| Abstract: | Topological graph theory seeks to find answers to the question of how graphs map into surfaces. This paper surveys the information now available about the range of a graph, namely, the set of surfaces on which the graph can be “neatly” embedded. Several other closely related topics, such as irreducible graphs, coloring problems, and crossing numbers, are ignored. As is quite often the case with mathematical theories, this discipline developed in a rather haphazard manner. Many isolated results existed before the practitioners became aware of the fact that they were developing a theory. The turning point occurred in 1968, when Ringel and Youngs completed their proof of the Heawood conjecture. Their proof, in addition to settling an old unsolved problem, also reinforced the significance of the rotation systems. It is the author's belief that these rotation systems, together with the generalized embedding schemes can, and should, become the main tool in all investigations concerning the embeddings of a graph. This survey is written from that point of view. After defining the scope of the area surveyed, this paper proceeds to discuss the significance of the rotation systems and embedding schemes. Several theorems of a general nature are listed. Attention is then focused on the maximum and minimum genera of a graph. Discussion of the first of these is deferred to another survey article by R. Ringeisen to appear in a subsequent issue. The various methods developed by researchers in this area for determining the (minimum) genus are then described. This is followed by a listing of all the theoretical information that is available about the genus parameter. The paper includes two tables that exhibit most of the graphs with known genus. |
