Randomness is an invaluable resource in theoretical computer science. However, pure random bits are hard to obtain. Quasi-randomness is a tool that has been widely used in eliminating/reducing the randomness from randomized algorithms. In this thesis, we study some aspects of quasi-randomness in graphs. Specifically, we provide an algorithm and a lower bound for two different kinds of regularity lemmas. Our algorithm for FK-regularity is derived using a spectral characterization of quasi-randomness. We also use a similar spectral connection to also answer an open question about quasi-random tournaments. We then provide a "Wowzer" type lower bound (for the number of parts required) for the strong regularity lemma. Finally, we study the derandomization of complexity classes using Turing machine simulations. 1. Connections between quasi-randomness and graph spectra. Quasi-random (or pseudo-random) objects are deterministic objects that behave almost like truly random objects. These objects have been widely studied in various settings (graphs, hypergraphs, directed graphs, set systems, etc.). In many cases, quasi-randomness is very closely related to the spectral properties of the combinatorial object that is under study. In this thesis, we discover the spectral characterizations of quasi-randomness in two different cases to solve open problems. A Deterministic Algorithm for Frieze-Kannan Regularity: The Frieze-Kannan regularity lemma asserts that any given graph of large enough size can be partitioned into a number of parts such that, across parts, the graph is quasi-random. . It was unknown if there was a deterministic algorithm that could produce a parition satisfying the conditions of the Frieze-Kannan regularity lemma in deterministic sub-cubic time. In this thesis, we answer this question by designing an O(n[superscript]w) time algorithm for constructing such a partition, where w is the exponent of fast matrix multiplication. Even Cycles and Quasi-Random Tournaments: Chung and Graham in had provided several equivalent characterizations of quasi-randomness in tournaments. One of them is about the number of "even" cycles where even is defined in the following sense. A cycle is said to be even, if when walking along it, an even number of edges point in the wrong direction. Chung and Graham showed that if close to half of the 4-cycles in a tournament T are even, then T is quasi-random. They asked if the same statement is true if instead of 4-cycles, we consider k-cycles, for an even integer k. We resolve this open question by showing that for every fixed even integer k geq 4, if close to half of the k-cycles in a tournament T are even, then T must be quasi-random. 2. A Wowzer type lower bound for the strong regularity lemma. The regularity lemma of Szemeredi asserts that one can partition every graph into a bounded number of quasi-random bipartite graphs. Alon, Fischer, Krivelevich and Szegedy obtained a variant of the regularity lemma that allows one to have an arbitrary control on this measure of quasi-randomness. However, their proof only guaranteed to produce a partition where the number of parts is given by the Wowzer function, which is the iterated version of the Tower function. We show here that a bound of this type is unavoidable by constructing a graph H, with the property that even if one wants a very mild control on the quasi-randomness of a regular partition, then any such partition of H must have a number of parts given by a Wowzer-type function. 3. How fast can we deterministically simulate nondeterminism? We study an approach towards derandomizing complexity classes using Turing machine simulations. We look at the problem of deterministically counting the exact number of accepting computation paths of a given nondeterministic Turing machine. We provide a deterministic algorithm, which runs in time roughly O(sqrt(S)), where S is the size of the configuration graph. The best of the previously known methods required time linear in S. Our result implies a simulation of probabilistic time classes like PP, BPP and BQP in the same running time. This is an improvement over the currently best known simulation by van Melkebeek and Santhanam.

In dieser Arbeit wird zuerst das Theorem von Chung, Graham und Wilson über quasi-zufällige Graphen zur sogenannten schwachen Quasi-Zufälligkeit für k-uniforme Hypergraphen verallgemeinert und somit eine Reihe äquivalenter Eigenschaften bestimmt. Basierend auf diesen Resultaten werden nichtbipartite Graphen gefunden, welche die Quasi-Zufälligkeit für Graphen ``forcieren''''. Zuvor waren nur bipartite Graphen mit dieser Eigenschaft bekannt. Desweiteren ist ein konzeptionell einfacher Algorithmus zum Verifizieren nicht erfüllbarer zufälliger k-SAT Formeln angegeben. Dann richtet sich der Fokus auf Anwendungen verschiedener Regularitätslemmata für Hypergraphen. Zuerst wird die Menge aller bezeichneten 3-uniformen Hypergraphen auf n Knoten, die keine Kopie des Hypergraphen der Fano Ebene enthalten, studiert. Es wird gezeigt, dass fast jedes Element aus dieser Menge ein bipartiter Hypergraph ist. Dies führt zu einem Algorithmus, der in polynomiell erwarteter Zeit einen zufälligen Fano-freien (und somit einen zufälligen bipartiten 3-uniformen) Hypergraphen richtig färbt. Schließlich wird die folgende extremale Funktion studiert. Es sind r Farben gegeben sowie ein k-uniformer Hypergraph F. Auf wie viele verschiedene Arten kann man die Kanten eines k-uniformen Hypergraphen H färben, so dass keine monochromatische Kopie von F entsteht? Welche Hypergraphen H maximieren die Anzahl erlaubter Kantenfärbungen? Hier wird ein strukturelles Resultat für eine natürliche Klasse von Hypergraphen bewiesen. Es wird für viele Hypergraphen F, deren extremaler Hypergraph bekannt ist, gezeigt, dass im Falle von zwei oder drei Farben die extremalen Hypergraphen die oben beschriebene Funktion maximieren, während für vier oder mehr Farben andere Hypergraphen mehr Kantenfärbungen zulassen. / This thesis presents first one possible generalization of the result of Chung, Graham and Wilson to k-uniform hypergraphs, and studies the so-called weak quasi-randomness. As applications we obtain a simple strong refutation algorithm for random sparse k-SAT formulas and we identify first non-bipartite forcing pairs for quasi-random graphs. Our focus then shifts from the study of quasi-random objects to applications of different versions of the hypergraph regularity lemmas; all these versions assert decompositions of hypergraphs into constantly many quasi-random parts, where the meaning of ``quasi-random'''' takes different contexts in different situations. We study the family of hypergraphs not containing the hypergraph of the Fano plane as a subhypergraph, and show that almost all members of this family are bipartite. As a consequence an algorithm for coloring bipartite 3-uniform hypergraphs with average polynomial running time is given. Then the following combinatorial extremal problem is considered. Suppose one is given r colors and a fixed hypergraph F. The question is: In at most how many ways can one color the hyperedges of a hypergraph H on n vertices such that no monochromatic copy of F is created? What are the extremal hypergraphs for this function? Here a structural result for a natural family of hypergraphs F is proven. For some special classes of hypergraphs we show that their extremal hypergraphs (for large n) maximize the number of edge colorings for 2 and 3 colors, while for at least 4 colors other hypergraphs are optimal.

Einst als Hilfssatz für Szemerédis Theorem entwickelt, hat sich das Regularitätslemma in den vergangenen drei Jahrzehnten als eines der wichtigsten Werkzeuge der Graphentheorie etabliert. Im Wesentlichen hat das Lemma zum Inhalt, dass dichte Graphen durch eine konstante Anzahl quasizufälliger, bipartiter Graphen approximiert werden können, wodurch zwischen deterministischen und zufälligen Graphen eine Brücke geschlagen wird. Da letztere viel einfacher zu handhaben sind, stellt diese Verbindung oftmals eine wertvolle Zusatzinformation dar. Vom Regularitätslemma ausgehend gliedert sich die vorliegende Arbeit in zwei Teile. Mit Fragestellungen der Extremalen Hypergraphentheorie beschäftigt sich der erste Teil der Arbeit. Es wird zunächst eine Version des Regularitätslemmas Hypergraphen angewandt, um asymptotisch scharfe Schranken für das Auftreten von Hamiltonkreisen in uniformen Hypergraphen mit hohem Minimalgrad herzuleiten. Nachgewiesen werden des Weiteren asymptotisch scharfe Schranken für die Existenz von perfekten und nahezu perfekten Matchings in uniformen Hypergraphen mit hohem Minimalgrad. Im zweiten Teil der Arbeit wird ein neuer, Szemerédis ursprüngliches Konzept generalisierender Regularitätsbegriff eingeführt. Diesbezüglich wird ein Algorithmus vorgestellt, welcher zu einem gegebenen Graphen ohne zu dichte induzierte Subgraphen eine reguläre Partition in polynomieller Zeit berechnet. Als eine Anwendung dieses Resultats wird gezeigt, dass das Problem MAX-CUT für die oben genannte Graphenklasse in polynomieller Zeit bis auf einen multiplikativen Faktor von (1+o(1)) approximierbar ist. Der Untersuchung von Chung, Graham und Wilson zu quasizufälligen Graphen folgend wird ferner der sich aus dem neuen Regularitätskonzept ergebende Begriff der Quasizufälligkeit studiert und in Hinsicht darauf eine Charakterisierung mittels Eigenwertseparation der normalisierten Laplaceschen Matrix angegeben. / Once invented as an auxiliary lemma for Szemerédi''s Theorem the regularity lemma has become one of the most powerful tools in graph theory in the last three decades which has been widely applied in several fields of mathematics and theoretical computer science. Roughly speaking the lemma asserts that dense graphs can be approximated by a constant number of bipartite quasi-random graphs, thus, it narrows the gap between deterministic and random graphs. Since the latter are much easier to handle this information is often very useful. With the regularity lemma as the starting point two roads diverge in this thesis aiming at applications of the concept of regularity on the one hand and clarification of several aspects of this concept on the other. In the first part we deal with questions from extremal hypergraph theory and foremost we will use a generalised version of Szemerédi''s regularity lemma for uniform hypergraphs to prove asymptotically sharp bounds on the minimum degree which ensure the existence of Hamilton cycles in uniform hypergraphs. Moreover, we derive (asymptotically sharp) bounds on minimum degrees of uniform hypergraphs which guarantee the appearance of perfect and nearly perfect matchings. In the second part a novel notion of regularity will be introduced which generalises Szemerédi''s original concept. Concerning this new concept we provide a polynomial time algorithm which computes a regular partition for given graphs without too dense induced subgraphs. As an application we show that for the above mentioned class of graphs the problem MAX-CUT can be approximated within a multiplicative factor of (1+o(1)) in polynomial time. Furthermore, pursuing the line of research of Chung, Graham and Wilson on quasi-random graphs we study the notion of quasi-randomness resulting from the new notion of regularity and concerning this we provide a characterisation in terms of eigenvalue separation of the normalised Laplacian matrix.