The problem of defining cycles in infinite graphs has received much attention in the literature. Diestel and Kuhn have proposed viewing a graph as 1-complex, and defining a topology on the point set of the graph together with its ends. In this setting, a circle in the graph is a homeomorph of the unit circle S^1 in this topological space. For locally finite graphs this setting appears to be natural, as many classical theorems on cycles in finite graphs extend to the infinite setting. A Hamilton circle in a graph is a circle containing all the vertices of the graph. We exhibit a necessary and sufficient condition that a countable graph contain a Hamilton circle in terms of the existence of Hamilton cycles in an increasing sequence of finite graphs. As corollaries, we obtain extensions to locally finite graphs of Zhan's theorem that all 7-connected line graphs are hamiltonian (confirming a conjecture of Georgakopoulos), and Ryjacek's theorem that all 7-connected claw-free graphs are hamiltonian. A third corollary of our main result is Georgakopoulos' theorem that the square of every two-connected locally finite graph contains a Hamilton circle (an extension of Fleischner's theorem that the square of every two-connected finite graph is Hamiltonian).

Savoir comment transmettre une information est fondamental dans un réseau. Il est essentiel que chaque entité du réseau soit capable de décider localement, avec sa vue du réseau, du chemin par lequel l'information doit passer. Ainsi, il est souvent utile d'étudier la topologie du réseau, modélisée par un graphe, pour répondre à ces exigences. Nous nous intéressons dans un premier temps, à la décomposition arborescente des graphes planaires. En effet, comme dans beaucoup de problèmes de graphes, l'étude de la topologie des graphes nous conduit à procéder à une décomposition du graphe afin d'exploiter les propriétés structurelles qui en découlent. En suite, nous nous sommes aussi intéressés à la structure des graphes qui excluent un mineur H, en particulier le graphe K_{2,r}. Ces travaux nous ont permis d'améliorer les bornes actuelles connues sur la largeur arborescente de ces graphes. Dans la dernière partie, nous abordons le problème du routage compact. Nous nous sommes intéressés aux schémas de routage de plus courts chemins utilisant des adresses, des tables de routage de tailles optimales de O(log n) bits, où n est le nombre de sommets du graphe. Nous proposons un tel schéma de routage pour une famille de graphes valués contenant les arbres et les graphes planaire-extérieurs. / In a network, it is crucial to know how to construct an efficent routing scheme. It is fundamental for each entity with its local knowledge of the network, to be able to decide on which link to forward messages. Thus, it is important to sutdy the underlying network topology in order to design routing schemes. In the first part of this thesis, we construct a new tree-decomposition for planar graphs. In fact, as in many graph problems, the study of the graph structure leads to do a tree-decomposition for exploiting structural propertys of the graphs. In second part, we studied the structure of H-minor free graphs, in particular whenever H = K_{2,r}. Our results improve upon previous known bounds about the tree-width of K_{2,r}-minor free graphs. At last, we treat the problème of compact routing scheme. More precisely, we are interested in shortest-path routing schemes that use O(\log n) bits for addresses, headers and routing tables, where n is the number of vertices in the graph. We propose such a routing scheme for a large family of weighted graphs including outerplanar graphs.