Recent Results in Topological Graph Theory*

A graph G is usually defined as a finite collection V of points together with a collection X of lines, each of which joins two distinct points and no two of which join the same pair of points. This combinatorial definition asserts nothing about drawing graphs on surfaces such as the plane, sphere, torus, projective plane etc. The purpose of this lecture is to explore some of these topological aspects of graph theory and to describe a few unsolved problems concerning them. In order to fix the terminology of this iecture, we begin by drawing all the graphs with four points: