dc.contributor.advisor | Cavenagh, Nicholas J. | |
dc.contributor.advisor | Stokes, Tim E. | |
dc.contributor.advisor | Broughan, Kevin A. | |
dc.contributor.author | Raass, Petelo Vaipuna | |
dc.date.accessioned | 2016-07-25T23:42:52Z | |
dc.date.available | 2016-07-25T23:42:52Z | |
dc.date.issued | 2016 | |
dc.identifier.citation | Raass, 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/10289/10557 | en |
dc.identifier.uri | https://hdl.handle.net/10289/10557 | |
dc.description.abstract | This 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.mimetype | application/pdf | |
dc.language.iso | en | |
dc.publisher | University of Waikato | |
dc.rights | All 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.title | Critical sets of full Latin squares | |
dc.type | Thesis | |
thesis.degree.grantor | University of Waikato | |
thesis.degree.level | Doctoral | |
thesis.degree.name | Doctor of Philosophy (PhD) | |
dc.date.updated | 2016-07-18T22:25:44Z | |
pubs.place-of-publication | Hamilton, New Zealand | en_NZ |