The Foundations of Topological Graph Theory

This is a text on topological graph theory written from a purely combinatorial viewpoint. The aim is to develop a rigorous approach to the foundation of the subject, and the book should therefore appeal to graduate students and researchers in topological graph theory. The basic tool used is the idea of a 3-graph which is a cubic graph endowed with a proper edge colouring in three colours. A special case of a 3-graph, called a gem, provides a model for a cellular imbedding of a graph in a surface. Thus theorems about imbeddings of graphs become theorems about gems. The book shows many of these theorems generalize to theorems about 3-graphs.