In (J Graph Theory 33 (2000), 14-24), Hell and Zhu proved that if a series-parallel graph G has girth at least 2⌊(3k-1)/2⌋, then χc(G)≤4k/(2k-1). In (J Graph Theory 33 (2000), 185-198), Chien and Zhu proved that the girth condition given in (J Graph Theory 33 (2000), 14-24) is sharp. Short proofs of both results are given in this note.

Let G and H be graphs. We say G is H-critical, if every proper subgraph of G except G itself is homomorphic to H. This generalizes the widely known concept of k-color-critical graphs, as they are the case H = Kk - 1. In 1963 [T. Gallai, Kritiche Graphen, I., Magyar Tud. Akad. Mat. Kutató Int. Közl. 8 (1963), 373-395], Gallai proved that the vertices of degree k in a Kk-critical graph induce a subgraph whose blocks are either odd cycles or complete graphs. We generalize Gallai's Theorem for every H-critical graph, where H = Kk - 2 + H′, (the join of a complete graph Kk - 2 with any graph H′). This answers one of the two unknown cases of a problem given in [J. Nešetřil, Y. Nigussie, Finite dualities and map-critical graphs on a fixed surface. (Submitted to Journal of Combin. Theory, Series B)]. We also propose an open question, which may be a characterization of all graphs for which Gallai's Theorem holds.

Let G be a graph. For k ≥ d ≥ 1, a k/d -coloring of G is a coloring c of vertices of G with colors 0, 1, 2, . . ., k - 1, such that d ≤ | c(x) - c(y) | ≤ k - d, whenever xy is an edge of G. We say that the circular chromatic number of G, denoted χc(G), is equal to the smallest k/d where a k/d -coloring exists. In [6], Pan and Zhu have given a function μ(g) that gives an upper bound for the circular-chromatic number for every K4-minor-free graph Gg of odd girth at least g, g ≥ 3. In [7], they have shown that their upper bound in [6] can not be improved by constructing a sequence of graphs approaching μ(g) asymptotically. We prove that for every odd integer g = 2k + 1, there exists a graph Gg ∈ G/K4 of odd girth g such that χc(Gg) = μ(g) if and only if k is not divisible by 3. In other words, for any odd g, the question of attainability of μ(g) is answered for all g by our results. Furthermore, the proofs [6] and [7] are long and tedious. We give simpler proofs for both of their results.

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We define a variation of injective oriented colouring as reflexive injective oriented colouring, or rio-colouring for short, which requires an oriented colouring to be injective on the neighbourhoods of the underlying graph, without requiring the colouring to be proper. An analysis of the complexity of the problem of deciding whether an oriented graph has a k-rio-colouring is considered, and we find a dichotomy for the values of k below 3 and above, being in P or NP-complete, respectively. Characterizations are given for the oriented graphs resulting from the polynomial-time solvable cases, and bounds are given for the rio-chromatic number in terms of maximum in-degree and out-degree, in general, and for oriented trees. Also, a polynomial-time algorithm is developed to aid in the rio-colouring of oriented trees.

We find an orientation of a tree with 20 vertices such that the corresponding fixed-template constraint satisfaction problem (CSP) is NP-complete, and prove that for every orientation of a tree with fewer vertices the corresponding CSP can be solved in polynomial time. We also compute the smallest tree that is NL-hard (assuming L≠NL), the smallest tree that cannot be solved by arc consistency, and the smallest tree that cannot be solved by Datalog. Our experimental results also support a conjecture of Bulín concerning a question of Hell, Nešetřil and Zhu, namely that ‘easy trees lack the ability to count’. Most proofs are computer-based and make use of the most recent universal-algebraic theory about the complexity of finite-domain CSPs. However, further ideas are required because of the huge number of orientations of trees. In particular, we use the well-known fact that it suffices to study orientations of trees that are cores and show how to efficiently decide whether a given orientation of a tree is a core using the arc-consistency procedure. Moreover, we present a method to generate orientations of trees that are cores that works well in practice. In this way we found interesting examples for the open research problem to classify finite-domain CSPs in NL.

The aim of this thesis is to bring together two areas of the graph theory. We first give a brief exposition of graph homomorphisms and related notions that directs to the definition of the replacement operation by using an appropriate replacement graph. The tree-depth is investigated as one of considerable chara- cteristics of each graph. Finally we focus on category representations in the category Graph of all finite graphs and Graphk of all finite graphs with tree-depth at most k.

Le Cloud est devenu une plate-forme très répandue pour le déploiement d'applications distribuées. Beaucoup d'entreprises peuvent sous-traiter leurs infrastructures d'hébergement et, ainsi, éviter des dépenses provenant d'investissements initiaux en infrastructure et de maintenance.Des petites et moyennes entreprises, en particulier, attirés par le modèle de coûts sur demande du Cloud, ont désormais accès à des fonctionnalités comme le passage à l'échelle, la disponibilité et la fiabilité, qui avant le Cloud étaient presque réservées à de grandes entreprises.Les services du Cloud peuvent être offerts aux utilisateurs de plusieurs façons. Dans cette thèse, nous nous concentrons sur le modèle d'Infrastructure sous Forme de Service. Ce modèle permet aux utilisateurs d’accéder à des ressources de calcul virtualisés sous forme de machine virtuelles (MVs).Pour installer une application distribuée, un client du Cloud doit d'abord définir l'association entre son application et l'infrastructure. Il est nécessaire de prendre en considération des contraintesde coût, de ressource et de communication pour pouvoir choisir un ensemble de MVs provenant d'opérateurs de Cloud publiques et privés le plus adaptés. Cependant, étant donné la quantité exponentiel de configurations, la définition manuelle de l'association entre application et infrastructure peut être un challenge dans des scénarios à large échelle ou ayant des contraintes importantes de temps. En effet, ce problème est une généralisation du problème de calcul de homomorphisme de graphes, qui est NP-complet.Dans cette thèse, nous adressons le problème de calculer des placements initiaux et de reconfiguration pour des applications distribuées sur potentiellement de multiples Clouds. L'objectif est de minimiser les coûts de location et de migration en satisfaisant des contraintes de ressources et communications. Pour cela, nous proposons des heuristiques performantes capables de calculer des placements de bonne qualité très rapidement pour des scénarios à petite et large échelles. Ces heuristiques, qui sont basées sur des algorithmes de partition de graphes et de vector packing, ont été évaluées en les comparant avec des approches de l'état de l'art comme des solveurs exactes et des méta-heuristiques. Nous montrons en utilisant des simulations que les heuristiques proposées arrivent à calculer des solutions de bonne qualité en quelques secondes tandis que des autres approches prennent des heures ou jours pour les calculer. / The Cloud has become a very popular platform for deploying distributed applications. Today, virtually any credit card holder can have access to Cloud services. There are many different ways of offering Cloud services to customers. In this thesis we especially focus on theInfrastructure as a Service (IaaS), a model that, usually, proposes virtualized computing resources to costumers in the form of virtual machines (VMs). Thanks to its attractive pay-as-you-use cost model, it is easier for customers, specially small and medium companies, to outsource hosting infrastructures and benefit of savings related to upfront investments and maintenance costs. Also, customers can have access to features such as scalability, availability, and reliability, which previously were almost exclusive for large companies. To deploy a distributed application, a Cloud customer must first consider the mapping between her application (or its parts) to the target infrastructure. She needs to take into consideration cost, resource, and communication constraints to select the most suitable set of VMs, from private and public Cloud providers. However, defining a mapping manually may be a challenge in large-scale or time constrained scenarios since the number of possible configuration explodes. Furthermore, when automating this process, scalability issues must be taken into account given that this mapping problem is a generalization of the graph homomorphism problem, which is NP-complete.In this thesis we address the problem of calculating initial and reconfiguration placements for distributed applications over possibly multiple Clouds. Our objective is to minimize renting and migration costs while satisfying applications' resource and communication constraints. We concentrate on the mapping between applications and Cloud infrastructure. Using an incremental approach, we split the problem into three different parts and propose efficient heuristics that can compute good quality placements very quickly for small and large scenarios. These heuristics are based on graph partition and vector packing heuristics and have been extensively evaluated against state of the art approaches such as MIP solvers and meta-heuristics. We show through simulations that the proposed heuristics manage to compute solutions in a few seconds that would take many hours or days for other approaches to compute.