2^w, w≥2: a polynomial P(x)=a₀+a₁x +···+a_dx^d with integral coefficients is a permutation polynomial modulo n if and only if a₁ is odd, (a₂+a₄+a₆+···) is even, and (a₃+a₅+a₇+···) is even. We also characterize polynomials defining latin squares modulo n=2^w, but prove that polynomial multipermutations (that is, a pair of polynomials defining a pair of orthogonal latin squares) modulo n=2^wdo not exist.     

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.     

Embedding a latin square in a pair of orthogonal latin squares

In this paper, it is shown that a latin square of order n with n ≥ 3 and n ≠ 6 can be embedded in a latin square of order n² which has an orthogonal mate. A similar result for idempotent latin squares is also presented. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 270–276, 2006     

Construction of a Jacobi matrix from spectral data

A proof is given for the existence and uniqueness of a correspondence between two pairs of sequences {a},{b} and {ω},{μ}, satisfying b_i>0 for i=1,…,n?1 and ω₁<μ₁<?<μ_n?1<ω_n, under which the symmetric Jacobi matrices J(n,a,b) and J(n?1,a,b) have eigenvalues {ω} and {μ} respectively. Here J(m,a,b) is symmetric and tridiagonal with diagonal elements a_i (i=1,…,m) and off diagonal elements b_i (i=1,…,m?1). A new concise proof is given for the known uniqueness result. The existence result is new.     

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     

A class of facet producing graphs for vertex packing polyhedra

We examine a family of graphs called webs. For integers n ? 2 and $\text{k, 1 ? k ?}\text{1}\text{2}\text{n}$, the web W(n, k) has vertices V_n = {1, …, n} and edges {(i, j): j = i+k, …, i+n ? k, for i?V_n (sums mod n)}. A characterization is given for the vertex packing polyhedron of W(n, k) to contain a facet, none of whose projections is a facet for the lower dimensional vertex packing polyhedra of proper induced subgraphs of W(n, k). Simple necessary and sufficient conditions are given for W(n, k) to contain W(n′, k′) as an induced subgraph; these conditions are used to show that webs satisfy the Strong Perfect Graph Conjecture. Complements of webs are also studied and it is shown that if both a graph and its complement are webs, then the graph is either an odd hole or its complement.     

Some isomorphisms between pairs of latin squares

Let A, B, C, D be latin squares with A orthogonal to B and C orthogonal to D. The pair A, B is isomorphic with the pair C, D if the graph of A, B is graph-isomorphic with the graph of C, D. A characterization is given for determining when a pair A, B of latin squares is isomorphic with a self-orthogonal square C and its transpose. Self-orthogonal squares are important because they are both abundant and easy to store. An algorithm either displays a self-orthogonal square C and an isomorphism from A, B to C, C^T or, if none exists, gives a small set of blocks to the existence of such a square isomorphism.     

Simultaneous block diagonalization of two real symmetric matrices

The simultaneous diagonalization of two real symmetric (r.s.) matrices has long been of interest. This subject is generalized here to the following problem (this question was raised by Dr. Olga Taussky-Todd, my thesis advisor at the California Institute of Technology): What is the first simultaneous block diagonal structure of a nonsingular pair of r.s. matrices ? For example, given a nonsingular pair of r.s. matrices S and T, which simultaneous block diagonalizations X′SX = diag(A₁, , A_k), X′TX = diag(B₁,, B_k) with dim A_i = dim B_i and X nonsingular are possible for 1 ? k ? n; and how well defined is a simultaneous block diagonalization for which k, the number of blocks, is maximal? Here a pair of r.s. matrices S and T is called nonsingular if S is nonsingular.If the number of blocks k is maximal, then one can speak of the finest simultaneous block diagonalization of S and T, since then the sizes of the blocks A_i are uniquely determined (up to permutations) by any set of generators of the pencil P(S, T) = {aS + bT|a, tb ε R} via the real Jordan normal form of S^?1T. The proof uses the canonical pair form theorem for nonsingular pairs of r.s. matrices. The maximal number k and the block sizes dim A_i are also determined by the factorization over C of ? (λ, μ) = det(λS + μT) for λ, μ ε R.     

Orthogonal latin square graphs

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.