The list of excluded minors for any minor-closed class is always finite. This is a consequence of the Graph Minor Theorem. However the lists are huge in general and impossible to find even with computers. Similarly to the Ph.D. Thesis of the applicant. We would like to refine the minor containment respect to toroidal graphs. Thereby we can get closer to a structural characterization. Tree-width is a useful (and today central) measure of graph. A graph contains a big grid minor if and only if it has large tree-width. Our goal is to study possible extensions for directed graphs. Hadwiger conjectured a surprising connection between the largest clique-minor and the chromatic number of a graph. We propose to develop some relaxations and variations of it, Which are hopefully of independent interest. These directions are connected to acyclic vertex-coloring and the list-chromatic number of a graph. The applicant would gain experience about graph embeddings by collaborating with an expert of the field. He gets the possibility to develop new areas in the theory of graphs using previous work of the host researchers and him. The benefits would include the continuation of the joint work between the fellow and C.Thomassen about graph colorings. The applicant will make a significant contribution to combinatorics within the profile of the Institute. Moreover he will join the graph theory group supported by the Danish Natural Science Research Council to carry out research with graph theorists in Odenseand Aalborg.