In the ordering game on a graph G, Alice and Bob take turns placing the vertices of G into a linear ordering. The score of the game is the maximum number of neighbors that any vertex has before it in the ordering. Alice's goal in the ordering game is to minimize the score, while Bob's goal is to maximize it. The game coloring number is the least score that Alice can always guarantee in the ordering game, no matter how Bob plays. This paper examines what happens to the game coloring number if Alice or Bob skip turns on the ordering game. Preliminary definitions and examples are provided to give context to the ordering game. These include a polynomial time algorithm to compute the coloring number, a non-competitive version of the game coloring number. The notion of the preordered game is introduced as the primary tool of the paper, along with its naturally defined preordered game coloring number. To emphasize the complex relationship between the coloring number and the preordered game coloring number, a non-polynomial time strategy is given to Alice and Bob that yields the preordered game coloring number on any graph. Additionally, the preordered game coloring number is shown to be monotonic, one of the most useful properties for turn-skipping. Techniques developed throughout the paper are then used to determine that Alice cannot reduce the score and Bob cannot improve the score by skipping any number of their respective turns. This paper can hopefully be used as a stepping stone towards bounding the score on graphs when vertices are removed, as well as in deciphering further properties of the asymmetric marking game.

Since the seminal work of Tur ́an, the forbidden subgraph problem has been among the central questions in extremal graph theory. Let ex(n;F) be the smallest number m such that any graph on n vertices with m edges contains F as a subgraph. Then the forbidden subgraph problem asks to find ex(n; F ) for various graphs F . The question can be further generalized by asking for the extreme values of other graph parameters like minimum degree, maximum degree, or connectivity. We call this type of question a Tura ́n-type problem. In this thesis, we will study Tura ́n-type problems and their variants for graphs and hypergraphs.

Chapter 2 contains a Tura ́n-type problem for cycles in dense graphs. The main result in this chapter gives a tight bound for the minimum degree of a graph which guarantees existence of disjoint cycles in the case of dense graphs. This, in particular, answers in the affirmative a question of Faudree, Gould, Jacobson and Magnant in the case of dense graphs.

In Chapter 3, similar problems for trees are investigated. Recently, Faudree, Gould, Jacobson and West studied the minimum degree conditions for the existence of certain spanning caterpillars. They proved certain bounds that guarantee existence of spanning caterpillars. The main result in Chapter 3 significantly improves their result and answers one of their questions by proving a tight minimum degree bound for the existence of such structures.

Chapter 4 includes another Tur ́an-type problem for loose paths of length three in a 3-graph. As a corollary, an upper bound for the multi-color Ramsey number for the loose path of length three in a 3-graph is achieved.

This project created a tool for visualizing constructive solid geometry (CSG) using an HTC Vive virtual reality
headset. This tool provides functionality for surface triangulation
of a variety of three-dimensional primitive solids. Then with those
solids it can perform the core CSG operations—intersection,
union and complement—to create more complex objects. This
tool also parses in Silo data files to allow the visualization
of scientific models like the Annular Core Research Reactor.
This project is useful for both education and visualization. This
project will be used by scientists to visualize and understand
their simulation results, and used as a museum exhibit to engage
the next generation of scientists in computer modeling.

Analytic research on basketball games is growing quickly, specifically in the National Basketball Association. This paper explored the development of this analytic research and discovered that there has been a focus on individual player metrics and a dearth of quantitative team characterizations and evaluations. Consequently, this paper continued the exploratory research of Fewell and Armbruster's "Basketball teams as strategic networks" (2012), which modeled basketball teams as networks and used metrics to characterize team strategy in the NBA's 2010 playoffs. Individual players and outcomes were nodes and passes and actions were the links. This paper used data that was recorded from playoff games of the two 2012 NBA finalists: the Miami Heat and the Oklahoma City Thunder. The same metrics that Fewell and Armbruster used were explained, then calculated using this data. The offensive networks of these two teams during the playoffs were analyzed and interpreted by using other data and qualitative characterization of the teams' strategies; the paper found that the calculated metrics largely matched with our qualitative characterizations of the teams. The validity of the metrics in this paper and Fewell and Armbruster's paper was then discussed, and modeling basketball teams as multiple-order Markov chains rather than as networks was explored.

This paper focuses on the Szemerédi regularity lemma, a result in the field of extremal graph theory. The lemma says that every graph can be partitioned into bounded equal parts such that most edges of the graph span these partitions, and these edges are distributed in a fairly uniform way. Definitions and notation will be established, leading to explorations of three proofs of the regularity lemma. These are a version of the original proof, a Pythagoras proof utilizing elemental geometry, and a proof utilizing concepts of spectral graph theory. This paper is intended to supplement the proofs with background information about the concepts utilized. Furthermore, it is the hope that this paper will serve as another resource for students and others to begin study of the regularity lemma.

Higher-rank graphs, or k-graphs, are higher-dimensional analogues of directed graphs, and as with ordinary directed graphs, there are various C*-algebraic objects that can be associated with them. This thesis adopts a functorial approach to study the relationship between k-graphs and their associated C*-algebras. In particular, two functors are given between appropriate categories of higher-rank graphs and the category of C*-algebras, one for Toeplitz algebras and one for Cuntz-Krieger algebras. Additionally, the Cayley graphs of finitely generated groups are used to define a class of k-graphs, and a functor is then given from a category of finitely generated groups to the category of C*-algebras. Finally, functoriality is investigated for product systems of C*-correspondences associated to k-graphs. Additional results concerning the structural consequences of functoriality, properties of the functors, and combinatorial aspects of k-graphs are also included throughout.

Measuring node centrality is a critical common denominator behind many important graph mining tasks. While the existing literature offers a wealth of different node centrality measures, it remains a daunting task on how to intervene the node centrality in a desired way. In this thesis, we study the problem of minimizing the centrality of one or more target nodes by edge operation. The heart of the proposed method is an accurate and efficient algorithm to estimate the impact of edge deletion on the spectrum of the underlying network, based on the observation that the edge deletion is essentially a local, sparse perturbation to the original network. Extensive experiments are conducted on a diverse set of real networks to demonstrate the effectiveness, efficiency and scalability of our approach. In particular, it is average of 260.95%, in terms of minimizing eigen-centrality, better than the standard matrix-perturbation based algorithm, with lower time complexity.

A tiling is a collection of vertex disjoint subgraphs called tiles. If the tiles are all isomorphic to a graph $H$ then the tiling is an $H$-tiling. If a graph $G$ has an $H$-tiling which covers all of the vertices of $G$ then the $H$-tiling is a perfect $H$-tiling or an $H$-factor. A goal of this study is to extend theorems on sufficient minimum degree conditions for perfect tilings in graphs to directed graphs. Corrádi and Hajnal proved that every graph $G$ on $3k$ vertices with minimum degree $delta(G)ge2k$ has a $K_3$-factor, where $K_s$ is the complete graph on $s$ vertices. The following theorem extends this result to directed graphs: If $D$ is a directed graph on $3k$ vertices with minimum total degree $delta(D)ge4k-1$ then $D$ can be partitioned into $k$ parts each of size $3$ so that all of parts contain a transitive triangle and $k-1$ of the parts also contain a cyclic triangle. The total degree of a vertex $v$ is the sum of $d^-(v)$ the in-degree and $d^+(v)$ the out-degree of $v$. Note that both orientations of $C_3$ are considered: the transitive triangle and the cyclic triangle. The theorem is best possible in that there are digraphs that meet the minimum degree requirement but have no cyclic triangle factor. The possibility of added a connectivity requirement to ensure a cycle triangle factor is also explored. Hajnal and Szemerédi proved that if $G$ is a graph on $sk$ vertices and $delta(G)ge(s-1)k$ then $G$ contains a $K_s$-factor. As a possible extension of this celebrated theorem to directed graphs it is proved that if $D$ is a directed graph on $sk$ vertices with $delta(D)ge2(s-1)k-1$ then $D$ contains $k$ disjoint transitive tournaments on $s$ vertices. We also discuss tiling directed graph with other tournaments. This study also explores minimum total degree conditions for perfect directed cycle tilings and sufficient semi-degree conditions for a directed graph to contain an anti-directed Hamilton cycle. The semi-degree of a vertex $v$ is $min{d^+(v), d^-(v)}$ and an anti-directed Hamilton cycle is a spanning cycle in which no pair of consecutive edges form a directed path.

A central concept of combinatorics is partitioning structures with given constraints. Partitions of on-line posets and on-line graphs, which are dynamic versions of the more familiar static structures posets and graphs, are examined. In the on-line setting, vertices are continually added to a poset or graph while a chain partition or coloring (respectively) is maintained. %The optima of the static cases cannot be achieved in the on-line setting. Both upper and lower bounds for the optimum of the number of chains needed to partition a width $w$ on-line poset exist. Kierstead's upper bound of $\frac{5^w-1}{4}$ was improved to $w^{14 \lg w}$ by Bosek and Krawczyk. This is improved to $w^{3+6.5 \lg w}$ by employing the First-Fit algorithm on a family of restricted posets (expanding on the work of Bosek and Krawczyk) . Namely, the family of ladder-free posets where the $m$-ladder is the transitive closure of the union of two incomparable chains $x_1\le\dots\le x_m$, $y_1\le\dots\le y_m$ and the set of comparabilities $\{x_1\le y_1,\dots, x_m\le y_m\}$. No upper bound on the number of colors needed to color a general on-line graph exists. To lay this fact plain, the performance of on-line coloring of trees is shown to be particularly problematic. There are trees that require $n$ colors to color on-line for any positive integer $n$. Furthermore, there are trees that usually require many colors to color on-line even if they are presented without any particular strategy. For restricted families of graphs, upper and lower bounds for the optimum number of colors needed to maintain an on-line coloring exist. In particular, circular arc graphs can be colored on-line using less than 8 times the optimum number from the static case. This follows from the work of Pemmaraju, Raman, and Varadarajan in on-line coloring of interval graphs.

In a large network (graph) it would be desirable to guarantee the existence of some local property based only on global knowledge of the network. Consider the following classical example: how many connections are necessary to guarantee that the network contains three nodes which are pairwise adjacent? It turns out that more than n^2/4 connections are needed, and no smaller number will suffice in general. Problems of this type fall into the category of ``extremal graph theory.'' Generally speaking, extremal graph theory is the study of how global parameters of a graph are related to local properties. This dissertation deals with the relationship between minimum degree conditions of a host graph G and the property that G contains a specified spanning subgraph (or class of subgraphs). The goal is to find the optimal minimum degree which guarantees the existence of a desired spanning subgraph. This goal is achieved in four different settings, with the main tools being Szemeredi's Regularity Lemma; the Blow-up Lemma of Komlos, Sarkozy, and Szemeredi; and some basic probabilistic techniques.