A demonstração feita por Erdos da existência de grafos com cintura e número cromático grandes é uma das primeiras aplicações do método probabilístico. Essa demonstração fornece um limite para o número de vértices de um grafo desse tipo, que é exponencial na cintura quando o número cromático é fixado. O foco deste texto, no entanto, são as construções determinísticas de grafos com cintura e número cromático grandes e os números de vértices dos grafos obtidos. As construções elementares conhecidas fornecem apenas grafos com um número Ackermanniano de vértices. O texto começa com uma breve repetição das demonstrações probabilísticas da existência de grafos e hipergrafos com cintura e número cromático grandes. Depois, a busca por construções determinísticas é motivada apresentando-se algumas construções para o caso particular de grafos livres de triângulo e com número cromático grande. São construídos os grafos de Tutte, Zykov, Mycielski e Kneser, os grafos de shift e os de planos projetivos finitos. Os números de vértices dessas construções são computados e comparados. De fato, a construção a partir de planos projetivos finitos tem um número polinomial de vértices. A parte principal do texto são as construções de grafos e hipergrafos com cintura e número cromático grandes. A primeira construção apresentada foi feita por Kriz. Ela foi a primeira construção para grafos com cintura e número cromático grandes que não envolvia hipergrafos. A segunda construção apresentada foi feita por Nesetril e Rödl. Essa construção antecede a de Kriz. Ela utiliza a amalgamação entre grafos e hipergrafos para obter um hipergrafo uniforme com cintura e número cromático grandes. A terceira e última construção apresentada foi encontrada por Alon, Kostochka, Reiniger, West e Zhu. Essa construção consegue obter hipergrafos uniformes com cintura e número cromático grandes diretamente a partir de um grafo, que é uma certa árvore aumentada. Em particular, essa construção obtém grafos com cintura e número cromático grandes sem envolver hipergrafos. Os números de vértices dos hipergrafos obtidos por essas construções são computados e comparados. / The proof by Erdos of the existence of graphs with high girth and high chromatic number is one of the first applications of the probabilistic method. This proof gives a bound on the number of vertices of such graphs, which is exponential on the girth if the chromatic number is fixed. The focus of this text is however on the deterministic construction of graphs with high girth and high chromatic number and on the number of vertices of the obtained graphs. The elementary known constructions can only give us graphs with an Ackermannian number of vertices. We begin by briefly repeating the probabilistic proofs of the existence of graphs and hypergraphs with high girth and high chromatic number. Then we motivate the search for deterministic constructions of such graphs by showing some constructions for the special case of triangle-free graphs with high chromatic number. We construct Tutte, Zykov, Mycielski and Kneser graphs, the shift graphs and graphs built from finite projective planes. We count and compare the number of vertices of the graphs obtained by each of these constructions. In fact, the construction based on finite projective planes gives us graphs with a polynomial number of vertices. The main part of the text consists of constructions of graphs and hypergraphs with high girth and high chromatic number. The first construction we present is due to Kriz. This was the first construction to give graphs with high girth and high chromatic number without using hypergraphs. The second construction we present is due to Nesetril and Rödl. This construction precedes the one by Kriz. It uses amalgamations between graphs and hypergraphs to obtain uniform hypergraphs with high girth and high chromatic number. The third and last construction we show was found by Alon, Kostochka, Reiniger, West and Zhu. This construction manages to build uniform hypergraphs with high girth and high chromatic number directly from a single graph, which is an augmented-tree. In particular, it constructs graphs with high girth and high chromatic number without using hypergraphs. We count and compare the number of vertices of the hypergraphs obtained by these constructions.

Grid Representations and the Chromatic Number Martin Balko August 2, 2012 Department: Department of Applied Mathematics Supervisor: doc. RNDr. Pavel Valtr Dr. Supervisor's email address: valtr@kam.mff.cuni.cz Abstract In the thesis we study grid drawings of graphs and their connections with graph colorings. A grid drawing of a graph maps vertices to distinct points of the grid Zd and edges to line segments that avoid grid points representing other vertices. We show that a graph G is qd -colorable, d, q ≥ 2, if and only if there is a grid drawing of G in Zd in which no line segment intersects more than q grid points. Second, we study grid drawings with bounded number of columns, introducing some new NP- complete problems. We also show a sharp lower bound on the area of plane grid drawings of balanced complete k-partite graphs, proving a conjecture of David R. Wood. Finally, we show that any planar graph has a planar grid drawing where every line segment contains exactly two grid points. This result proves conjectures of D. Flores Pe˝naloza and F. J. Zaragoza Martinez. Keywords: graph representations, grid, chromatic number, plane

The alphabet overlap graph is a modification of the well known de Bruijn graph. De Bruijn graphs have been highly studied and hence many properties of these graphs have been determined. However, very little is known about alphabet overlap graphs. In this work we determine the chromatic number for a special case of these graphs. We define the alphabet overlap graph by G = AO(a, k, t, where a, k and t are positive integers such that 0 ≤ t ≤ k. The vertex set of G is the set of all k-letter sequences over an alphabet of size a. Also there is an edge between vertices u, v if and only if the last t letters in u match the first t letters in v or the first t letters in u match the last t letters in v. We consider the chromatic number for the AO(a, k, t graphs when k > 2, t = k - 1 and a = 2.

The entropy of a graph is a functional depending both on the graph itself and on a probability distribution on its vertex set. This graph functional originated from the problem of source coding in information theory and was introduced by J. K\"{o}rner in 1973. Although the notion of graph entropy has its roots in information theory, it was proved to be closely related to some classical and frequently studied graph theoretic concepts. For example, it provides an equivalent definition for a graph to be perfect and it can also be applied to obtain lower bounds in graph covering problems. In this thesis, we review and investigate three equivalent definitions of graph entropy and its basic properties. Minimum entropy colouring of a graph was proposed by N. Alon in 1996. We study minimum entropy colouring and its relation to graph entropy. We also discuss the relationship between the entropy and the fractional chromatic number of a graph which was already established in the literature. A graph $G$ is called \emph{symmetric with respect to a functional $F_G(P)$} defined on the set of all the probability distributions on its vertex set if the distribution $P^*$ maximizing $F_G(P)$ is uniform on $V(G)$. Using the combinatorial definition of the entropy of a graph in terms of its vertex packing polytope and the relationship between the graph entropy and fractional chromatic number, we prove that vertex transitive graphs are symmetric with respect to graph entropy. Furthermore, we show that a bipartite graph is symmetric with respect to graph entropy if and only if it has a perfect matching. As a generalization of this result, we characterize some classes of symmetric perfect graphs with respect to graph entropy. Finally, we prove that the line graph of every bridgeless cubic graph is symmetric with respect to graph entropy.

A demonstração feita por Erdos da existência de grafos com cintura e número cromático grandes é uma das primeiras aplicações do método probabilístico. Essa demonstração fornece um limite para o número de vértices de um grafo desse tipo, que é exponencial na cintura quando o número cromático é fixado. O foco deste texto, no entanto, são as construções determinísticas de grafos com cintura e número cromático grandes e os números de vértices dos grafos obtidos. As construções elementares conhecidas fornecem apenas grafos com um número Ackermanniano de vértices. O texto começa com uma breve repetição das demonstrações probabilísticas da existência de grafos e hipergrafos com cintura e número cromático grandes. Depois, a busca por construções determinísticas é motivada apresentando-se algumas construções para o caso particular de grafos livres de triângulo e com número cromático grande. São construídos os grafos de Tutte, Zykov, Mycielski e Kneser, os grafos de shift e os de planos projetivos finitos. Os números de vértices dessas construções são computados e comparados. De fato, a construção a partir de planos projetivos finitos tem um número polinomial de vértices. A parte principal do texto são as construções de grafos e hipergrafos com cintura e número cromático grandes. A primeira construção apresentada foi feita por Kriz. Ela foi a primeira construção para grafos com cintura e número cromático grandes que não envolvia hipergrafos. A segunda construção apresentada foi feita por Nesetril e Rödl. Essa construção antecede a de Kriz. Ela utiliza a amalgamação entre grafos e hipergrafos para obter um hipergrafo uniforme com cintura e número cromático grandes. A terceira e última construção apresentada foi encontrada por Alon, Kostochka, Reiniger, West e Zhu. Essa construção consegue obter hipergrafos uniformes com cintura e número cromático grandes diretamente a partir de um grafo, que é uma certa árvore aumentada. Em particular, essa construção obtém grafos com cintura e número cromático grandes sem envolver hipergrafos. Os números de vértices dos hipergrafos obtidos por essas construções são computados e comparados. / The proof by Erdos of the existence of graphs with high girth and high chromatic number is one of the first applications of the probabilistic method. This proof gives a bound on the number of vertices of such graphs, which is exponential on the girth if the chromatic number is fixed. The focus of this text is however on the deterministic construction of graphs with high girth and high chromatic number and on the number of vertices of the obtained graphs. The elementary known constructions can only give us graphs with an Ackermannian number of vertices. We begin by briefly repeating the probabilistic proofs of the existence of graphs and hypergraphs with high girth and high chromatic number. Then we motivate the search for deterministic constructions of such graphs by showing some constructions for the special case of triangle-free graphs with high chromatic number. We construct Tutte, Zykov, Mycielski and Kneser graphs, the shift graphs and graphs built from finite projective planes. We count and compare the number of vertices of the graphs obtained by each of these constructions. In fact, the construction based on finite projective planes gives us graphs with a polynomial number of vertices. The main part of the text consists of constructions of graphs and hypergraphs with high girth and high chromatic number. The first construction we present is due to Kriz. This was the first construction to give graphs with high girth and high chromatic number without using hypergraphs. The second construction we present is due to Nesetril and Rödl. This construction precedes the one by Kriz. It uses amalgamations between graphs and hypergraphs to obtain uniform hypergraphs with high girth and high chromatic number. The third and last construction we show was found by Alon, Kostochka, Reiniger, West and Zhu. This construction manages to build uniform hypergraphs with high girth and high chromatic number directly from a single graph, which is an augmented-tree. In particular, it constructs graphs with high girth and high chromatic number without using hypergraphs. We count and compare the number of vertices of the hypergraphs obtained by these constructions.

We present a novel theorem of Borel Combinatorics that sheds light on the types of continuous functions that can be defined on the Cantor space. We specifically consider the part X=F(2ᴳ) from the Cantor space, where the group G is the additive group of integer pairs ℤ². That is, X is the set of aperiodic {0,1} labelings of the two-dimensional infinite lattice graph. We give X the Bernoulli shift action, and this action induces a graph on X in which each connected component is again a two-dimensional lattice graph. It is folklore that no continuous (indeed, Borel) function provides a two-coloring of the graph on X, despite the fact that any finite subgraph of X is bipartite. Our main result offers a much more complete analysis of continuous functions on this space. We construct a countable collection of finite graphs, each consisting of twelve "tiles", such that for any property P (such as "two-coloring") that is locally recognizable in the proper sense, a continuous function with property P exists on X if and only if a function with a corresponding property P' exists on one of the graphs in the collection. We present the theorem, and give several applications.

No description available.

A (closed) neighborhood-restricted [≤2]-coloring of a graph G is an assignment of colors to the vertices of G such that no more than two colors are assigned in any closed neighborhood, that is, for every vertex v in G, the vertex v and its neighbors are in at most two different color classes. The [≤2]-achromatic number is defined as the maximum number of colors in any [≤2]-coloring of G. We study the [≤2]-achromatic number. In particular, we improve a known upper bound and characterize the extremal graphs for some other known bounds.

Let G = (V (G), E(G)) be a graph, and let k ≥ 1 be an integer. A set S ⊆⊆ V (G) is called a global offensive k-alliance if ΙN(ν) ∩ SΙ ≥ ΙN(ν)-SΙ+k for every ν ε V (G)-S, where N(v) is the neighborhood of ν. The global offensive k-alliance number γko(G) is the minimum cardinality of a global o ensive k-alliance in G. We present di erent bounds on γko(G) in terms of order, maximum degree, independence number, chromatic number and minimum degree.

An E-graph is constructed by replacing each edge in a core graph G with a copy of a graph H. An important property of E-graphs is that their invariant values can be determined from parameters of the original graphs G and H. We determine chromatic number, clique number, vertex and edge cover numbers, vertex and edge independence numbers, circumference, and girth of E-graphs. A characterization of hamiltonian E-graphs is also given.