Idempotent Latin Squares Research Articles

The spectrum of resolvable Bose triple systems

A classical construction of Bose produces a Steiner triple system of order 3n from a symmetric, idempotent latin square of order n, which exists whenever n is odd. In an application to access-balancing in storage systems, certain Bose triple systems play a central role. A natural question arises: For which orders v does there exist a resolvable Bose triple system? Elementary counting establishes the necessary condition that v≡9(mod18). For specific Bose triple systems that optimize an access metric, we show that v≡9(mod18) is also sufficient.

The spectrum for large sets of resolvable idempotent Latin squares

AbstractAn idempotent Latin square of order is called resolvable and denoted by RILS(v) if the off‐diagonal cells can be resolved into disjoint transversals. A large set of resolvable idempotent Latin squares of order , briefly LRILS(v), is a collection of RILS(v)s pairwise agreeing on only the main diagonal. In this article, an LRILS(v) is constructed for by using multiplier automorphism groups. Hence, there exists an LRILS(v) for any positive integer , except .

Enumeration of Latin squares with conjugate symmetry

AbstractA 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.

The existence of r‐golf designs

AbstractAn r‐golf design of order , briefly by r‐G(v), is a large set of idempotent Latin squares of order (ILSs) which contains symmetric ILSs and transposed pairs of ILSs. In this paper, we mainly consider the existence problem of r‐G(v)s. We present several recursive constructions and also display some direct constructions. As an application, several infinite classes of r‐G(v)s are determined, including the existence of 0‐G(v)s for more than half of admissible parameters and the existence of r‐G(v)s where and is any admissible integer ( odd and ).

On the spectrum of mutually r-orthogonal idempotent Latin squares

Two Latin squares of order v are r-orthogonal if their superposition produces exactly r distinct ordered pairs. The two squares are said to be r-orthogonal idempotent Latin squares and denoted by r-MOILS(v) if they are all idempotent. In this paper, we show that for any integer v ≥ 28, there exists an r-MOILS(v) if and only if r ∈ [v, v2] \ {v + 1, v2 - 1}.

Completing a solution of the embedding problem for incomplete idempotent latin squares when numerical conditions suffice

Building upon the work of several previous papers, necessary and sufficient conditions are obtained for an incomplete idempotent latin square R of order n to be embedded in an idempotent latin square of order 2n. This work extends known necessary and sufficient conditions for embeddings into idempotent latin squares of order t≥2n+1, thereby pushing the bounds down to the point where subsequent developments must handle additional conditions that involve the arrangement of symbols on the given square.

Further Results on Large Sets of Resolvable Idempotent Latin Squares

AbstractAn idempotent Latin square of order v is called resolvable and denoted by RILS(v) if the off‐diagonal cells can be resolved into disjoint transversals. A large set of resolvable idempotent Latin squares of order v, briefly LRILS(v), is a collection of RILS(v)s pairwise agreeing on only the main diagonal. In this paper, it is established that there exists an LRILS(v) for any positive integer , except for , and except possibly for .

Large sets of resolvable idempotent Latin squares

An idempotent Latin square of order v is called resolvable and denoted by RILS ( v ) if the v ( v − 1 ) off-diagonal cells can be resolved into v − 1 disjoint transversals. A large set of resolvable idempotent Latin squares of order v , briefly LRILS ( v ) , is a collection of v − 2 RILS ( v ) s pairwise agreeing on only the main diagonal. In this paper we display some recursive and direct constructions for LRILSs.

The recognition of symmetric latin squares

AbstractA 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, Kn + 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 Kn + 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

Embedding a latin square in a pair of orthogonal latin squares

AbstractIn 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

Latin Squares with Self-Orthogonal Conjugates

In this paper, we investigate the existence of idempotent Latin squares for which each conjugate is orthogonal to precisely its own transpose. We show that the spectrum of Latin squares with this desired property contains all integers v⩾8, with the possible exception of 10 and 11. As an application of our results, it is shown that for all integers v⩾8, with the possible exception of 10 and 11, there exists an idempotent Latin square of order v that realizes the one-regular graph on 6 vertices as a conjugate orthogonal Latin square graph.

V(m,t) and its variants

Abstract A V ( m , t ) leads to m idempotent pairwise orthogonal Latin squares of order ( m +1) t +1 with one common hole of order t . For m =3,4,5 and 6 the spectrum for V ( m , t ) has been determined recently by Ling et al. In this article, Weil's theorem on character sums is used to get the spectra for V ( m , t )'s, for m =7. For variant V ( m , t )'s, such as V (2) ( m , t ) with m =2,4,6 and V (4) ( m , t ) with m =2,4, the spectrums are also determined. Three infinite families of V λ ( m , t )'s with λ=2, m=2; λ=2, m=3 and λ=3, m=2 are proved to exist.

Constructions for Generalized Group Divisible Designs in Settings Admitting Symmetry

SYNOPTIC ABSTRACTSeveral methods for constructing generalized group divisible designs with two classes and blocksize three are presented. Two constructions of completely symmetric designs are also given. All of the constructed designs achieve a lower bound on the diagonal elements of their information matrices required for E- and MV-optimality arguments; for most of the designs these optimalities are established. Difference techniques and the finite fields play key roles, as do idempotent and half-idempotent Latin squares.

Truncated Transversal Designs: A New Lower Bound on the Number of Idempotent MOLS of Side n

A truncated transversal design TTD of type gkm1 is a {k,k+1}-GDD of type gkm1 in which each point on the group of size m lies only in blocks of size k+1. Thus a TTD of type gkm1 is equivalent to a transversal design TD (k,g) having m disjoint parallel classes of blocks. We employ a new construction developed by the author (1993, J. Combin. Des.1, 15–26) to show that if g1<g2 and if there exists a TD (k,g1) and a TD (k+1,g2), then there exists a TTD of type (g1g2)km1 for any 0⩽m⩽(g2divg1)g21. As a corollary, we obtain a new lower bound on the number of mutually orthogonal idempotent latin squares of side g: if g1<g2 and there exist r MOLS of side g1 and r+1 MOLS of side g2 , then N(1 g1g2)⩾r.

V(m,t)'s for m=4,5,6

The spectrum for V(m,t) is solved for m=4,5 and 6 using ad hoc methods for m= 4 and using exponential sums for m=5 and 6. A V(m,t) leads to m idempotent pairwise orthogonal Latin squares of order (m+1)t+1 with one common hole of order t.

An [formula omitted] to [formula omitted] embedding of incomplete idempotent latin squares for small values of [formula omitted

In 1983, necessary and sufficient conditions were obtained for an incomplete idempotent latin square of order n to be embedded in an idempotent latin square of order 2n, providing n>16. In this paper we consider the case where n⩽16.

Three mutually orthogonal idempotent Latin squares of orders 22 and 26

Three mutually orthogonal idempotent Latin squares of orders 22 and 26 are constructed, which can be used to obtain 3 HMOLS of type 5 22 and type 23 22 and to obtain a (110, 5, 1)-PMD and a (130 5, 1)-PMD.

Small embeddings of partial directed cycle systems

In this paper, a generalisation of the Andersen, Hilton, Rodger Theorem for embedding partial idempotent latin squares is proved. This result is then used to prove that a partial directed m-cycle system of order n can be embedded in a directed m-cycle system of order (2n + 1)m if m is odd, of order 2nm if m ≥ 8 is even, 12n + 1 if m = 6 and approximately 2n + √2n if m = 4.

More Results on the Intersections of Latin Squares

Abstract In this paper, we determine some results regarding the parameter D i (n, k), i=1, 2, 3, 4 where D1(n, k) is the maximum number of latin squares such that any two of them intersect in k entries, which occurring in each of the latin squares, and D 2(n, k), D 3(n, k), D 4(n, k) are similar results for idempotent latin squares, idem-potent commutative latin squares, and latin cubes respectively. Moreover, we apply some of the above results to obtain some similar results in Steiner triple systems and Steiner quadruple systems [7, 12].

Further results on incomplete (3,2,1)- conjugate orthogonal idempotent Latin squares

Let us denote by COILS(υ) a (3,2,1)-conjugate orthogonal idempotent Latin square of order υ, and by ICOILS(υ, n) an incomplete COILS(υ) missing a sub-COILS( n). A necessary condition for the existence of an ICOILS(υ, n) is υ ⩾ 3 n + 1. An ICOILS(υ, 1) is equivalent to a COILS(υ), and the necessary condition for its existence has recently been shown by the authors to be sufficient for all υ ⩾ 4 with the exception of υ = 6 and the possible exception of υ = 12. Two of the above authors have previously shown that for n>1, an ICOILS(υ, n) exists if υ = 3 n + 1 or υ ⩾ 8 n + 42. Moreover, it was also shown that, for 2 ⩽ n ⩽ 6, an ICOILS( v,n) exists for all υ ⩾ 3 n + 1 with some possible exceptions. The main purpose of this paper is two-fold. First of all, for 2 ⩽ n ⩽ 6, we substantially reduce the number of possible exceptions and show that, in particular, the necessary condition is sufficient for n = 4,5 and 6 except possibly when (υ, n) = (30, 5). Secondly, we show that for n ⩾ 1, an ICOILS(υ, n) exists for all υ ⩾ ( 13 4 )n + 88 , which gives a general bound much closer to the necessary condition.