Ramsey theory has to do with order within disorder. This thesis studies two Ramsey numbers, R(3; 3) and R(3; 4), to see if they can provide insight into finding larger Ramsey numbers. The numbers are studied with the help of computer programs. In the second part of the thesis we try to create a coloring of K45 which lacks monochromatic K5 and where each vertex has an equal degree for both color of edges. The results from studying R(3; 3) and R(3; 4) fail to give any further insight into larger Ramsey numbers. Every coloring of K45 we produce contains a monochromatic K5.

Given a connected oriented graph D, we say that a subset S of V(D) is convex in D if, for every pair of vertices x, y in S, the vertex set of every x-y geodesic (x-y shortest dipath) and y-x geodesic in D is contained in S. The convexity number con (D) of a nontrivial connected oriented graph D is the maximum cardinality of a proper convex set of D. Let S_{C}(K_{n})={con(D)|D is an orientation of K_{n}} and S_{SC}(K_{n})={con(D)|D is a strong orientation of K_{n}}. We show that S_{C}(K_{3})={1,2} and S_{C}(K_{n})={1,3,4,...,n-1} if n >= 4. We also have that S_{SC}(K_{3})={1} and S_{SC}(K_{n})={1,3,4,...,n-2} if n >= 4 . We also show that every triple n, m, k of integers with n >= 5, 3 <= k <= n-2, and n+1 <= m <= n(n-1)/2, there exists a strong connected oriented graph D of order n with |E(D)|=m and con (D)=k.

Given a configuration of pebbles on the vertices of a graph, a pebbling move is defined by removing two pebbles from some vertex and placing one pebble on an adjacent vertex. The cover pebbling number of a graph, γ (G), is the smallest number of pebbles such that through a sequence of pebbling moves, a pebble can eventually be placed on every vertex simultaneously, no matter how the pebbles are initially distributed. We determine Bose-Einstein and Maxwell-Boltzmann cover pebbling thresholds for the complete graph. Also, we show that the cover pebbling decision problem is NP-complete.

Effective visualization of graphs is a powerful tool to help understand the relationships among the graph's underlying objects and to interact with them. Several styles for drawing graphs have emerged over the last three decades. Polyline drawing is a widely used style for drawing graphs, where each node is mapped to a distinct point in the plane and each edge is mapped to a polygonal chain between their corresponding nodes. Some common optimization criteria for such a drawing are defined in terms of area requirement, number of bends per edge, angular resolution, number of distinct line segments, edge crossings, and number of planar layers. In this thesis we develop algorithms for drawing graphs that optimize different aesthetic qualities of the drawing. Our algorithms seek to simultaneously optimize multiple drawing aesthetics, reveal potential trade-offs among them, and improve many previous graph drawing algorithms. We start by exploring probable trade-offs in the context of planar graphs. We prove that every $n$-vertex planar triangulation $G$ with maximum degree $\Delta$ can be drawn with at most $2n+t-3$ segments and $O(8^t \cdot \Delta^{2t})$ area, where $t$ is the number of leaves in a Schnyder tree of $G$. We then show that one can improve the area by allowing the edges to have bends. Since compact drawings often suffer from bad angular resolution, we seek to compute polyline drawings with better angular resolution. We develop a polyline drawing algorithm that is simple and intuitive, yet implies significant improvement over known results. At this point we move our attention to drawing nonplanar graphs. We prove that every thickness-$t$ graph can be drawn on $t$ planar layers with $\min\{O(2^{t/2} \cdot n^{1-1/\beta}), 2.25n +O(1)\}$ bends per edge, where $\beta = 2^{\lceil (t-2)/2 \rceil }$. Previously, the bend complexity, i.e., the number of bends per edge, was not known to be sublinear for $t>2$. We then examine the case when the number of available layers is restricted. The layers may now contain edge crossings. We develop a technique to draw complete graphs on two layers, which improves previous upper bounds on the number of edge crossings in such drawings. / October 2016

In this paper we extend the study of Heffter arrays and the biembedding of graphs on orientable surfaces first discussed by Archdeacon in 2014. We begin with the definitions of Heffter systems, Heffter arrays, and their relationship to orientable biembeddings through current graphs. We then focus on two specific cases. We first prove the existence of embeddings for every K_(6n+1) with every edge on a face of size 3 and a face of size n. We next present partial results for biembedding K_(10n+1) with every edge on a face of size 5 and a face of size n. Finally, we address the more general question of ordering subsets of Z_n take away {0}. We conclude with some open conjectures and further explorations.

Necessary and sufficient conditions are given for the existence of isomorphic decompositions of the complete bipartite graph, the complete graph with a hole, and the λ-fold complete graph into copies of a 4-cycle with a pendant edge.

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No description available.

<p>Denote by $k_t(G)$ the number of cliques of order $t$ in a graph $G$ having $n$ vertices. Let $k_t(n) = \min\{k_t(G)+k_t(\overline{G}) \}$ where $\overline{G}$ denotes the complement of $G$. Let $c_t(n) = {k_t(n)}/{\tbinom{n}{t}}$ and $c_t$ be the limit of $c_t(n)$ for $n$ going to infinity. A 1962 conjecture of Erd\H{o}s stating that $c_t = 2^{1-\tbinom{t}{2}}$ was disproved by Thomason in 1989 for all $t\geq 4$. Tighter counterexamples have been constructed by Jagger, {\v S}{\v t}ov{\' \i}{\v c}ek and Thomason in 1996, by Thomason for $t\leq 6$ in 1997, and by Franek for $t=6$ in 2002. Further tightenings $t=6,7$ and $8$ was recently obtained by Deza, Franek, and Liu.</p> <p>We investigate the computational framework used by Deza, Franek, and Liu. In particular, we present the benefits and limitations of different parallel computer memory architectures and parallel programming models. We propose a functional decomposition approach which is implemented in C++ with POSIX thread (Pthread) libraries for multi-threading. Computational benchmarking on the parallelized framework and a performance analysis including a comparison with the original computational framework are presented.</p> / Master of Science (MSc)

Nesta dissertacao estudamos problemas de empacotamento de arvores em grafos, com enfase no caso de grafos completos. Denotamos por Ti uma arvore de ordem i. Dizemos que existe um empacotamento de arvores T1, . . . , Tn num grafo G se e possivel encontrar em G subgrafos H1, . . . , Hn, dois a dois disjuntos nas arestas, tais que Hi e isomorfo a Ti. Em 1976, A. Gyarfas e J. Lehel levantaram a seguinte questao, que conjecturaram ter uma resposta positiva: e possivel empaco- tar qualquer sequencia de arvores T1, . . . , Tn no Kn? Esta dissertacao tem como tema principal os estudos realizados por diversos pesquisadores na busca de uma resposta para esta pergunta, que permanece ainda em aberto. Tendo em vista a dificuldade para tratar esta questao, surge natural- mente a pergunta sobre a existencia de classes de arvores para as quais a resposta e afirmativa. Nessa linha, existem diversos resultados positivos, como por exemplo quando queremos empacotar estrelas e caminhos, ou estrelas e biestrelas. Por outro lado, em vez de restringir a classe das arvores, faz sentido restringir o tamanho da sequencia e reformular a pergunta. Por exemplo, dado s < n, e possivel empacotar qualquer sequencia de arvores T1, . . . , Ts no Kn? Em 1983, Bollobas mostrou ? que a resposta e afirmativa se s <= n / sqrt(2). Na primeira parte deste trabalho focamos nosso estudo em questoes desse tipo. Na segunda parte desta dissertacao investigamos algumas conjecturas que foram motivadas pela pergunta levantada por Gyarfas & Lehel. Por exemplo, Hobbs, Bourgeois e Kasiraj formularam a seguinte questao: para n par, e possivel empacotar qualquer sequencia de arvores T1, . . . , Tn no grafo bipartido Kn/2,n-1? Para essa pergunta apresentamos alguns resultados conhecidos analogos aos obtidos para a conjectura de Gyarfas & Lehel. Mais recentemente, Gerbner, Keszegh e Palmer estudaram a seguinte generalizacao da conjectura original: e possivel empacotar qualquer sequencia de arvores T1, . . . , Tk num grafo k-cromatico? Neste trabalho estudamos essas e outras questoes relacionadas e apresentamos os principais resultados que encontramos na literatura. / In this dissertation we address the problem of packing trees into graphs, with focus on complete graphs. We denote by Ti a tree of order i. We say that there exists a packing of trees T1,...,Tn in a graph G if its possible to find in G pairwise edge-disjoint subgraphs H1, . . . , Hn such that Hi is isomorphic to Ti. In 1976, A. Gyárfás and J. Lehel raised the following question, that they conjectured to have an affirmative answer: is it possible to pack any sequence of trees T1, . . . , Tn into the complete graph Kn? In this dissertation, we study a number of contributions made by various researchers in the search for an answer to this question, that is still open. In view of the difficulty of this question, it is natural to look for the existence of classes of trees for which the answer is affirmative. In this direction, some positive results have been found, as for example, when the sequences of trees are restricted to stars and paths, or stars and bistars. On the other hand, instead of restricting the classes of trees, it makes sense to restrict the length of the sequence and reformulate the question. For example, given s < n, is it possible to pack any sequence of trees T1, . . . , Ts into Kn? In 1983, Bollobás showed that the answer is affirmative if s <= n/sqrt(2). In the first part of this work, we focus on such kind of questions. In the second part of this dissertation we investigate some other conjectures that were motivated by the conjecture of Gyárfás & Lehel. For example, Hobbs, Bourgeois and Kasiraj formulated the following question: For n even, is it possible to pack any sequence of trees T1, . . . , Tn into the complete bipartite graph Kn/2,n-1? For this question, we present some known results analogous to those obtained for the conjecture of Gyárfás & Lehel. More recently, Gerbner, Keszegh and Palmer studied the following generalization of the of former conjecture: is it possible to pack any sequence of trees T1,...,Tk in a k-chromatic graph? In this dissertation, we study this and other related questions and present the main results we found in the literature.