The three-way intersection problem for latin squares

Adams, Peter; Billington, Elizabeth J.; Bryant, Darryn; Mahmoodian, E. S.

doi:10.1016/s0012-365x(00)00454-4

Cited by 15 publications

(37 citation statements)

References 4 publications

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“…As IS There does not exist a large set of idempotent latin squares of order n = 6, however there does exist a (4,5,6)-latin trade given by: (2,3,4,5) • (1, 4, 5, 3) (5, 2, 1, 4) (3, 5, 2, 1) (4,1,3,2) • (3, 2, 5, 4) (6, 3, 4, 5) (4, 5, 2, 6) (5, 6, 3, 2) (2,4,6,3) (1, 4, 5, 3) (4, 5, 3, 6) (5, 1, 6, 4) • (6, 3, 1, 5) (3, 6, 4, 1) Applying Theorem 7 of [7] to the combination of a (3,5,6) …”

Section: Resultsmentioning

confidence: 99%

“…We can thus interpret the results of [1] in terms of 3-way latin trades and combine them with Theorem 3.4.6 to yield: • k ∈ {0, 9}, for λ = 3;…”

Section: Packing Constructionmentioning

confidence: 96%

“…A modification to this problem is finding the spectrum of sizes of the intersection of more than two latin squares, which is called the µ-way intersection problem for latin squares. In this case, we say a set of three or more latin squares have common intersection S if any pair of the latin squares from the set of latin squares has intersection S. The spectrum of possible common intersection sizes |S| has been investigated for sets of three and four latin squares (for three latin squares with common intersection, see [1] [46]; for four latin squares with common intersection, see [4]). Another modification to the initial problem that has found some attention is asking for the spectrum of possible sizes of the intersection of two or more latin squares with certain properties (i.e.…”

Section: µ-Way K-homogeneous Latin Tradesmentioning

confidence: 99%

“…The corresponding partial transversals have been found by a computer search, and have been written in reduced form in the appendix, in respectively Tables A. 1,A.2,and A.3 ,and Tables A.5,A.6,and A.7.…”

Section: Existence Of Partial Transversals In Subsquaresmentioning

confidence: 99%

“…An extension of this is to consider the µ-way intersections of the structures, and work has been done taking the underlying structure to be Steiner Triple Systems in [73], m-cycle systems in [2], and latin squares in [3] and [1].…”

Section: Introductionmentioning

confidence: 99%

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Latin Squares and Related Structures

Marbach¹

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A latin square of order n is an n × n array of cells, each filled with one of n symbols such that each row and each column contain each symbol precisely once. This thesis contributes three new results from three different topics within the study of latin squares.In doing this, we give a broad overview of three distinct areas of the study of latin squares, and the results that have been accomplished in the literature so far.The first study presents new results pertaining to transversals in latin squares. Previous work on transversals has investigated the spectrum of intersection sizes of two transversals within the back-circulant latin squares. A natural extension to this work is to investigate the spectrum of intersection sizes of more than two transversals within the back-circulant latin squares. In this thesis we will accomplish this for three and four transversals, and for all but a finite list of exceptions give a design theoretic construction that recursively builds from base designs that we found by a computational search.The second study investigates µ-way k-homogeneous latin trades. These structures have been extensively studied when µ = 2, but much less is known when µ > 2. Previous investigation had filled in a fraction of the spectrum when µ = 3. We continue this study giving new constructions and show that 3-way k-homogeneous latin trades of order m exist for all but 196 possible exceptions.The third study investigates mutually nearly orthogonal latin squares (MNOLS). These MNOLS are similar to mutually orthogonal latin squares, and can also be used in the design of experiments. Continuing from previous investigations, we enumerate the number of collections of three cyclic MNOLS for latin squares with order up to 16. This required using computational enumeration techniques and a large optimised computer search, as the search space was incredibly large. We present the number of collections of three MNOLS for latin squares with order up to 16 under a variety of equivalences, where we take the collections to be either sets or ordered lists.ii Declaration by authorThis thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis.

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