This thesis puts forward the conjecture that for <i>n</i> > 3<i>k</i> with <i>k</i> > 2, the generalized Petersen graph, <i>GP</i>(<i>n,k</i>) is Hamilton-laceable if <i>n</i> is even and <i>k</i> is odd, and it is Hamilton-connected otherwise. We take the first step in the proof of this conjecture by proving the case <i>n</i> = 3<i>k</i> + 1 and <i>k</i> greater than or equal to 1. We do this mainly by means of an induction which takes us from <i>GP</i>(3<i>k</i> + 1, <i>k</i>) to <i>GP</i>(3(<i>k</i> + 2) + 1, <i>k</i> + 2). The induction takes the form of mapping a Hamilton path in the smaller graph piecewise to the larger graph an inserting subpaths we call <i>rotors</i> to obtain a Hamilton path in the larger graph.

Modelling mechanical systems using symbolic equations can provide many advantages over the more widely-used numerical methods of modelling these systems. The use of symbolic equations produces more efficient models, which can be used for many purposes such as real-time simulation and control. However, the number, complexity, and computational efficiency of these equations is highly dependent on which coordinate set was used to model the system. One method of modelling a mechanism's topology and formulating its symbolic equations is to model the system using a graph-theoretical approach. This approach models mechanisms using a linear graph, from which spanning trees can be used to define a mechanism's coordinate set. This report develops two tree selection algorithms capable of estimating the tree set, and hence coordinate set, that produces models having the fastest forward dynamic simulation times. The first tree selection algorithm is a heuristic-based algorithm that tries to find the coordinate set containing the minimal possible number of modelling variables. Most of this algorithm's heuristics are based on tree selection criteria found in the literature and on observations of a series of benchmark problems. It uses the topology information provided by a system's graph to find the coordinates set for the given system that produce very low simulation times of the system. The second tree selection algorithm developed in this report also uses graph theory. It bases most of its heuristics on observations of one of the methods developed to obtain a mechanical system's symbolic equations using graph theory. This second algorithm also makes use of, and improves upon, a few of the heuristics developed in the first tree selection algorithm. A series of examples for both algorithms will demonstrate the computational efficiency obtained by using the modelling variables found by the automated tree selection algorithms that are proposed in this report.

This thesis explores two infinite families of self-complementary arc-transitive graphs: the familiar Paley graphs and the newly discovered Peisert graphs. After studying both families, we examine a result of Peisert which proves the Paley and Peisert graphs are the only self-complementary arc transitive graphs other than one exceptional graph. Then we consider other families of graphs which share many properties with the Paley and Peisert graphs. In particular, we construct an infinite family of self-complementary strongly regular graphs from affine planes. We also investigate the pseudo-Paley graphs of Weng, Qiu, Wang, and Xiang. Finally, we prove a lower bound on the number of maximal cliques of certain pseudo-Paley graphs, thereby distinguishing them from Paley graphs of the same order.

In this thesis we take a specialized approach to edge-colouring by focusing exclusively on multigraphs with high chromatic index. The bulk of our results can be classified into three categories. First, we prove results which aim to characterize those multigraphs achieving known upper bounds. For example, Goldberg's Theorem says that χ'≤ Δ+1+(Δ-2}/(g₀+1) (where χ' denotes chromatic index, Δ denotes maximum degree, and g₀ denotes odd girth). We characterize this bound by proving that for a connected multigraph G, χ'= Δ+1+(Δ-2}/(g₀+1) if and only if G=μC_g₀ and (g₀+1)|2(μ-1) (where μ denotes maximum edge-multiplicity). Our second category of results are new upper bounds for chromatic index in multigraphs, and accompanying polynomial-time edge-colouring algorithms. Our bounds are all approximations to the famous Seymour-Goldberg Conjecture, which asserts that χ'≤ max{⌈ρ⌉, Δ+1} (where ρ=max{(2|E[S]|)/(|S|-1): S⊆V, |S|≥3 and odd}). For example, we refine Goldberg's classical Theorem by proving that χ'≤ max{⌈ρ⌉, Δ+1+(Δ-3)/(g₀+3)}. Our third category of results are characterizations of high chromatic index in general, with particular focus on our approximation results. For example, we completely characterize those multigraphs with χ'> Δ+1+(Δ-3)/(g₀+3). The primary method we use to prove results in this thesis is the method of Tashkinov trees. We first solidify the theory behind this method, and then provide general edge-colouring results depending on Tashkinov trees. We also explore the limits of this method, including the possibility of vertex-colouring graphs which are not line graphs of multigraphs, and the importance of Tashkinov trees with regard to the Seymour-Goldberg Conjecture.

This thesis puts forward the conjecture that for <i>n</i> > 3<i>k</i> with <i>k</i> > 2, the generalized Petersen graph, <i>GP</i>(<i>n,k</i>) is Hamilton-laceable if <i>n</i> is even and <i>k</i> is odd, and it is Hamilton-connected otherwise. We take the first step in the proof of this conjecture by proving the case <i>n</i> = 3<i>k</i> + 1 and <i>k</i> greater than or equal to 1. We do this mainly by means of an induction which takes us from <i>GP</i>(3<i>k</i> + 1, <i>k</i>) to <i>GP</i>(3(<i>k</i> + 2) + 1, <i>k</i> + 2). The induction takes the form of mapping a Hamilton path in the smaller graph piecewise to the larger graph an inserting subpaths we call <i>rotors</i> to obtain a Hamilton path in the larger graph.

This thesis will address several problems in discrete optimization. These problems are considered hard to solve. However, good approximation algorithms for these problems may be helpful in approximating problems in computational biology and computer science. Given an undirected graph G=(V,E) and a family of subsets of vertices S, the minimum crossing spanning tree is a spanning tree where the maximum number of edges crossing any single set in S is minimized, where an edge crosses a set if it has exactly one endpoint in the set. This thesis will present two algorithms for special cases of minimum crossing spanning trees. The first algorithm is for the case where the sets of S are pairwise disjoint. It gives a spanning tree with the maximum crossing of a set being 2OPT+2, where OPT is the maximum crossing for a minimum crossing spanning tree. The second algorithm is for the case where the sets of S form a laminar family. Let b_i be a bound for each S_i in S. If there exists a spanning tree where each set S_i is crossed at most b_i times, the algorithm finds a spanning tree where each set S_i is crossed O(b_i log n) times. From this algorithm, one can get a spanning tree with maximum crossing O(OPT log n). Given an undirected graph G=(V,E), and a family of subsets of vertices S, the minimum crossing perfect matching is a perfect matching where the maximum number of edges crossing any set in S is minimized. A proof will be presented showing that finding a minimum crossing perfect matching is NP-hard, even when the graph is bipartite and the sets of S are pairwise disjoint.

Modelling mechanical systems using symbolic equations can provide many advantages over the more widely-used numerical methods of modelling these systems. The use of symbolic equations produces more efficient models, which can be used for many purposes such as real-time simulation and control. However, the number, complexity, and computational efficiency of these equations is highly dependent on which coordinate set was used to model the system. One method of modelling a mechanism's topology and formulating its symbolic equations is to model the system using a graph-theoretical approach. This approach models mechanisms using a linear graph, from which spanning trees can be used to define a mechanism's coordinate set. This report develops two tree selection algorithms capable of estimating the tree set, and hence coordinate set, that produces models having the fastest forward dynamic simulation times. The first tree selection algorithm is a heuristic-based algorithm that tries to find the coordinate set containing the minimal possible number of modelling variables. Most of this algorithm's heuristics are based on tree selection criteria found in the literature and on observations of a series of benchmark problems. It uses the topology information provided by a system's graph to find the coordinates set for the given system that produce very low simulation times of the system. The second tree selection algorithm developed in this report also uses graph theory. It bases most of its heuristics on observations of one of the methods developed to obtain a mechanical system's symbolic equations using graph theory. This second algorithm also makes use of, and improves upon, a few of the heuristics developed in the first tree selection algorithm. A series of examples for both algorithms will demonstrate the computational efficiency obtained by using the modelling variables found by the automated tree selection algorithms that are proposed in this report.

This thesis explores two infinite families of self-complementary arc-transitive graphs: the familiar Paley graphs and the newly discovered Peisert graphs. After studying both families, we examine a result of Peisert which proves the Paley and Peisert graphs are the only self-complementary arc transitive graphs other than one exceptional graph. Then we consider other families of graphs which share many properties with the Paley and Peisert graphs. In particular, we construct an infinite family of self-complementary strongly regular graphs from affine planes. We also investigate the pseudo-Paley graphs of Weng, Qiu, Wang, and Xiang. Finally, we prove a lower bound on the number of maximal cliques of certain pseudo-Paley graphs, thereby distinguishing them from Paley graphs of the same order.

In this thesis we take a specialized approach to edge-colouring by focusing exclusively on multigraphs with high chromatic index. The bulk of our results can be classified into three categories. First, we prove results which aim to characterize those multigraphs achieving known upper bounds. For example, Goldberg's Theorem says that χ'≤ Δ+1+(Δ-2}/(g₀+1) (where χ' denotes chromatic index, Δ denotes maximum degree, and g₀ denotes odd girth). We characterize this bound by proving that for a connected multigraph G, χ'= Δ+1+(Δ-2}/(g₀+1) if and only if G=μC_g₀ and (g₀+1)|2(μ-1) (where μ denotes maximum edge-multiplicity). Our second category of results are new upper bounds for chromatic index in multigraphs, and accompanying polynomial-time edge-colouring algorithms. Our bounds are all approximations to the famous Seymour-Goldberg Conjecture, which asserts that χ'≤ max{⌈ρ⌉, Δ+1} (where ρ=max{(2|E[S]|)/(|S|-1): S⊆V, |S|≥3 and odd}). For example, we refine Goldberg's classical Theorem by proving that χ'≤ max{⌈ρ⌉, Δ+1+(Δ-3)/(g₀+3)}. Our third category of results are characterizations of high chromatic index in general, with particular focus on our approximation results. For example, we completely characterize those multigraphs with χ'> Δ+1+(Δ-3)/(g₀+3). The primary method we use to prove results in this thesis is the method of Tashkinov trees. We first solidify the theory behind this method, and then provide general edge-colouring results depending on Tashkinov trees. We also explore the limits of this method, including the possibility of vertex-colouring graphs which are not line graphs of multigraphs, and the importance of Tashkinov trees with regard to the Seymour-Goldberg Conjecture.

We discuss the problem of embedding planar graphs onto the plane with pre-specified vertex locations. In particular, we introduce a method for constructing such an embedding for both the case where the mapping from the vertices onto the vertex locations is fixed and the case where this mapping can be chosen. Moreover, the technique we present is sufficiently abstract to generalize to a method for constructing simultaneous planar embeddings with fixed vertex locations. In all cases, we are concerned with minimizing the number of bends per edge in the embeddings we produce. In the case where the mapping is fixed, our technique guarantees embeddings with at most 8n - 7 bends per edge in the worst case and, on average, at most 16/3n - 1 bends per edge. This result improves previously known techniques by a significant constant factor. When the mapping is not pre-specified, our technique guarantees embeddings with at most O(n^(1 - 2^(1-k))) bends per edge in the worst case and, on average, at most O(n^(1 - 1/k)) bends per edge, where k is the number of graphs in the simultaneous embedding. This improves upon the previously known O(n) bound on the number of bends per edge for k at least 2. Moreover, we give an average-case lower bound on the number of bends that has similar asymptotic behaviour to our upper bound.