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dc.contributor.advisorCavenagh, Nicholas J.
dc.contributor.advisorStokes, Tim E.
dc.contributor.advisorBroughan, Kevin A.
dc.contributor.authorRaass, Petelo Vaipuna
dc.date.accessioned2016-07-25T23:42:52Z
dc.date.available2016-07-25T23:42:52Z
dc.date.issued2016
dc.identifier.citationRaass, P. V. (2016). Critical sets of full Latin squares (Thesis, Doctor of Philosophy (PhD)). University of Waikato, Hamilton, New Zealand. Retrieved from https://hdl.handle.net/10289/10557en
dc.identifier.urihttps://hdl.handle.net/10289/10557
dc.description.abstractThis thesis explores the properties of critical sets of the full n-Latin square and related combinatorial structures including full designs, (m,n,2)-balanced Latin rectangles and n-Latin cubes. In Chapter 3 we study known results on designs and the analogies between critical sets of the full n-Latin square and minimal defining sets of the full designs. Next in Chapter 4 we fully classify the critical sets of the full (m,n,2)-balanced Latin square, by describing the precise structures of these critical sets from the smallest to the largest. Properties of different types of critical sets of the full n-Latin square are investigated in Chapter 5. We fully classify the structure of any saturated critical set of the full n-Latin square. We show in Theorem 5.8 that such a critical set has size exactly equal to n³ - 2n² - n. In Section 5.2 we give a construction which provides an upper bound for the size of the smallest critical set of the full n-Latin square. Similarly in Section 5.4, another construction gives a lower bound for the size of the largest non-saturated critical set. We conjecture that these bounds are best possible. Using the results from Chapter 5, we obtain spectrum results on critical sets of the full n-Latin square in Chapter 6. In particular, we show that a critical set of each size between (n - 1)³ + 1 and n(n - 1)² + n - 2 exists. In Chapter 7, we turn our focus to the completability of partial k-Latin squares. The relationship between partial k-Latin squares and semi-k-Latin squares is used to show that any partial k-Latin square of order n with at most (n - 1) non-empty cells is completable. As Latin cubes generalize Latin squares, we attempt to generalize some of the results we have established on k-Latin squares so that they apply to k-Latin cubes. These results are presented in Chapter 8.
dc.format.mimetypeapplication/pdf
dc.language.isoen
dc.publisherUniversity of Waikato
dc.rightsAll items in Research Commons are provided for private study and research purposes and are protected by copyright with all rights reserved unless otherwise indicated.
dc.titleCritical sets of full Latin squares
dc.typeThesis
thesis.degree.grantorUniversity of Waikato
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy (PhD)
dc.date.updated2016-07-18T22:25:44Z
pubs.place-of-publicationHamilton, New Zealanden_NZ


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