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As stressed by the Society for Industrial and Applied Mathematics (SIAM): Applied mathematics, in partnership with computational science, is essential in solving many real-world problems. Combinatorial optimization focuses on problems arising from discrete structures such as graphs and polyhedra. This thesis deals with extremal graphs and strings and focuses on two problems: the Erdős ’ problem on multiplicities of complete subgraphs and the maximum number of distinct squares in a string. The first part of the thesis deals with strengthening the bounds for the minimum proportion of monochromatic t cliques and t cocliques for all 2-colourings of the edges of the complete graph on n vertices. Denote by kt(G) the number of cliques of order t in a graph G. Let kt(n) = min{kt(G) + kt(G)} where G denotes the complement of G of order n. Let ct(n) = kt(n) / () n and ct be the limit of ct(n) for n going to infinity. A t t 2 was disproved by Thomason in 1989 1962 conjecture of Erdős stating that ct = 2 1− for all t ≥ 4. Tighter counterexamples have been constructed by Jagger, ˇ Sˇtovíček and Thomason in 1996, by Thomason for t ≤ 6 in 1997, and by Franek for t = 6 in 2002. We present a computational framework to investigate tighter upper bounds for small t yielding the following improved upper bounds for t = 6, 7 and 8: c6 ≤ 0.7445×2 1− 6

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We investigate the function ρd(n) = max { r(x) | x is a (d, n)-string} where r(x) is the number of runs in the string x, and a (d, n)-string is a string with length n and exactly d distinct symbols. Our investigation is motivated by the conjecture that ρd(n) ≤ n − d. We present and discuss fundamental properties of the ρd(n) function. The values of ρd(n) are presented in the (d, n−d)-table with rows indexed by d and columns indexed by n − d which reveals the regularities of the function. We introduce the concepts of the r-cover and core vector of a string, yielding a novel computational framework for determining ρd(n) values. The computation of the previously intractable instances is achieved via first computing a lower bound, and then using the structural properties to limit our exhaustive search only to strings that can possibly exceed this number of runs. Using this approach, we extended the known maximum number of runs in binary string from 60 to 74. In doing so, we find the first examples of run-maximal strings containing four consecutive identical symbols. Our framework

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This thesis discusses and describes empirical comparisons of execution times of three programs for computing runs in strings. Since two of the programs were thought to be of O(n log n) algorithms (crochB and crochB7) and the third is an implementation of a linear algorithm (runFinder), it was expected that for larger strings runFinder() will strongly outperform the other two programs in the processing of long strings. The aim of this study is thus manifold. We establish the upper limits of lengths of strings for which the performances of crochB and crochB7 are faster or comparable to the performance of runFinder; we also investigate what kind of penalty in performance crochB7 incurs for the memory saving implementation; furthermore, we wish to explore the relative trade-offs of using one technique (represented through the programs with which experimentation was gone about) over another: within what context would it be advantageous to utilize one program over another of those that are being investigated. The motivation for this work is the continuation of work of Franek, Jiang, Smyth, Weng, and Xiao, who implemented a space efficient version of Crochemore’s repetition algorithm [6], and then extended it to compute runs [4, 5]. The three programs tested are: 1. crochB – a direct C++ implementation of the extension of Crochemore’s algorithm for runs by Franek, Jiang, and Weng without any space savings techniques; 2. crochB7 – a space efficient version of crochB by the same authors, 3. runFinder – an efficient C++ implementation by Hideo Bannai from the

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Fraenkel and Simpson conjectured in 1998 that the number of distinct squares in a string is at most its length. Similarly, Kolpakov and Kucherov conjectured in 1999 that the number of runs in a string is at most its length. Since then, both conjectures attracted the attention of many researchers and many results have been presented, including asymptotic lower bounds for both, asymptotic upper bounds for runs, and universal upper bounds for distinct squares. We consider the role played by the size of the alphabet of the string in both problems and investigate the functions σd(n) and ρd(n), i.e. the maximum number of distinct primitively rooted squares, respectively runs, over all strings of length n containing exactly d distinct symbols. We revisit earlier results and conjectures and express them in terms of singularities of the two functions where a pair (d, n) is a singularity if σd(n) − σd−1(n − 2) ≥ 2, or ρd(n) − ρd−1(n − 2) ≥ 2 respectively.