| dc.description.abstract | Generalised t-designs, defined by Cameron, describe a generalisation of many
combinatorial objects including: Latin squares, 1-factorisations of K2n (the
complete graph on 2n vertices), and classical t-designs.
This new relationship raises the question of how their respective theory
would fare in a more general setting. In 1991, Jacobson and Matthews published
an algorithm for generating uniformly distributed random Latin squares and
Cameron conjectures that this work extends to other generalised 2-designs with
block size 3.
In this thesis, we divide Cameron’s conjecture into three parts. Firstly, for
constants RC, RS and CS, we study a generalisation of Latin squares, which
are (r c) grids whose cells each contain RC symbols from the set f1;2; : : : ; sg
such that each symbol occurs RS times in each column and CS times in each
row. We give fundamental theory about these objects, including an enumeration
for small parameter values. Further, we prove that Cameron’s conjecture is true
for these designs, for all admissible parameter values, which provides the first
method for generating them uniformly at random.
Secondly, we look at a generalisation of 1-factorisations of the complete
graph. For constants NN and NC, these graphs have n vertices, each incident
with NN coloured edges, such that each colour appears at each vertex NC
times. We successfully show how to generate these designs uniformly at random
when NC 0 (mod 2) and NN NC.
Finally, we observe the difficulties that arise when trying to apply Jacobson
and Matthews’ theory to the classical triple systems. Cameron’s conjecture
remains open for these designs, however, there is mounting evidence which
suggests an affirmative result.
A function reference for DesignMC, the bespoke software that was used
during this research, is provided in an appendix. | en_US |
