Some Unsolved Problems in Map Enumeration

For many years, people have considered various problems in the enumeration of graphs, beginning with the enumeration of trees. Extensive work on the enumeration of maps did not begin until Tutte's work in the 1960's; however, one map enumeration problem| the determination of the number of combinatorially inequivalent 3-dimensional convex polytopes|is over a century old. Traditionally an map has been thought of as a connected unlabeled graph which has been embedded in (= continuous injection into) a sphere. In the 1960's various classes of (rooted) maps were enumerated thanks to the pioneering work of Tutte. His approach consists of three main steps: