The thesis describes new results for several problems in random graph theory. The first problem relates to the uniform random graph model in the supercritical phase; i.e. a graph, uniformly distributed, on $n$ vertices and $M=n/2+s$ edges for $s=s(n)$ satisfying $n^{2/3}=o(s)$ and $s=o(n)$. The property studied is the length of the longest cycle in the graph. We give a new upper bound, which holds asymptotically almost surely, on this length. As part of our proof we establish a result about the heaviest cycle in a certain randomly-edge-weighted nearly-3-regular graph, which may be of independent interest. Our second result is a new contiguity result for a random $d$-regular graph. Let $j=j(n)$ be a function that is linear in $n$. A $(d,d-1)$-irregular graph is a graph which is $d$-regular except for $2j$ vertices of degree $d-1$. A $j$-edge matching in a graph is a set of $j$ independent edges. In this thesis we prove the new result that a random $(d,d-1)$-irregular graph plus a random $j$-edge matching is contiguous to a random $d$-regular graph, in the sense that in the two spaces, the same events have probability approaching 1 as $n\to\infty$. This allows one to deduce properties, such as colourability, of the random irregular graph from the corresponding properties of the random regular one. The proof applies the small subgraph conditioning method to the number of $j$-edge matchings in a random $d$-regular graph. The third problem is about the 3-colourability of a random 5-regular graph. Call a colouring balanced if the number of vertices of each colour is equal, and locally rainbow if every vertex is adjacent to vertices of all the other colours. Using the small subgraph conditioning method, we give a condition on the variance of the number of locally rainbow balanced 3-colourings which, if satisfied, establishes that the chromatic number of the random 5-regular graph is asymptotically almost surely equal to 3. We also describe related work which provides evidence that the condition is likely to be true. The fourth problem is about the chromatic number of a random $d$-regular graph for fixed $d$. Achlioptas and Moore recently announced a proof that a random $d$-regular graph asymptotically almost surely has chromatic number $k-1$, $k$, or $k+1$, where $k$ is the smallest integer satisfying $d < 2(k-1)\log(k-1)$. In this thesis we prove that, asymptotically almost surely, it is not $k+1$, provided a certain second moment condition holds. The proof applies the small subgraph conditioning method to the number of balanced $k$-colourings, where a colouring is balanced if the number of vertices of each colour is equal. We also give evidence that suggests that the required second moment condition is true.

Cyclic sieving phenomenon (CSP) is a generalization by Reiner, Stanton, White of Stembridge's q=-1 phenomenon. When CSP is exhibited, orbits of a cyclic action on combinatorial objects show a nice structure and their sizes can be encoded by one polynomial. In this thesis we study various proofs of a very interesting cyclic sieving phenomenon, that jeu-de-taquin promotion on rectangular Young tableaux exhibits CSP. The first proof was obtained by Rhoades, who used Kazhdan-Lusztig representation. Purbhoo's proof uses Wronski map to equate tableaux with points in the fibre of the map. Finally, we consider Petersen, Pylyavskyy, Rhoades's proof on 2 and 3 row tableaux by bijecting the promotion of tableaux to rotation of webs. This thesis also propose a combinatorial approach to prove the CSP for square tableaux. A variation of jeu-de-taquin move yields a way to count square tableaux which has minimal orbit under promotion. These tableaux are then in bijection to permutations. We consider how this can be generalized.

Given two set systems $\mathscr{A}$ and $\mathscr{B}$ over an $n$-element set, we say that $(\mathscr{A,B})$ forms a recovering pair if the following conditions hold: \\ $ \forall A, A' \in \mathscr{A}$ and $ \forall B, B' \in \mathscr{B}$, $A \setminus B = A' \setminus B' \Rightarrow A=A'$ \\ $ \forall A, A' \in \mathscr{A}$ and $ \forall B, B' \in \mathscr {B}$, $B \setminus A = B' \setminus A' \Rightarrow B=B'$ \\ In 1989, G\'bor Simonyi conjectured that if $(\mathscr)$ forms a recovering pair, then $|\mathscr||\mathscr|\leq 2^n$. This conjecture is the focus of this thesis. This thesis contains a collection of proofs of special cases that together form a complete proof that the conjecture holds for all values of $n$ up to 8. Many of these special cases also verify the conjecture for certain recovering pairs when $n>8$. We also present a result describing the nature of the set of numbers over which the conjecture in fact holds. Lastly, we present a new problem in graph theory, and discuss a few cases of this problem.

Cyclic sieving phenomenon (CSP) is a generalization by Reiner, Stanton, White of Stembridge's q=-1 phenomenon. When CSP is exhibited, orbits of a cyclic action on combinatorial objects show a nice structure and their sizes can be encoded by one polynomial. In this thesis we study various proofs of a very interesting cyclic sieving phenomenon, that jeu-de-taquin promotion on rectangular Young tableaux exhibits CSP. The first proof was obtained by Rhoades, who used Kazhdan-Lusztig representation. Purbhoo's proof uses Wronski map to equate tableaux with points in the fibre of the map. Finally, we consider Petersen, Pylyavskyy, Rhoades's proof on 2 and 3 row tableaux by bijecting the promotion of tableaux to rotation of webs. This thesis also propose a combinatorial approach to prove the CSP for square tableaux. A variation of jeu-de-taquin move yields a way to count square tableaux which has minimal orbit under promotion. These tableaux are then in bijection to permutations. We consider how this can be generalized.

In this dissertation we consider several extremal and probabilistic problems in bootstrap percolation on various families of graphs, including grids, hypercubes and trees. Bootstrap percolation is one of the simplest cellular automata. The most widely studied model is the so-called r-neighbour bootstrap percolation, in which we consider the spread of infection on a graph G according to the following deterministic rule: infected vertices of G remain infected forever and in successive rounds healthy vertices with at least r already infected neighbours become infected. Percolation is said to occur if eventually every vertex is infected. In Chapter 1 we consider a particular extremal problem in 2-neighbour bootstrap percolation on the n \times n square grid. We show that the maximum time an infection process started from an initially infected set of size n can take to infect the entire vertex set is equal to the integer nearest to (5n^2-2n)/8. In Chapter 2 we relax the condition on the size of the initially infected sets and show that the maximum time for sets of arbitrary size is 13n^2/18+O(n). In Chapter 3 we consider a similar problem, namely the maximum percolation time for 2-neighbour bootstrap percolation on the hypercube. We give an exact answer to this question showing that this time is \lfloor n^2/3 \rfloor. In Chapter 4 we consider the following probabilistic problem in bootstrap percolation: let T be an infinite tree with branching number \br(T) = b. Initially, infect every vertex of T independently with probability p > 0. Given r, define the critical probability, p_c(T,r), to be the value of p at which percolation becomes likely to occur. Answering a problem posed by Balogh, Peres and Pete, we show that if b \geq r then the value of b itself does not yield any non-trivial lower bound on p_c(T,r). In other words, for any \varepsilon > 0 there exists a tree T with branching number \br(T) = b and critical probability p_c(T,r) < \varepsilon. However, in Chapter 5 we prove that this is false if we limit ourselves to the well-studied family of Galton--Watson trees. We show that for every r \geq 2 there exists a constant c_r>0 such that if T is a Galton--Watson tree with branching number \br(T) = b \geq r then \[ p_c(T,r) > \frac{c_r}{b} e^{-\frac{b}{r-1}}. \] We also show that this bound is sharp up to a factor of O(b) by describing an explicit family of Galton--Watson trees with critical probability bounded from above by C_r e^{-\frac{b}{r-1}} for some constant C_r>0.

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The study of combinatorics on words dates back at least to the beginning of the 20th century and the work of Axel Thue. Thue was the first to give an example of an infinite word over a three letter alphabet that contains no squares (identical adjacent blocks) <i>xx</i>. This result was eventually used to solve some longstanding open problems in algebra and has remarkable connections to other areas of mathematics and computer science as well. In this thesis we primarily study several variations of the problems studied by Thue in his work on repetitions in words, including some recent connections to other areas, such as graph theory. In Chapter 1 we give a brief introduction to the subject of combinatorics on words. In Chapter 2 we use uniform morphisms to construct an infinite binary word that contains no cubes <i>xxx</i> and no squares <i>yy</i> with |<i>y</i>| &#8805; 4, thus giving a simpler construction than that of Dekking. We also use uniform morphisms to construct an infinite binary word avoiding all squares except 0??, 1??, and (01)??, thus giving a simpler construction than that of Fraenkel and Simpson. We give some new enumeration results for these avoidance properties and solve an open problem of Prodinger and Urbanek regarding the perfect shuffle of infinite binary words that avoid arbitrarily large squares. In Chapter 3 we examine ternary squarefree words in more detail, and in Chapter 4 we study words <i>w</i> satisfying the property that for any sufficiently long subword <i>w'</i> of <i>w</i>, <i>w</i> does not contain the reversal of <i>w'</i> as a subword. In Chapter 5 we discuss an application of the property of squarefreeness to colourings of graphs. In Chapter 6 we study strictly increasing sequences (<i>a</i>(<i>n</i>))<i>n</i>&#8805;0 of non-negative integers satisfying the equation <i>a</i>(<i>a</i>(<i>n</i>)) = <i>dn</i>. Finally, in Chapter 7 we give a brief conclusion and present some open problems.

The pfaffian is a classical tool which can be regarded as a generalization of the determinant. The hyperpfaffian, which was introduced by Barvinok, generalizes the pfaffian to higher dimension. This was further developed by Luque, Thibon and Abdesselam. There are several non-equivalent definitions for the hyperpfaffian, which are discussed in the introduction of this thesis. Following this we examine the extension of the Matrix-Tree theorem to the Hyperpfaffian-Cactus theorem by Abdesselam, proving it without the use of the Grassman-Berezin Calculus and with the new terminology of the non-uniform hyperpfaffian. Next we look at the extension of pfaffian orientations for counting matchings on graphs to hyperpfaffian orientations for counting matchings on hypergraphs. Finally pfaffian rings and ideal s are extended to hyperpfaffian rings and ideals, but we show that under reason able assumptions the algebra with straightening law structure of these rings cannot be extended.

The hypercube, Qd, is a natural and much studied combinatorial object, and we discuss various extremal problems related to it. A subgraph of the hypercube is said to be (Qd; F)-saturated if it contains no copies of F, but adding any edge forms a copy of F. We write sat(Qd; F) for the saturation number, that is, the least number of edges a (Qd; F)-saturated graph may have. We prove the upper bound sat(Qd;Q2) < 10 2d, which strongly disproves a conjecture of Santolupo that sat(Qd;Q2) = 1 4 + o(1) d2d 1. We also prove upper bounds on sat(Qd;Qm) for general m. Given a down-set A and an up-set B in the hypercube, Bollobás and Leader conjectured a lower bound on the number of edge-disjoint paths between A and B in the directed hypercube. Using an unusual form of the compression argument, we confirm the conjecture by reducing the problem to a the case of the undirected hypercube. We also prove an analogous conjecture for vertex-disjoint paths using the same techniques, and extend both results to the grid. Additionally, we deal with subcube intersection graphs, answering a question of Johnson and Markström of the least r = r(n) for which all graphs on n vertices may be represented as subcube intersection graph where each subcube has dimension exactly r. We also contribute to the related area of biclique covers and partitions, and study relationships between various parameters linked to such covers and partitions. Finally, we study topological properties of uniformly random simplicial complexes, employing a characterisation due to Korshunov of almost all down-sets in the hypercube as a key tool.