Oblique Graphs

The main issue of this work is to investigate asymmetric structures in graphs. While symmetry structures in graphs are well observed, the opposite question has not been investigated deeply so far. It is known from a theorem of Wright, that almost all graphs are asymmetric. The class of asymmetric graphs is restricted further by forbidding even local symmetries. The main question is to determine how many graphs do have this stronger so-called obliqueness property. On the one hand the superclass of asymmetric graphs is not only infinite, but contains almost all graphs. On the other hand many Ramsey type results indicate, that growing structures enforce at least local symmetries or regularities. In the first part of the work oblique structures in polyhedral graphs are investigated. Several type definitions for faces and edges of a polyhedral graph are introduced. A polyhedral graph is called oblique with respect to a type definition if it does not contain two faces or edges, respectively, which are of the same type. The main results of that part are theorems that show the finiteness of the class of such oblique polyhedral graphs. In some cases it is shown that a large enough polyhedral graph always contains more than z faces or edges of a common type, where z is an arbitrary positive integer. The finiteness results on oblique polyhedral graphs are generalized for maps on orientable surfaces. In the second part of the work oblique structures in arbitrary graphs are investigated. A graph is called vertex-oblique if any two vertices differ in the degree sequence of their neighbors. It is shown that even under additional restrictive properties for the considered graphs the set of vertex-oblique graphs is infinite. Moreover, it is proved, that the probability of a random graph to be vertex-oblique tends to 1 as the order of the graph tends to infinity if the probability p of an edge to belong to the edge set of the graph is within given bounds.