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Let v [greater than or equal to] k [greater than or equal to] 1 and lamda [greater than or equal to] 0 be integers and G be a graph with n vertices, m edges, and no multiple edges. A (v, k, lambda) block design is a collection Beta of k-subsets of a v-set X in which every unordered pair of elements in X is contained in exactly lambda of the subsets in Beta. A (G-decomposition, or (v, G, lambda) graph design, is a collection H1, H2, ..., Ht of subgraphs of Kv (the complete graph on v vertices) such that each edge of Kv is an edge of exactly lambda of the subgraphs Hi and each of the subgraphs Hi is isomorphic to G. A famous result by Wilson says that for a fixed graph G and integer lambda, there exists a (v, G, lambda) graph design for all sufficiently large integers v satisfying certain necessary conditions. In this thesis, we extend this result to include the case of loops in G. As a consequence, one obtains asymptotic existence of equireplicate graph designs for values of v satisfying certain necessary conditions, where a graph design is called equireplicate if each vertex of Kv occurs in the same number of subgraphs Hi of the decomposition.

Given a graph G, a K_3-decomposition of G, also called a triangle decomposition, is a set of subgraphs isomorphic to K_3 whose edges partition the edge set of G. Further, a rational K_3-decomposition of G is a non-negative rational weighting of the copies of K_3 in G such that the total weight on any edge of G equals one. In this thesis, we explore the problem of rational triangle decompositions of dense graphs. We start by considering necessary conditions for a rational triangle decomposition, which can be represented by facets of a convex cone generated by a certain incidence matrix. We identify several infinite families of these facets that represent meaningful obstructions to rational triangle decomposability of a graph. Further, we classify all facets on up to 9 vertices and check all 8-vertex graphs of degree at least four for rational triangle decomposability. As the study of graph decompositions is closely related to design theory, we also prove the existence of certain types of designs. We then explore sufficient conditions for rational triangle decomposability. A famous conjecture in the area due to Nash-Williams states that any sufficiently large graph (satisfying some divisibility conditions) with minimum degree at least 3/4v is K_3-decomposable; the same conjecture stands for rational K_3-decomposability (no divisibility conditions required). By perturbing and restricting the coverage matrix of a complete graph, we show that minimum degree of at least 22/23v is sufficient to guarantee that the given graph is rationally triangle decomposable. This density bound is a great improvement over the previously known results and is derived using estimates on the matrix norms and structures originating from association schemes. We also consider applications of rational triangle decompositions. The method we develop in the search for sufficient conditions provides an efficient way to generate certain sampling plans in statistical experimental design. Furthermore, rational graph decompositions serve as building blocks within certain design-theoretic proofs and we use them to prove that it is possible to complete partial designs given certain constraints. / Graduate / 0405

La quantité d’informations stockée dans les bases de données est en constante augmentation rendant ainsi nécessaire la mise au point de systémes d’analyse et de visualisation. Nous nous intéressons dans cette thèse aux données relationnelles et plus particulièrement aux données biologiques. Cette thèse s’oriente autour de trois axes principaux : tout d’abord, la décomposition de graphes en groupes d’éléments ”similaires” a?n de détecter d’éventuelles structures de communauté ; le deuxième aspect consiste à mettre en évidence ces structures dans un système de visualisation, et dans un dernier temps, nous nous intéressons à l’utilisabilité de l’un de ces systèmes de visualisation via une évaluation expérimentale. Les travaux de cette thèse ont été appliqués sur des données réelles provenant de deux domaines de la biologie : les réseaux métaboliques et les réseaux d’interactions génes- protéines. / The amount of information stored in databases is constantly increasing making necessary to develop systems for analysis and visualization. In this thesis, we are interested in relational data and in particular, in biological data. This thesis focuses on three main axes : ?rstly, the decomposition of graph into clusters of ”similar” elements in order to detect the community structures ; the second aspect is to highlight these structures in a visualization system; and thirdly, we are interested in the usability of one of these visualization systems through an experimental evaluation. The work presented in this thesis was applied on real data from two ?elds of biology : the metabolic networks and the gene-protein interaction networks.

Aquest treball és una contribució a l'estudi de diferents problemes que sorgeixen de dues àrees fortament connexes de la Teoria de Grafs: etiquetaments i descomposicions. Molts etiquetaments de grafs deuen el seu origen als presentats l'any 1967 per Rosa. Un d'aquests etiquetaments, àmpliament conegut com a etiquetament graceful, va ser definit originalment com a eina per atacar la conjectura de Ringel, la qual diu que el graf complet d'ordre 2m+1 pot ser descompost en m copies d'un arbre donat de mida m. Aquí, estudiem etiquetaments relacionats que ens donen certes aproximacions a la conjectura de Ringel, així com també a una altra conjectura de Graham i Häggkvist que, en una forma dèbil, demana la descomposició d'un graf bipartit complet per un arbre donat de mida apropiada. Les principals contribucions que hem fet en aquest tema són la prova de la darrera conjectura per grafs bipartits complets del doble de mida essent descompostos per arbres de gran creixement i un nombre primer d'arestes, i la prova del fet que cada arbre és un subarbre gran de dos arbres pels quals les dues conjectures es compleixen respectivament. Aquests resultats estan principalment basats en una aplicació del mètode polinomial d'Alon. Un altre tipus d'etiquetaments, els etiquetaments magic, també són tractats aquí. Motivats per la noció de quadrats màgics de Teoria de Nombres, en aquest tipus d'etiquetaments volem asignar nombres enters a parts del graf (vèrtexs, arestes, o vèrtexs i arestes) de manera que la suma de les etiquetes assignades a certes subestructures del graf sigui constant. Desenvolupem tècniques basades en particions de certs conjunts d'enters amb algunes condicions additives per construir etiquetaments cycle-magic, un nou tipus d'etiquetament introduït en aquest treball i que estén la noció clàssica d'etiquetament magic. Els etiquetaments magic no donen cap descomposició de grafs, però les tècniques usades per obtenir-los estan al nucli d'un altre problema de descomposició, l'ascending subgraph decomposition (ASD). Alavi, Boals, Chartrand, Erdös i Oellerman, van conjecturar l'any 1987 que tot graf té un ASD. Aquí, estudiem l'ASD per grafs bipartits, una classe de grafs per la qual la conjectura encara no ha estat provada. Donem una condició necessària i una de suficient sobre la seqüència de graus d'un estable del graf bipartit de manera que admeti un ASD en que cada factor sigui un star forest. Les tècniques utilitzades estan basades en l'existència de branca-acoloriments en multigrafs bipartits. També tractem amb el sumset partition problem, motivat per la conjectura ASD, que demana una partició de [n] de manera que la suma dels elements de cada part sigui igual a un valor prescrit. Aquí donem la millor condició possible per la versió modular del problema que ens permet provar els millors resultats ja coneguts en el cas enter per n primer. La prova està de nou basada en el mètode polinomial. / This work is a contribution to the study of various problems that arise from two strongly connected areas of the Graph Theory: graph labelings and graph decompositions. Most graph labelings trace their origins to the ones presented in 1967 by Rosa. One of these labelings, widely known as the graceful labeling, originated as a means of attacking the conjecture of Ringel, which states that the complete graph of order 2m+1 can be decomposed into m copies of a given tree of size m. Here, we study related labelings that give some approaches to Ringel's conjecture, as well as to another conjecture by Graham and Häggkvist that, in a weak form, asks for the decomposition of a complete bipartite graph by a given tree of appropriate size. Our main contributions in this topic are the proof of the latter conjecture for double sized complete bipartite graphs being decomposed by trees with large growth and prime number of edges, and the proof of the fact that every tree is a large subtree of two trees for which both conjectures hold respectively. These results are mainly based on a novel application of the so-called polynomial method by Alon. Another kind of labelings, the magic labelings, are also treated. Motivated by the notion of magic squares in Number Theory, in these type of labelings we want to assign integers to the parts of a graph (vertices, edges, or vertices and edges) in such a way that the sums of the labels assigned to certain substructures of the graph remain constant. We develop techniques based on partitions of certain sets of integers with some additive conditions to construct cycle-magic labelings, a new brand introduced in this work that extends the classical magic labelings. Magic labelings do not provide any graph decomposition, but the techniques that we use to obtain them are the core of another decomposition problem, the ascending subgraph decomposition (ASD). In 1987, was conjectured by Alavi, Boals. Chartrand, Erdös and Oellerman that every graph has an ASD. Here, we study ASD of bipartite graphs, a class of graphs for which the conjecture has not been shown to hold. We give a necessary and a sufficient condition on the one sided degree sequence of a bipartite graph in order that it admits an ASD by star forests. Here the techniques are based on the existence of edge-colorings in bipartite multigraphs. Motivated by the ASD conjecture we also deal with the sumset partition problem, which asks for a partition of [n] in such a way that the sum of the elements of each part is equal to a prescribed value. We give a best possible condition for the modular version of the sumset partition problem that allows us to prove the best known results in the integer case for n a prime. The proof is again based on the polynomial method.

Les décompositions de graphes, lorsqu’elles sont de petite largeur, sont souvent utilisées pour résoudre plus efficacement des problèmes étant difficiles dans le cas de graphes quelconques. Dans ce travail de thèse, nous nous intéressons aux limites liées à ces décompositions, et à la construction d’obstructions certifiant leur grande largeur. Dans une première partie, nous donnons un algorithme généralisant et unifiant la construction d’obstructions pour différentes largeurs de graphes, en temps XP lorsque paramétré par la largeur considérée. Nous obtenons en particulier le premier algorithme permettant de construire efficacement une obstruction à la largeur arborescente en temps O(ntw+4). La seconde partie de notre travail porte sur l’étude du problème ENSEMBLE [σ, ρ]-DOMINANT, une généralisation des problèmes de domination sur les graphes et caractérisée par deux ensembles d’entiers σ et ρ. Les diverses études de ce problème apparaissant dans la littérature concernent uniquement les cas ou le problème est FPT, lorsque paramétré par la largeur arborescente. Nous montrons que ce problème ne l’est pas toujours, et que pour certains cas d’ensembles σ et ρ, il devient W[1]-difficile lorsque paramétré par la largeur arborescente. Dans la dernière partie, nous étudions la complexité d’un nouveau problème de coloration appelé k-COLORATION ADDITIVE, combinant théorie des graphes et théorie des nombres. Nous montrons que ce nouveau problème est NP-complet pour tout k ≥ 4 fixé, tandis qu’il peut être résolu en temps polynomial sur les arbres pour k quelconque et non fixé. / Graphs decompositions of small width are usually used to solve efficiently problems which are difficult in general. In this thesis, we focus on some limits of these decompositions, and the construction of some obstructions certifying a large width. First, we give a generic algorithm unifying obstructions’ construction for several graph widths, in XP time when parameterized by the considered width. In particular, it gives the first algorithm computing efficiently an obstruction to tree-width in time O(ntw+4). Secondly, we study the parameterized complexity of [σ, ρ]-DOMINATING SET, a generalization of some domination problems characterized by two sets of integers σ and ρ. All known studies focused only on cases where this problem is FPT when parameterized by tree-width. In this work, we show that there are some cases where the problem is no longer FPT, and become W[1]-hard instead. Finally, we study the computational complexity of a new coloration problem, named k-ADDITIVE COLORING, which combines both graph theory and number theory. We show that this new problem is NP-complete for any fixed number k ≥ 4, while it can be solved in polynomial time on trees for any k.

In the study of mixed graphs, a common question is: What are the necessary and suffcient conditions for the existence of a decomposition of the complete mixed graph into isomorphic copies of a given mixed graph? Since the complete mixed graph has twice as many arcs as edges, then an obvious necessary condition is that the isomorphic copies have twice as many arcs as edges. We will prove necessary and suffcient conditions for the existence of a decomposition of the complete mixed graphs into mixed stars with two edges and four arcs. We also consider some special cases of decompositions of the complete mixed graph into partially oriented stars with twice as many arcs as edges. We employ difference methods in most of our constructions when showing suffciency. 2

Let $M_v$ denotes a complete mixed graph on $v$ vertices, and let $S_6^i$ denotes the partial orientation of the 6-star with twice as many arcs as edges. In this work, we state and prove the necessary and sufficient conditions for the existence of $\lambda$-fold decomposition of a complete mixed graph into $S_6^i$ for $i\in\{1,2,3,4\}$. We used the difference method for our proof in some cases. We also give some general sufficient conditions for the existence of $S_6^i$-decomposition of the complete bipartite mixed graph for $i\in\{1,2,3,4\}$. Finally, this work introduces the decomposition of a complete mixed graph with a hole into mixed stars.

Indiana University-Purdue University Indianapolis (IUPUI) / Structural analysis and mining of large and complex graphs for describing the characteristics of a vertex or an edge in the graph have widespread use in graph clustering, classification, and modeling. There are various methods for structural analysis of graphs including the discovery of frequent subgraphs or network motifs, counting triangles or graphlets, spectral analysis of networks using eigenvectors of graph Laplacian, and finding highly connected subgraphs such as cliques and quasi cliques. Unfortunately, the algorithms for solving most of the above tasks are quite costly, which makes them not-scalable to large real-life networks. Two such very popular decompositions, k-core and k-truss of a graph give very useful insight about the graph vertex and edges respectively. These decompositions have been applied to solve protein functions reasoning on protein-protein networks, fraud detection and missing link prediction problems. k-core decomposition with is linear time complexity is scalable to large real-life networks as long as the input graph fits in the main memory. k-truss on the other hands is computationally more intensive due to its definition relying on triangles and their is no linear time algorithm available for it. In this paper, we propose distributed algorithms on Apache Spark for k-truss and k-core decomposition of a graph. We also compare the performance of our algorithm with state-of-the-art Map-Reduce and parallel algorithms using openly available real world network data. Our proposed algorithms have shown substantial performance improvement.