This thesis touches three different topics in graph theory, namely, rainbow colouring, product dimension and boxicity. Rainbow colouring An edge colouring of a graph is called a rainbow colouring, if every pair of vertices is connected by atleast one path in which no two edges are coloured the same. The rainbow connection number of a graph is the minimum number of colours required to rainbow colour it. In this thesis we give upper bounds on rainbow connection number based on graph invariants like minimum degree, vertex connectivity, and radius. We also give some computational complexity results for special graph classes. Product dimension The product dimension or Prague dimension of a graph G is the smallest natural number k such that G is an induced subgraph of a direct product of k complete graphs. In this thesis, we give upper bounds on the product dimension for forests, bounded tree width graphs and graphs of bounded degeneracy. Boxicity and cubicity The boxicity (cubicity of a graph G is the smallest natural number k such that G can be represented as an intersection graph of axis-parallel rectangular boxes(axis-parallel unit cubes) in Rk .In this thesis, we study the boxicity and the cubicity of Cartesian, strong and direct products of graphs and give estimates on the boxicity and the cubicity of a product graph based on invariants of the component graphs. Separation dimension The separation dimension of a hypergraph H is the smallest natural number k for which the vertices of H can be embedded in Rk such that any two disjoint edges of H can be separated by a hyper plane normal to one of the axes. While studying the boxicity of line graphs, we noticed that a box representation of the line graph of a hypergraph has a nice geometric interpretation. Hence we introduced this new parameter and did an extensive study of the same.

The minimum number of colors required to color the edges of a graph so that any two distinct vertices are connected by at least one path in which no two edges are colored the same is called its rainbow connection number. This graph parameter was introduced by Chartrand et al. in 2008. The problem has garnered considerable interest and several variants of the initial version have since been introduced. The rainbow connection number of a connected graph G is denoted by rc(G). It can be shown that the rainbow connection number of a tree on n vertices is n -1. Hence |G|-1 is an upper bound for rc(G)of any non-trivial graph G. For all non-trivial, bridge-less and connected graphs G, Basavaraju etal. Showed that rc(G) can be upper-bounded by a quadratic function of its radius. In addition they also proved the tightness of the bound. It is clear that we cannot hope to get an upper-bound better than |G| - 1 in the case of graphs with bridges. An immediate and natural question is the following: Are there classes of bridge-less graphs whose rainbow connection numbers are linear functions of their radii? This question is of particular interest since the diameter is a trivial lower bound for rc(G). We answer in affirmative to the above question. In particular we studied three (graph) product operations (Cartesian, Lexicographic and Strong) and the graph powering operation. We were able to show that the rainbow connection number of the graph resulting from any of the above graph operations is upper-bounded by 2r(G)+c, where r(G) is radius of the resultant graph and c ε {0, 1, 2}.

The vertex coloring problem is a classic problem in graph theory that asks for a partition of the vertex set into a minimum number of stable sets. This thesis presents our studies on three vertex (re)coloring problems on graphs and on a problem related to a long-standing conjecture on subdivision of digraphs. Firstly, we address the convex recoloring problem in which an arbitrarily colored graph G is given and one wishes to find a minimum weight recoloring such that each color class induces a connected subgraph of G. We show inapproximability results, introduce an integer linear programming (ILP) formulation that models the problem and present some computational experiments using a column generation approach. The k-fold coloring problem is a generalization of the classic vertex coloring problem and consists in covering the vertex set of a graph by a minimum number of stable sets in such a way that every vertex is covered by at least k (possibly identical) stable sets. We present an ILP formulation for this problem and show a detailed polyhedral study of the polytope associated with this formulation. The last coloring problem studied in this thesis is the proper orientation problem. It consists in orienting the edge set of a given graph so that adjacent vertices have different in-degrees and the maximum in-degree is minimized. Clearly, the in-degrees induce a partition of the vertex set into stable sets, that is, a coloring (in the conventional sense) of the vertices. Our contributions in this problem are on hardness and upper bounds for bipartite graphs. Finally, we study a problem related to a conjecture of Mader from the eighties on subdivision of digraphs. This conjecture states that, for every acyclic digraph H, there exists an integer f(H) such that every digraph with minimum out-degree at least f(H) contains a subdivision of H as a subdigraph. We show evidences for this conjecture by proving that it holds for some particular classes of acyclic digraphs. / O problema de coloração de grafos é um problema clássico em teoria dos grafos cujo objetivo é particionar o conjunto de vértices em um número mínimo de conjuntos estáveis. Nesta tese apresentamos nossas contribuições sobre três problemas de coloração de grafos e um problema relacionado a uma antiga conjectura sobre subdivisão de digrafos. Primeiramente, abordamos o problema de recoloração convexa no qual é dado um grafo arbitrariamente colorido G e deseja-se encontrar uma recoloração de peso mínimo tal que cada classe de cor induza um subgrafo conexo de G. Mostramos resultados sobre inaproximabilidade, introduzimos uma formulação linear inteira que modela esse problema, e apresentamos alguns resultados computacionais usando uma abordagem de geração de colunas. O problema de k-upla coloração é uma generalização do problema clássico de coloração de vértices e consiste em cobrir o conjunto de vértices de um grafo com uma quantidade mínima de conjuntos estáveis de tal forma que cada vértice seja coberto por pelo menos k conjuntos estáveis (possivelmente idênticos). Apresentamos uma formulação linear inteira para esse problema e fazemos um estudo detalhado do politopo associado a essa formulação. O último problema de coloração estudado nesta tese é o problema de orientação própria. Ele consiste em orientar o conjunto de arestas de um dado grafo de tal forma que vértices adjacentes possuam graus de entrada distintos e o maior grau de entrada seja minimizado. Claramente, os graus de entrada induzem uma partição do conjunto de vértices em conjuntos estáveis, ou seja, induzem uma coloração (no sentido convencional) dos vértices. Nossas contribuições nesse problema são em complexidade computacional e limitantes superiores para grafos bipartidos. Finalmente, estudamos um problema relacionado a uma conjectura de Mader, dos anos oitenta, sobre subdivisão de digrafos. Esta conjectura afirma que, para cada digrafo acíclico H, existe um inteiro f(H) tal que todo digrafo com grau mínimo de saída pelo menos f(H) contém uma subdivisão de H como subdigrafo. Damos evidências para essa conjectura mostrando que ela é válida para classes particulares de digrafos acíclicos.

This thesis devote itself to use of information system for school agenda administration. Schools are forced to administer big amounts of informations, not only referred to their students. Broad issue is very extensive and disparate, so the most common types of data and demands on school information system operation are stated. The system for automatic generation of timetables is part of the school information system. At the first, basic conceptions of scheduling scope are defined and tied together with them are methods and algorithms for timetable creation problem solving. School timetabling is problem of scheduling lessons with certain limitative conditions. Further, thesis is engaged in design of school information system, data organization in such system and solving of system design problems. Designed information system accentuates on easy expandability and wide range of usage possibilities. Also suggested algorithm for solving of defined school timetabling is stated in this part of thesis.

This thesis is primarily focused on problems in extremal combinatorics, although we will also consider some questions of analytic and algorithmic nature. The d-dimensional hypercube is the graph with vertex set {0,1}^d where two vertices are adjacent if they differ in exactly one coordinate. In Chapter 2 we obtain an upper bound on the 'saturation number' of Q_m in Q_d. Specifically, we show that for m ≥ 2 fixed and d large there exists a subgraph G of Q_d of bounded average degree such that G does not contain a copy of Q_m but, for every G' such that G ⊊ G' ⊆ Q_d, the graph G' contains a copy of Q_m. This result answers a question of Johnson and Pinto and is best possible up to a factor of O(m). In Chapter 3, we show that there exists ε > 0 such that for all k and for n sufficiently large there is a collection of at most 2^{(1-ε)k} subsets of [n] which does not contain a chain of length k+1 under inclusion and is maximal subject to this property. This disproves a conjecture of Gerbner, Keszegh, Lemons, Palmer, Pálvölgyi and Patkós. We also prove that there exists a constant c ∈ (0,1) such that the smallest such collection is of cardinality 2^{(1+o(1))^ck} for all k. In Chapter 4, we obtain an exact expression for the 'weak saturation number' of Q_m in Q_d. That is, we determine the minimum number of edges in a spanning subgraph G of Q_d such that the edges of E(Q_d)\E(G) can be added to G, one edge at a time, such that each new edge completes a copy of Q_m. This answers another question of Johnson and Pinto. We also obtain a more general result for the weak saturation of 'axis aligned' copies of a multidimensional grid in a larger grid. In the r-neighbour bootstrap process, one begins with a set A₀ of 'infected' vertices in a graph G and, at each step, a 'healthy' vertex becomes infected if it has at least r infected neighbours. If every vertex of G is eventually infected, then we say that A₀ percolates. In Chapter 5, we apply ideas from weak saturation to prove that, for fixed r ≥ 2, every percolating set in Q_d has cardinality at least (1+o(1))(d choose r-1)/r. This confirms a conjecture of Balogh and Bollobás and is asymptotically best possible. In addition, we determine the minimum cardinality exactly in the case r=3 (the minimum cardinality in the case r=2 was already known). In Chapter 6, we provide a framework for proving lower bounds on the number of comparable pairs in a subset S of a partially ordered set (poset) of prescribed size. We apply this framework to obtain an explicit bound of this type for the poset 𝒱(q,n) consisting of all subspaces of 𝔽_qⁿordered by inclusion which is best possible when S is not too large. In Chapter 7, we apply the result from Chapter 6 along with the recently developed 'container method,' to obtain an upper bound on the number of antichains in 𝒱(q,n) and a bound on the size of the largest antichain in a p-random subset of 𝒱(q,n) which holds with high probability for p in a certain range. In Chapter 8, we construct a 'finitely forcible graphon' W for which there exists a sequence (ε_i)^∞_i=1 tending to zero such that, for all i ≥ 1, every weak ε_i-regular partition of W has at least exp(ε_i^-2/2^{5log∗ε_i^-2}) parts. This result shows that the structure of a finitely forcible graphon can be much more complex than was anticipated in a paper of Lovász and Szegedy. For positive integers p,q with p/q ❘≥ 2, a circular (p,q)-colouring of a graph G is a mapping V(G) → ℤ_p such that any two adjacent vertices are mapped to elements of ℤ_p at distance at least q from one another. The reconfiguration problem for circular colourings asks, given two (p,q)-colourings f and g of G, is it possible to transform f into g by recolouring one vertex at a time so that every intermediate mapping is a p,q-colouring? In Chapter 9, we show that this question can be answered in polynomial time for 2 ≤ p/q < 4 and is PSPACE-complete for p/q ≥ 4.

Mes travaux de thèse s’inscrivent dans le cadre du projet ANR GRAPHEM1 (Graphemebased Retrieval and Analysis for PaleograpHic Expertise of Middle Age Manuscripts). Ilsprésentent une contribution méthodologique applicable à l'analyse automatique des écrituresanciennes pour assister les experts en paléographie dans le délicat travail d’étude et dedéchiffrage des écritures.L’objectif principal est de contribuer à une instrumetation du corpus des manuscritsmédiévaux détenus par l’Institut de Recherche en Histoire des Textes (IRHT – Paris) en aidantles paléographes spécialisés dans ce domaine dans leur travail de compréhension de l’évolutiondes formes de l’écriture par la mise en place de méthodes efficaces d’accès au contenu desmanuscrits reposant sur une analyse fine des formes décrites sous la formes de petits fragments(les graphèmes). Dans mes travaux de doctorats, j’ai choisi d’étudier la dynamique del’élément le plus basique de l’écriture appelé le ductus2 et qui d’après les paléographes apportebeaucoup d’informations sur le style d’écriture et l’époque d’élaboration du manuscrit.Mes contributions majeures se situent à deux niveaux : une première étape de prétraitementdes images fortement dégradées assurant une décomposition optimale des formes en graphèmescontenant l’information du ductus. Pour cette étape de décomposition des manuscrits, nousavons procédé à la mise en place d’une méthodologie complète de suivi de traits à partir del’extraction d’un squelette obtenu à partir de procédures de rehaussement de contraste et dediffusion de gradients. Le suivi complet du tracé a été obtenu à partir de l’application des règlesfondamentales d’exécution des traits d’écriture, enseignées aux copistes du Moyen Age. Il s’agitd’information de dynamique de formation des traits portant essentiellement sur des indicationsde directions privilégiées.Dans une seconde étape, nous avons cherché à caractériser ces graphèmes par desdescripteurs de formes visuelles compréhensibles à la fois par les paléographes et lesinformaticiens et garantissant une représentation la plus complète possible de l’écriture d’unpoint de vue géométrique et morphologique. A partir de cette caractérisation, nous avonsproposé une approche de clustering assurant un regroupement des graphèmes en classeshomogènes par l’utilisation d’un algorithme de classification non-supervisé basée sur lacoloration de graphe. Le résultat du clustering des graphèmes a conduit à la formation dedictionnaires de formes caractérisant de manière individuelle et discriminante chaque manuscrittraité. Nous avons également étudié la puissance discriminatoire de ces descripteurs afin d’obtenir la meilleure représentation d’un manuscrit en dictionnaire de formes. Cette étude a étéfaite en exploitant les algorithmes génétiques par leur capacité à produire de bonne sélection decaractéristiques.L’ensemble de ces contributions a été testé à partir d’une application CBIR sur trois bases demanuscrits dont deux médiévales (manuscrits de la base d’Oxford et manuscrits de l’IRHT, baseprincipale du projet), et une base comprenant de manuscrits contemporains utilisée lors de lacompétition d’identification de scripteurs d’ICDAR 2011. L’exploitation de notre méthode dedescription et de classification a été faite sur une base contemporaine afin de positionner notrecontribution par rapport aux autres travaux relevant du domaine de l’identification d’écritures etétudier son pouvoir de généralisation à d’autres types de documents. Les résultats trèsencourageants que nous avons obtenus sur les bases médiévales et la base contemporaine, ontmontré la robustesse de notre approche aux variations de formes et de styles et son caractèrerésolument généralisable à tout type de documents écrits. / My thesis work is part of the ANR GRAPHEM Project (Grapheme based Retrieval andAnalysis for Expertise paleographic Manuscripts of Middle Age). It represents a methodologicalcontribution applicable to the automatic analysis of ancient writings to assist the experts inpaleography in the delicate work of the studying and deciphering the writing.The main objective is to contribute to an instrumentation of the corpus of medievalmanuscripts held by “Institut de Recherche en Histoire de Textes” (IRHT-Paris), by helping thepaleographers specialized in this field in their work of understanding the evolution of forms inthe writing, with the establishment of effective methods to access the contents of manuscriptsbased on a fine analysis of the forms described in the form of small fragments (graphemes). Inmy PhD work, I chose to study the dynamic of the most basic element of the writing called theductus and which according to the paleographers, brings a lot of information on the style ofwriting and the era of the elaboration of the manuscript.My major contribution is situated at two levels: a first step of preprocessing of severelydegraded images to ensure an optimal decomposition of the forms into graphemes containingthe ductus information. For this decomposition step of manuscripts, we have proceeded to theestablishment of a complete methodology for the tracings of strokes by the extraction of theskeleton obtained from the contrast enhancement and the diffusion of the gradient procedures.The complete tracking of the strokes was obtained from the application of fundamentalexecution rules of the strokes taught to the scribes of the Middle Ages. It is related to thedynamic information of the formation of strokes focusing essentially on indications of theprivileged directions.In a second step, we have tried to characterize the graphemes by visual shape descriptorsunderstandable by both the computer scientists and the paleographers and thus unsuring themost complete possible representation of the wrting from a geometrical and morphological pointof view. From this characterization, we have have proposed a clustering approach insuring agrouping of graphemes into homogeneous classes by using a non-supervised classificationalgorithm based on the graph coloring. The result of the clustering of graphemes led to theformation of a codebook characterizing in an individual and discriminating way each processedmanuscript. We have also studied the discriminating power of the descriptors in order to obtaina better representation of a manuscript into a codebook. This study was done by exploiting thegenetic algorithms by their ability to produce a good feature selection.The set of the contributions was tested from a CBIR application on three databases ofmanuscripts including two medieval databases (manuscripts from the Oxford and IRHTdatabases), and database of containing contemporary manuscripts used in the writersidentification contest of ICDAR 2011. The exploitation of our description and classificationmethod was applied on a cotemporary database in order to position our contribution withrespect to other relevant works in the writrings identification domain and study itsgeneralization power to other types of manuscripts. The very encouraging results that weobtained on the medieval and contemporary databases, showed the robustness of our approachto the variations of the shapes and styles and its resolutely generalized character to all types ofhandwritten documents.