The foundations of topological graph theory

the long version

This book on topological graph theory is written from a purely combinatorial viewpoint. Its aim is to develop a rigorous approach to the foundations of the subject. 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 coloring in three colors. A special case of a 3-graph, called a gem, provides a model for a cellular embedding of a graph in a surface. Thus, theorems about embeddings of graphs become theorems about gems. The authors show that many of these theorems generalize to theorems about 3-graphs. Thus, results such as the classification of surfaces, and the theorem that the first Betti number of a surface is the largest number of closed curves that can be drawn on the surface without dividing it into two or more regions, find a general setting in the theory of 3-graphs. The book therefore uses 3-graphs to develop the foundations of topological graph theory and differs in this way from other books on this subject. Readers should find in its pages a fresh approach to a subject with which they may already have some familiarity.