Abstract
The theory of graph minors appeared in the first part of the twentieth century with the characterization of planar graphs by Kuratowski and Wagner. The study of classes of graphs closed under minors has applications to many areas of graph theory (graphs embedded in surfaces, coloring, extremal graph theory, theory of rigidity, ...). The first part of this thesis is devoted to prove the existence of minors of complete graphs in graphs where each edge belongs to a certain number of triangles. This property has applications to the theory of rigidity and to the coloration of some minor-closed classes. A second part is devoted to the generalization this property from graphs to matroids. Matroids are combinatorial objects introduced in 1935 by Whitney to axiomatize the concept of linear independence. In particular, the notions of triangle and minor can be generalized to these objects. We will study matroids in which every element belongs to a certain number of triangles and show that we can find some particular minors in these matroids. Finally, the last part of this thesis is devoted to the study of certain orientations of graphs embedded in surfaces.