Topics in Topological Graph Theory

The use of topological ideas to explore various aspects of graph theory, and vice versa, is a fruitful area of research. There are links with other areas of mathematics, such as design theory and geometry, and increasingly with such areas as computer networks where symmetry is an important feature. Other books cover portions of the material here, but there are no other books with such a wide scope. This book contains fifteen expository chapters written by acknowledged international experts in the field. Their well-written contributions have been carefully edited to enhance readability and to standardize the chapter structure, terminology and notation throughout the book. To help the reader, there is an extensive introductory chapter that covers the basic background material in graph theory and the topology of surfaces. Each chapter concludes with an extensive list of references.

Preface; Foreword Jonathan L. Gross and Thomas W. Tucker; Introduction Lowell W. Beineke and Robin J. Wilson; 1. Embedding graphs on surfaces Jonathan L. Gross and Thomas W. Tucker; 2. Maximum genus Jianer Chen and Yuanqiu Huang; 3. Distributions of embeddings Jonathan L. Gross; 4. Algorithms and obstructions for embeddings Bojan Mohar; 5. Graph minors: generalizing Kuratowski's theorem R. Bruce Richter; 6. Colouring graphs on surfaces Joan P. Hutchinson; 7. Crossing numbers R. Bruce Richter and G. Salazar; 8. Representing graphs and maps Tomaž Pisanski and Arjana Žitnik; 9. Enumerating coverings Jin Ho Kwak and Jaeun Lee; 10. Symmetric maps Jozef Širán and Thomas W. Tucker; 11. The genus of a group Thomas W. Tucker; 12. Embeddings and geometries Arthur T. White; 13. Embeddings and designs M. J. Grannell and T. S. Griggs; 14. Infinite graphs and planar maps Mark E. Watkins; 15. Open problems Dan Archdeacon; Notes on contributors; Index of definitions.

The use of topological ideas to explore various aspects of graph theory, and vice versa, is a fruitful area of research. There are links with other areas of mathematics, such as design theory and geometry, and increasingly with such areas as computer networks where symmetry is an important feature.

Lowell W. Beineke is Schrey Professor of Mathematics at Indiana University–Purdue University Fort Wayne, where he has been since receiving his Ph.D. from the University of Michigan under the guidance of Frank Harary. His graph theory interests are broad, and include topological graph theory, line graphs, tournaments, decompositions and vulnerability. With Robin Wilson he edited Selected Topics in Graph Theory (3 volumes), Applications of Graph Theory, Graph Connections and Topics in Algebraic Graph Theory. Until recently he was editor of the College Mathematics Journal.
Robin J. Wilson is Professor of Pure Mathematics at the Open University, UK, and Emeritus Professor of Geometry at Gresham College, London. After graduating from Oxford, he received his Ph.D. in number theory from the University of Pennsylvania. He has written and edited many books on graph theory and the history of mathematics, including Introduction to Graph Theory and Four Colours Suffice, and his research interests include graph colourings and the history of combinatorics, and he has Erdos number 1. He has won a Lester Ford Award and a George Pólya Award from the MAA for his expository writing.
Jonathan L. Gross is Professor of Computer Science at Columbia University. His mathematical work in topology and graph theory have earned him an Alfred P. Sloan Fellowship, an IBM Postdoctoral Fellowship, and numerous research grants. With Thomas Tucker, he wrote Topological Graph Theory and several fundamental pioneering papers on voltage graphs and on enumerative methods. He has written and edited eight books on graph theory and combinatorics, seven books on computer programming topics, and one book on cultural sociometry.
Thomas W. Tucker is the Charles Hetherington Professor of Mathematics at Colgate University, where he has been since 1973, after a PhD in 3-manifolds from Dartmouth in 1971 and a postdoc at Princeton (where his father A. W. Tucker was chairman and John Nash's thesis advisor). He is co-author (with Jonathan Gross) of Topological Graph Theory. His early publications were on non-compact 3-manifolds, then topological graph theory, but his recent work is mostly algebraic, especially distinguishability and the group-theoretic structure of symmetric maps.