The field of graph theory has undergone tremendous growth during the past century. As recently as the 1950s, the graph theory community had few members and most were in Europe and North America; today there are hundreds of graph theorists and they span the globe. By the mid 1970s, the subject had reached the point where we perceived a need for a collection of surveys of various areas of graph theory: the result was our three-volume series Selected Topics in Graph Theory, comprising articles written by distinguished experts and then edited into a common style. Since then, the transformation of the subject has continued, with individual branches (such as chromatic graph theory) expanding to the point of having important subdivisions themselves. This inspired us to conceive of a new series of books, each a collection of articles within a particular area of graph theory written by experts within that area. The first three of these books were the companion volumes to the present one, on algebraic graph theory, topological graph theory and structural graph theory. This is thus the fourth volume in the series.

A special feature of these books is the engagement of academic consultants (here, Bjarne Toft) to advise us on topics to be included and authors to be invited.We believe that this has been successful, with the result that the chapters of each book cover the full range of area within the given area. In the present case, the area is chromatic graph theory, with chapters written by authors from around the world. Another important feature is that, where possible, we have imposed uniform terminology and notation throughout, in the belief that this will aid readers in going from one chapter to another. For a similar reason, we have not tried to remove a small amount of material common to some of the chapters.

We hope that these features will facilitate usage of the book in advanced courses and seminars.

The origins of topological graph theory lie in the 19th century, largely with the four colour problem and its extension to higher-order surfaces – the Heawood map problem. With the explosive growth of topology in the early 20th century, mathematicians like Veblen, Rado and Papakyriakopoulos provided foundational results for understanding surfaces combinatorially and algebraically. Kuratowski, MacLane and Whitney in the 1930s approached the four colour problem as a question about the structure of graphs that can be drawn without edge-crossings in the plane. Kuratowski's theorem characterizing planarity by two obstructions is the most famous, and its generalization to the higher-order surfaces became an influential unsolved problem.

The second half of the 20th century saw the solutions of all three problems: the Heawood map problem by Ringel, Youngs et al. by 1968, the four colour problem by Appel and Haken in 1976, and finally the generalized Kuratowski problem by Robertson and Seymour in the mid-1990s. Each is a landmark of 20th-century mathematics. The Ringel–Youngs work led to an alliance between combinatorics and the algebraic topology of branched coverings. The Appel–Haken work was the first time that a mathematical theorem relied on exhaustive computer calculations. And the Robertson–Seymour work led to their solution of Wagner's conjecture, which provides a breathtaking structure for the collection of all finite graphs, a collection that would seem to have no structure at all.

Each of these problems centres on the question of which graphs can be embedded in which surfaces, with two complementary perspectives – fixing the graph or fixing the surface.

The field of graph theory has undergone tremendous growth during the past century. As recently as fifty years ago, the graph theory community had few members and most were in Europe and North America; today there are hundreds of graph theorists and they span the globe. By the mid-1970s, the field had reached the point where we perceived the need for a collection of surveys of the areas of graph theory: the result was our three-volume series Selected Topics in Graph Theory, comprising articles written by distinguished experts in a common style. During the past quarter-century, the transformation of the subject has continued, with individual areas (such as topological graph theory) expanding to the point of having important sub-branches themselves. This inspired us to conceive of a new series of books, each a collection of articles within a particular area written by experts within that area. The first of these books was our companion volume on algebraic graph theory, published in 2004. This is the second of these books.

One innovative feature of these volumes is the engagement of academic consultants (here, Jonathan Gross and Thomas Tucker) to advise us on topics to be included and authors to be invited. We believe that this has been successful, the result being chapters covering the full range of areas within topological graph theory written by authors from around the world. Another important feature is that we have imposed uniform terminology and notation throughout, as far as possible, in the belief that this will aid readers in going from one chapter to another.