Cette thèse traite de la coloration de graphe, de la coloration par liste,d'ensembles indépendants de poids maximum et de la théorie structurelle des graphes.Dans un premier temps, nous fournissons un algorithme s'exécutant en temps polynomial pour le problème de la 4-coloration dans des sous-classes de graphe sans $P_6$. Ces algorithmes se basent sur une compréhension précise de la structure de ces classes de graphes, pour laquelle nous donnons une description complète.Deuxièmement, nous étudions une conjecture portant sur la coloration par liste et prouvons que pour tout graphe parfait sans griffe dont la taille de la plus grande clique est bornée par 4, le nombre chromatique est égal au nombre chromatique par liste. Ce résultat est obtenu en utilisant un théorème de décomposition des graphes parfaits sans griffe, une description structurelle des graphes de base de cette décomposition et le célèbre théorème de Galvin.Ensuite, en utilisant la description structurelle élaborée dans le premier chapitre et en renforçant certains aspects de celle-ci, nous fournissons un algorithme s'exécutant en temps polynomial pour le problème d'indépendant de poids maximum dans des sous-classes de graphe sans $P_6$ et sans $P_7$. Dans le dernier chapitre de ce manuscrit, nous infirmons une conjecture datant de 1999 de De Simone et K"orner sur les graphes normaux. Notre preuve est probabiliste et est obtenue en utilisant les graphes aléatoires. / This thesis deals with graph coloring, list-coloring, maximum weightstable set (shortened as MWSS) and structural graph theory.First, we provide polynomial-time algorithms for the 4-coloring problem insubclasses of $P_6$-free graphs. These algorithms rely on a preciseunderstanding of the structure of these classes of graphs for which we give afull description.Secondly, we study the list-coloring conjecture and prove that for anyclaw-free perfect graph with clique number bounded by 4, the chromatic numberand the choice number are equal. This result is obtained by using adecomposition theorem for claw-free perfect graphs, a structural description ofthe basic graphs of this decomposition and by using Galvin's famous theorem.Next by using the structural description given in the first chapter andstrengthening other aspects of this structure, we provide polynomial-timealgorithms for the MWSS problem in subclasses of $P_6$-free and $P_7$-freegraphs.In the last chapter of the manuscript, we disprove a conjecture of De Simoneand K"orner made in 1999 related to normal graphs. Our proof is probabilisticand is obtained by the use of random graphs.

Dans cette thèse nous étudions quatre problèmes de théorie des graphes. En particulier,Nous étudions le problème du cycle hamiltonien dans les line graphes, et aussi nous prouvons l’existence de cycles hamiltoniens dans certains sous graphes couvrants d’un line graphe. Notre résultat principal est: Si L(G) est hamiltonien, alors SL(G) est hamiltonien. Grâce à ce résultat nous proposons une conjecture équivalente à des conjectures célèbres. Et nous obtenons deux résultats sur les cycles hamiltoniens disjoints dans les line graphes.Nous considérons alors la bipancyclicité résistante aux pannes des graphes de Cayley engendrés par transposition d’arbres. Nous prouvons que de tels graphes de Cayley excepté le “star graph” ont une bipancyclicité (n − 3)-arête résistante aux pannes.Ensuite nous introduisons la coloration des arcs d’un digraphe sommet distinguant. Nous étudions la relation entre cette notion et la coloration d’arêtes sommet distinguant dans les graphes non orientés. Nous obtenons quelques résultats sur le nombre arc chromatique des graphes orientés (semi-)sommet-distinguant et proposons une conjecture sur ce paramètre. Pour vérifier cette conjecture nous étudions la coloration des arcs d’un digraphe sommet distinguant des graphes orientés réguliers.Finalement nous introduisons la coloration acyclique des arcs d’un graphe orienté. Nous calculons le nombre chromatique acyclique des arcs de quelques familles de graphes orientés et proposons une conjecture sur ce paramètre. Nous considérons les graphes orientés de grande maille et utilisons le Lemme Local de Lovász; d’autre part nous considérons les graphes orientés réguliers aléatoires. Nous prouvons que ces deux classes de graphes vérifient la conjecture. / In this thesis, we study four problems in graph theory, the Hamiltonian cycle problem in line graphs, the edge-fault-tolerant bipancyclicity of Cayley graphs generated by transposition trees, the vertex-distinguishing arc colorings in digraph- s and the acyclic arc coloring in digraphs. The first two problems are the classic problem on the cycles in graphs. And the other two arc coloring problems are related to the modern graph theory, in which we use some probabilistic methods. In particular,We first study the Hamiltonian cycle problem in line graphs and find the Hamiltonian cycles in some spanning subgraphs of line graphs SL(G). We prove that: if L(G) is Hamiltonian, then SL(G) is Hamiltonian. Due to this, we propose a conjecture, which is equivalent to some well-known conjectures. And we get two results about the edge-disjoint Hamiltonian cycles in line graphs.Then, we consider the edge-fault-tolerant bipancyclicity of Cayley graphs generated by transposition trees. And we prove that the Cayley graph generated by transposition tree is (n − 3)-edge-fault-tolerant bipancyclic if it is not a star graph.Later, we introduce the vertex-distinguishing arc coloring in digraphs. We study the relationship between the vertex-distinguishing edge coloring in undirected graphs and the vertex-distinguishing arc coloring in digraphs. And we get some results on the (semi-) vertex-distinguishing arc chromatic number for digraphs and also propose a conjecture about it. To verify the conjecture we study the vertex-distinguishing arc coloring for regular digraphs.Finally, we introduce the acyclic arc coloring in digraphs. We calculate the acyclic arc chromatic number for some digraph families and propose a conjecture on the acyclic arc chromatic number. Then we consider the digraphs with high girth by using the Lovász Local Lemma and we also consider the random regular digraphs. And the results of the digraphs with high girth and the random regular digraphs verify the conjecture.

The bin packing problem has been the corner stone of approximation algorithms and has been extensively studied starting from the early seventies. In the classical bin packing problem, we are given a list of real numbers in the range (0, 1], the goal is to place them in a minimum number of bins so that no bin holds numbers summing to more than 1. In this thesis we study approximation algorithms for three generalizations of bin packing: geometric bin packing, vector bin packing and weighted bipartite edge coloring. In two-dimensional (2-D) geometric bin packing, we are given a collection of rectangular items to be packed into a minimum number of unit size square bins. Geometric packing has vast applications in cutting stock, vehicle loading, pallet packing, memory allocation and several other logistics and robotics related problems. We consider the widely studied orthogonal packing case, where the items must be placed in the bin such that their sides are parallel to the sides of the bin. Here two variants are usually studied, (i) where the items cannot be rotated, and (ii) they can be rotated by 90 degrees. We give a polynomial time algorithm with an asymptotic approximation ratio of $\ln(1.5) + 1 \approx 1.405$ for the versions with and without rotations. We have also shown the limitations of rounding based algorithms, ubiquitous in bin packing algorithms. We have shown that any algorithm that rounds at least one side of each large item to some number in a constant size collection values chosen independent of the problem instance, cannot achieve an asymptotic approximation ratio better than 3/2. In d-dimensional vector bin packing (VBP), each item is a d-dimensional vector that needs to be packed into unit vector bins. The problem is of great significance in resource constrained scheduling and also appears in recent virtual machine placement in cloud computing. Even in two dimensions, it has novel applications in layout design, logistics, loading and scheduling problems. We obtain a polynomial time algorithm with an asymptotic approximation ratio of $\ln(1.5) + 1 \approx 1.405$ for 2-D VBP. We also obtain a polynomial time algorithm with almost tight (absolute) approximation ratio of $1+\ln(1.5)$ for 2-D VBP. For $d$ dimensions, we give a polynomial time algorithm with an asymptotic approximation ratio of $\ln(d/2) + 1.5 \approx \ln d+0.81$. We also consider vector bin packing under resource augmentation. We give a polynomial time algorithm that packs vectors into $(1+\epsilon)Opt$ bins when we allow augmentation in (d - 1) dimensions and $Opt$ is the minimum number of bins needed to pack the vectors into (1,1) bins. In weighted bipartite edge coloring problem, we are given an edge-weighted bipartite graph $G=(V,E)$ with weights $w: E \rightarrow [0,1]$. The task is to find a proper weighted coloring of the edges with as few colors as possible. An edge coloring of the weighted graph is called a proper weighted coloring if the sum of the weights of the edges incident to a vertex of any color is at most one. This problem is motivated by rearrangeability of 3-stage Clos networks which is very useful in various applications in interconnected networks and routing. We show a polynomial time approximation algorithm that returns a proper weighted coloring with at most $\lceil 2.2223m \rceil$ colors where $m$ is the minimum number of unit sized bins needed to pack the weight of all edges incident at any vertex. We also show that if all edge weights are $>1/4$ then $\lceil 2.2m \rceil$ colors are sufficient.

ENGLISH ABSTRACT: Determining the (classical) chromatic number of a graph (i.e. finding the smallest number of colours with which the vertices of a graph may be coloured so that no two adjacent vertices receive the same colour) is a well known combinatorial optimization problem and is widely encountered in scheduling problems. Since the late 1960s the notion of the chromatic number has been generalized in several ways by relaxing the restriction of independence of the colour classes. / AFRIKAANSE OPSOMMING: Die bepaling van die (klassieke) chromatiese getal van ’n grafiek (naamlik die kleinste aantal kleure waarmee die punte van ’n grafiek gekleur kan word sodat geen twee naasliggende punte dieselfde kleur ontvang nie) is ’n bekende kombinatoriese optimeringsprobleem wat wyd in skeduleringstoepassings te¨egekom word. Sedert die laat 1960s is die definisie van die chromatiese getal op verskeie maniere veralgemeen deur die vereiste van onafhanklikheid van die kleurklasse te verslap. / Thesis (DPhil)--Stellenbosch University, 2007.

This thesis is devoted to two $\mathcal{NP}$-complete combinatorial optimization problems arising in computational biology, the well-studied \emph{multiple sequence alignment} problem and the new formulated \emph{interval constraint coloring} problem. It shows that advanced mathematical programming techniques are capable of solving large scale real-world instances from biology to optimality. Furthermore, it reveals alternative methods that provide approximate solutions. In the first part of the thesis, we present a \emph{Lagrangian relaxation} approach for the multiple sequence alignment (MSA) problem. The multiple alignment is one common mathematical abstraction of the comparison of multiple biological sequences, like DNA, RNA, or protein sequences. If the weight of a multiple alignment is measured by the sum of the projected pairwise weights of all pairs of sequences in the alignment, then finding a multiple alignment of maximum weight is $\mathcal{NP}$-complete if the number of sequences is not fixed. The majority of the available tools for aligning multiple sequences implement heuristic algorithms; no current exact method is able to solve moderately large instances or instances involving sequences exhibiting a lower degree of similarity. We present a branch-and-bound (B\&B) algorithm for the MSA problem.\ignore{the multiple sequence alignment problem.} We approximate the optimal integer solution in the nodes of the B\&B tree by a Lagrangian relaxation of an ILP formulation for MSA relative to an exponential large class of inequalities, that ensure that all pairwise alignments can be incorporated to a multiple alignment. By lifting these constraints prior to dualization the Lagrangian subproblem becomes an \emph{extended pairwise alignment} (EPA) problem: Compute the longest path in an acyclic graph, that is penalized a charge for entering ``obstacles''. We describe an efficient algorithm that solves the EPA problem repetitively to determine near-optimal \emph{Lagrangian multipliers} via subgradient optimization. The reformulation of the dualized constraints with respect to additionally introduced variables improves the convergence rate dramatically. We account for the exponential number of dualized constraints by starting with an empty \emph{constraint pool} in the first iteration to which we add cuts in each iteration, that are most violated by the convex combination of a small number of preceding Lagrangian solutions (including the current solution). In this \emph{relax-and-cut} scheme, only inequalities from the constraint pool are dualized. The interval constraint coloring problem appears in the interpretation of experimental data in biochemistry. Monitoring hydrogen-deuterium exchange rates via mass spectroscopy is a method used to obtain information about protein tertiary structure. The output of these experiments provides aggregate data about the exchange rate of residues in overlapping fragments of the protein backbone. These fragments must be re-assembled in order to obtain a global picture of the protein structure. The interval constraint coloring problem is the mathematical abstraction of this re-assembly process. The objective of the interval constraint coloring problem is to assign a color (exchange rate) to a set of integers (protein residues) such that a set of constraints is satisfied. Each constraint is made up of a closed interval (protein fragment) and requirements on the number of elements in the interval that belong to each color class (exchange rates observed in the experiments). We introduce a polyhedral description of the interval constraint coloring problem, which serves as a basis to attack the problem by integer linear programming (ILP) methods and tools, which perform well in practice. Since the goal is to provide biochemists with all possible candidate solutions, we combine related solutions to equivalence classes in an improved ILP formulation in order to reduce the running time of our enumeration algorithm. Moreover, we establish the polynomial-time solvability of the two-color case by the integrality of the linear programming relaxation polytope $\mathcal{P}$, and also present a combinatorial polynomial-time algorithm for this case. We apply this algorithm as a subroutine to approximate solutions to instances with arbitrary but fixed number of colors and achieve an order of magnitude improvement in running time over the (exact) ILP approach. We show that the problem is $\mathcal{NP}$-complete for arbitrary number of colors, and we provide algorithms that, given an instance with $\mathcal{P}\neq\emptyset$, find a coloring that satisfies all the coloring requirements within $\pm 1$ of the prescribed value. In light of our $\mathcal{NP}$-completeness result, this is essentially the best one can hope for. Our approach is based on polyhedral theory and randomized rounding techniques. In practice, data emanating from the experiments are noisy, which normally causes the instance to be infeasible, and, in some cases, even forces $\mathcal{P}$ to be empty. To deal with this problem, the objective of the ILP is to minimize the total sum of absolute deviations from the coloring requirements over all intervals. The combinatorial approach for the two-color case optimizes the same objective function. Furthermore, we use this combinatorial method to compute, in a Lagrangian way, a bound on the minimum total error, which is exploited in a branch-and-bound manner to determine all optimal colorings. Alternatively, we study a variant of the problem in which we want to maximize the number of requirements that are satisfied. We prove that this variant is $\mathcal{APX}$-hard even in the two-color case and thus does not admit a polynomial time approximation scheme (PTAS) unless $\mathcal{P}=\mathcal{NP}$. Therefore, we slightly (by a factor of $(1+\epsilon)$) relax the condition on when a requirement is satisfied and propose a \emph{quasi-polynomial time approximation scheme} (QPTAS) which finds a coloring that ``satisfies'' the requirements of as many intervals as possible.

Un stable (ou ensemble indépendant) est un ensemble de sommets qui sont deux à deux non adjacents. De nombreux résultats classiques en optimisation combinatoire portent sur le nombre de stabilité (défini comme la plus grande taille d'un stable), et les stables se classent certainement parmi les structures les plus simples et fondamentales en théorie des graphes. La thèse est divisée en deux parties, toutes deux liées à la notion de stables dans un graphe. Dans la première partie, nous étudions un problème de coloration de graphes, c'est à dire de partition en stables, où le but est de minimiser l'entropie de la partition. C'est une variante du problème classique de minimiser le nombre de couleurs utilisées. Nous considérons aussi une généralisation du problème aux couvertures d'ensembles. Ces deux problèmes sont appelés respectivement minimum entropy coloring et minimum entropy set cover, et sont motivés par diverses applications en théorie de l'information et en bioinformatique. Nous obtenons entre autres une caractérisation précise de la complexité de minimum entropy set cover : le problème peut être approximé à une constante lg e (environ 1.44) près, et il est NP-difficile de faire strictement mieux. Des résultats analogues sont prouvés concernant la complexité de minimum entropy coloring. Dans la deuxième partie de la thèse, nous considérons les graphes dont le nombre de stabilité augmente dès qu'une arête est enlevée. Ces graphes sont dit être "alpha-critiques", et jouent un rôle important dans de nombreux domaines, comme la théorie extrémale des graphes ou la combinatoire polyédrique. Nous revisitons d'une part la théorie des graphes alpha-critiques, donnant à cette occasion de nouvelles démonstrations plus simples pour certains théorèmes centraux. D'autre part, nous étudions certaines facettes du polytope des ordres totaux qui peuvent être vues comme une généralisation de la notion de graphe alpha-critique. Nous étendons de nombreux résultats de la théorie des graphes alpha-critiques à cette famille de facettes.

Hamming-distance graphs arise naturally in the study of error-correcting codes and have been utilized by several authors to provide new proofs for (and in some cases improve) known bounds on the size of block codes. We study various standard graph properties of the Hamming-distance graphs with special emphasis placed on the chromatic number. A notion of robustness is defined for colorings of these graphs based on the tolerance of swapping colors along an edge without destroying the properness of the coloring, and a complete characterization of the maximally robust colorings is given for certain parameters. Additionally, explorations are made into subgraph structures whose identification may be useful in determining the chromatic number.

Multiple original results in the theory of automata and formal languages are presented, dealing mainly with combinatorial problems and complexity questions related to reset words and road coloring. The other results concern jumping finite automata and related types of rewriting systems. Powered by TCPDF (www.tcpdf.org)

O objetivo deste trabalho é usar jogos e tópicos de Teoria dos Grafos como ferramenta para desenvolver a habilidade da enumeração, que está por trás dos cálculos combinatórios ensinados no Ensino Fundamental e Médio. Mais especificamente, neste trabalho são introduzidos os métodos mais comuns de contagem através de situacões-problema e jogos, como o Nim e o Dominó, que podem ser melhor explorados ao serem descritos atráves dos elementos de um grafo. Com essa motivacão são apresentados conceitos básicos da Teoria dos Grafos e tópicos de coloração de grafos, como o número cromático e os polinômios cromáticos. Esses tópicos fornecem exemplos ricos e motivacionais ao processo de ensino e aprendizagem dos raciocínios combinatórios. Por outro lado, os tópicos abordados contém em si a riqueza e a complexidade da Matemática, como é o caso do Teorema das 4 Cores, demonstrado com o uso da enumeração de todos os casos possíveis. Nesse contexto são apresentados os conceitos de coloração de vértices de grafos dando destaque principal para problemas combinatórios que envolvem o número cromático e o polinômio cromático de um grafo. Complementando o trabalho, são propostas atividades para serem desenvolvidas em sala de aula. / The purpose of this work is to use games and topics of Graph Theory as a tool to develop the ability of enumeration, which is behind combinatorial calculations taught in Elementary and High School. More specifically, in this work, the most common methods of counting through problem situations and games, such as Nim and Domino, which can be better explored when described through the elements of a graph. With this motivation are presented basic concepts of the Theory of Graphs and graph coloring topics such as chromatic number and chromatic polynomials. Those topics provide rich and motivational examples to the process of teaching and learning combinatorial reasoning. On the other hand, the topics approach contains in itself the richness and complexity of Mathematics, as is the case with the 4-Color Theorem, demonstrated with the use of the enumeration of all possible cases. In this context are presented concepts of coloring of vertices of graphs giving main highlight to combinatorial problems which involve the chromatic number and the chromatic polynomial of a graph. Complementing the work, activities are proposed to be developed in the classroom.

Níveis crescentes de zeolita (clinoptilolita) - 0%; 1,5%; 3,0%; 4,5% - foram utilizados com o intuito de se avaliar a qualidade e a palatabilidade de biscoitos para cães. No âmbito da qualidade de produto foi avaliada a atividade de água, através da mensuração da umidade relativa de equilíbrio, a coloração, utilizando-se colorímetro em sistema CIEL*a*b, a textura através de texturômetro com sonda específica e a ordenação de preferência por parte dos proprietários de cães. O ensaio de palatabilidade foi realizado com 14 cães adultos, sem raça definida, machos e fêmeas, com idade média de seis anos e peso médio de 14kg. Foi utilizado delineamento inteiramente casualizado. Nos níveis de inclusão de 3% e 4,5% a zeolita diminuiu (p<0,01) os valores de atividade de água e para todos os níveis testados não houve efeito (p<0,01) na coloração de biscoitos para cães. A pressão de cisalhamento foi crescente (p<0,01) nos tratamentos 0%; 1,5% e 3,0%, respectivamente. Porém, sofreu uma queda (p<0,01) no nível de inclusão de 4,5%. Na análise sensorial os proprietários de cães preferiram (p<0,05) para o parâmetro cor o nível de 0% de inclusão de zeolita, para o odor os níveis 0% e 1,5% e para a dureza os níveis 0% e 4,5%. No ensaio de palatabilidade, tanto para primeira escolha, como para a razão de ingestão, houve diferença (p<0,01) significativa, sendo o nível de 3% de Zeolita o preferido, seguido dos níveis 4,5%, 0% e 1,5%, respectivamente. Mesmo não havendo efeito na coloração, a adição de Zeolita (clinoptilolita) altera a qualidade de biscoitos para cães, causando uma diminuição na atividade de água, nos níveis de 3,0% e 4,5% de inclusão e modificando a textura nos níveis 1,5% e 3,0%. Diferentes níveis de zeolita na composição de petiscos podem ser identificados pelos cães / Increasing levels of zeolite (clinoptilolite) - 0%, 1.5%, 3.0%, 4.5% - were used in order to evaluate the quality and palatability of dog biscuits. Concerning product quality water activity was evaluated by measuring the equilibrium relative humidity, coloring by colorimeter on the CIEL *a*b system, texture through texturometer using three point bending rig probe and ordering of preference for the dog\'s owners. The palatability test was conducted with 14 adult dogs, mixed breed, male and female, with an average age of six years, and an average weight of 14kg. The statistical design was applied in a completely randomized study. In inclusion levels of 3.0% and 4.5% of zeolite decreased (p <0.01) the values of water activity and at all levels tested did not affect (p <0.01) in staining. The shear stress is increased (p <0.01) between treatments 0%, 1.5% and 3.0%, respectively. However, has declined (p <0.01) on level of inclusion of 4.5%. In the sensory analysis dog owners preferred (p <0.05) for the color parameter 0% of inclusion, for the odor 0% and 1.5% and hardness 0% and 4.5%. In palatability test, both first choice and ratio of ingestion had significant differences (p <0.01) between the levels tested and the level of 3% zeolite was preferred, followed by 4.5%, 0 % and 1.5% levels, respectively. Even without effecting coloring, adding zeolite (clinoptilolite) alters the quality of dog biscuits, causing a decrease in water activity on levels of 3.0%, and 4.5% inclusion and changing the texture on levels 1.5% and 3.0%. Different levels of zeolite in the composition of snacks can be identified by dogs