A simple algorithm for sampling colourings of G(N,D/N) up to Gibbs Uniqueness threshold

Approximate random $k$-colouring of a graph G is a well studied problem in computer science and statistical physics. It amounts to constructing a k-colouring of G which is distributed close to Gibbs distribution in polynomial time. Here, we deal with the problem when the underlying graph is an instance of Erdos-Renyi random graph G(n,d/n), where d is a sufficiently large constant. We propose a novel efficient algorithm for approximate random k-colouring G(n,d/n) for any k>(1+\epsilon)d. To be more specific, with probability at least 1-n^{-\Omega(1)} over the input instances G(n,d/n) and for k>(1+\epsilon)d, the algorithm returns a k-colouring which is distributed within total variation distance n^{-\Omega(1)} from the Gibbs distribution of the input graph instance. The algorithm we propose is neither a MCMC one nor inspired by the message passing algorithms proposed by statistical physicists. Roughly the idea is as follows: Initially we remove sufficiently many edges of the input graph. This results in a ``simple graph" which can be $k$-coloured randomly efficiently. The algorithm colours randomly this simple graph. Then it puts back the removed edges one by one. Every time a new edge is put back the algorithm updates the colouring of the graph so that the colouring remains random. The performance of the algorithm depends heavily on certain spatial correlation decay properties of the Gibbs distribution

Rotulações graciosas e rotulações semifortes em grafos

Orientador: Christiane Neme CamposTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Três problemas de rotulação em grafos são investigados nesta tese: a Conjetura das Árvores Graciosas, a Conjetura 1,2,3 e a Conjetura 1,2. Uma rotulação graciosa de um grafo simples G=(V(G),E(G)) é uma função injetora f de V(G) em {0,...,|E(G)|} tal que {|f(u)-f(v)|: uv em E(G)} = {1,...,|E(G)|}. A Conjetura das Árvores Graciosas, proposta por Rosa e Kotzig em 1967, afirma que toda árvore possui uma rotulação graciosa. Um problema relacionado à Conjetura das Árvores Graciosas consiste em determinar se, para todo vértice v de uma árvore T, existe uma rotulação graciosa de T que atribui o rótulo 0 a v. Árvores com tal propriedade são denominadas 0-rotativas. Nesta tese, apresentamos famílias infinitas de caterpillars 0-rotativos. Nossos resultados reforçam a conjetura de que todo caterpillar com diâmetro pelo menos cinco é 0-rotativo. Também investigamos uma rotulação graciosa mais restrita, chamada rotulação-alpha. Uma rotulação graciosa f de G é uma rotulação-alpha se existir um inteiro k, 0 <= k <= |E(G)|, tal que, para toda aresta uv em E(G), f(u) <= k < f(v) ou f(v) <= k < f(u). Nesta tese, apresentamos duas famílias de lobsters com grau máximo três que possuem rotulações-alpha. Nossos resultados contribuem para uma caracterização de todos os lobsters com grau máximo três que possuem rotulações-alpha. Na segunda parte desta tese, investigamos generalizações da Conjetura 1,2,3 e da Conjetura 1,2. Dado um grafo simples G = (V(G),E(G)) e um subconjunto L dos números reais, dizemos que uma função f de E(G) em L é uma L-rotulação de arestas de G e dizemos que uma função f da união de V(G) com E(G) em L é uma L-rotulação total de G. Para todo vértice v de G, a cor de v, C(v), é definida como a soma dos rótulos das arestas incidentes em v, se f for uma L-rotulação de arestas de G. Se f for uma L-rotulação total, C(v) é a soma dos rótulos das arestas incidentes no vértice v mais o valor f(v). O par (f,C) é uma L-rotulação de arestas semiforte (L-rotulação total semiforte) se f for uma rotulação de arestas (rotulação total) e C(u) for diferente de C(v) para quaisquer dois vértices adjacentes u,v de G. A Conjetura 1,2,3, proposta por Karónski et al. em 2004, afirma que todo grafo simples e conexo com pelo menos três vértices possui uma {1,2,3}-rotulação de arestas semiforte. A Conjetura 1,2, proposta por Przybylo e Wozniak em 2010, afirma que todo grafo simples possui uma {1,2}-rotulação total semiforte. Sejam a,b,c três reais distintos. Nesta tese, nós investigamos {a,b,c}-rotulações de arestas semifortes e {a,b}-rotulações totais semifortes para cinco famílias de grafos: as potências de caminho, as potências de ciclo, os grafos split, os grafos cobipartidos regulares e os grafos multipartidos completos. Provamos que essas famílias possuem tais rotulações para alguns valores reais a,b,c. Como corolário de nossos resultados, obtemos que a Conjetura 1,2,3 e a Conjetura 1,2 são verdadeiras para essas famílias. Além disso, também mostramos que nossos resultados em rotulações de arestas semifortes implicam resultados similares para outro problema de rotulação de arestas relacionadoAbstract: This thesis addresses three labelling problems on graphs: the Graceful Tree Conjecture, the 1,2,3-Conjecture, and the 1,2-Conjecture. A graceful labelling of a simple graph G=(V(G),E(G)) is an injective function f from V(G) to {0,...,|E(G)|} such that {|f(u)-f(v)| : uv in E(G)} = {1,...,|E(G)|}. The Graceful Tree Conjecture, posed by Rosa and Kotzig in 1967, states that every tree has a graceful labelling. A problem connected with the Graceful Tree Conjecture consists of determining whether, for every vertex v of a tree T, there exists a graceful labelling of T that assigns label 0 to v. Trees with such a property are called 0-rotatable. In this thesis, we present infinite families of 0-rotatable caterpillars. Our results reinforce a conjecture that states that every caterpillar with diameter at least five is 0-rotatable. We also investigate a stronger type of graceful labelling, called alpha-labelling. A graceful labelling f of G is an alpha-labelling if there exists an integer k with 0<= k <= |E(G)| such that, for each edge uv in E(G), either f(u) <= k < f(v) or f(v) <= k < f(u). In this thesis, we prove that the following families of lobsters have alpha-labellings: lobsters with maximum degree three, without Y-legs and with at most one forbidden ending; and lobsters T with a perfect matching M such that the contracted tree T/M has a balanced bipartition. These results point towards a characterization of all lobsters with maximum degree three that have alpha-labellings. In the second part of the thesis, we focus on generalizations of the 1,2,3-Conjecture and the 1,2-Conjecture. Given a simple graph G=(V(G),E(G)) and a subset L of real numbers, we call a function f from E(G) to L an L-edge-labelling of G, and we call a function f from V(G) union E(G) to L an L-total-labelling of G. For each vertex v of G, the colour of v, C(v), is defined as the sum of the labels of its incident edges, if f is an L-edge-labelling. If f is an L-total-labelling, C(v) is the sum of the labels of the edges incident with vertex v plus the label f(v). The pair (f,C) is a neighbour-distinguishing L-edge-labelling (neighbour-distinguishing L-total-labelling) if f is an edge-labelling (total-labelling) and C(u) is different from C(v), for every edge uv in E(G). The 1,2,3-Conjecture, posed by Kar\'onski et al. in 2004, states that every connected simple graph with at least three vertices has a neighbour-distinguishing {1,2,3}-edge-labelling. The 1,2-Conjecture, posed by Przybylo and Wozniak in 2010, states that every simple graph has a neighbour-distinguishing {1,2}-total-labelling. Let a,b,c be distinct real numbers. In this thesis, we investigate neighbour-distinguishing {a,b,c}-edge-labellings and neighbour-distinguishing {a,b}-total labellings for five families of graphs: powers of paths, powers of cycles, split graphs, regular cobipartite graphs and complete multipartite graphs. We prove that these families have such labellings for some real values a, b, and c. As a corollary of our results, we obtain that the 1,2,3-Conjecture and the 1,2-Conjecture are true for these families. Furthermore, we also show that our results on neighbour-distinguishing edge-labellings imply similar results on a closely related problem called detectable edge-labelling of graphsDoutoradoCiência da ComputaçãoDoutor em Ciência da Computação2014/16861-8FAPESPCAPE

Analyzing the Performance of the Fuzzy Inference System in Decision Making

Inference systems that are fuzzy, It is common practise to make use of models such as the Mamdani and Sugeno models in order to take into consideration the presence of uncertainty and imprecision in the decision-making process. MATLAB is a well-known programming environment that provides persons who are interested in developing and implementing fuzzy inference systems with the necessary tools and strategies to accomplish their goals. In order to evaluate Diabetes Mellitus (DM), the Mamdani and Sugeno fuzzy inference systems have been developed in MATLAB. This abstract provides a brief summary of how the evaluation was carried out.The Mamdani model provides a description of uncertain data through the utilisation of fuzzy sets and is founded on language standards. Through the utilisation of the Fuzzy Logic Toolbox, users of MATLAB are able to rapidly construct and simulate Mamdani fuzzy systems. Membership functions, fuzzy rule sets, simulations, and Mamdani system optimisations can all be defined and created by users without any restrictions that are placed on them. The visualisation options that are available in MATLAB, such as the surface plot and the rules plot, help to make the behaviour of the system more understandable.For the purpose of producing inferences and predictions, the Sugeno model, also known as the Takagi-Sugeno-Kang (TSK) model, combines fuzzy principles with linear calculations. It is possible to implement Sugeno fuzzy systems by utilising the Fuzzy Logic Toolbox that is included in MATLAB. The user is able to create the linear functions that are associated to each rule after the information regarding the input-output relationships has been specified through the utilisation of linguistic variables and membership functions. Evaluation, simulation, and visualisation of the rule surfaces and output response curves of Sugeno fuzzy systems can be accomplished in MATLAB in a short amount of time. To summarise, the Mamdani and Sugeno fuzzy inference systems are capable of being constructed in an efficient manner by utilising MATLAB. There is software available for rapid system modelling, simulation, and analysis applications. The techniques of fuzzy logic that are available in MATLAB can be utilised by both professionals and academics in order to address the issue of uncertainty and imprecision in decision-making processes

Algebraic number-theoretic properties of graph and matroid polynomials

PhDThis thesis is an investigation into the algebraic number-theoretical properties of certain polynomial invariants of graphs and matroids. The bulk of the work concerns chromatic polynomials of graphs, and was motivated by two conjectures proposed during a 2008 Newton Institute workshop on combinatorics and statistical mechanics. The first of these predicts that, given any algebraic integer, there is some natural number such that the sum of the two is the zero of a chromatic polynomial (chromatic root); the second that every positive integer multiple of a chromatic root is also a chromatic root. We compute general formulae for the chromatic polynomials of two large families of graphs, and use these to provide partial proofs of each of these conjectures. We also investigate certain correspondences between the abstract structure of graphs and the splitting fields of their chromatic polynomials. The final chapter concerns the much more general multivariate Tutte polynomials—or Potts model partition functions—of matroids. We give three separate proofs that the Galois group of every such polynomial is a direct product of symmetric groups, and conjecture that an analogous result holds for the classical bivariate Tutte polynomial