Publisher: Springer (2009)Details: 795 pages, Hardcover
Edition: 2 Series: Springer Monographs in Mathematics
Price: $149.00
ISBN: 9781848009974
Category: Monograph
Topics: Graph Theory, Algorithms
Jǿrgen Bang-Jensen and Gregory Gutin
Publisher: Springer (2009)
Details: 795 pages, Hardcover
Edition: 2 Series: Springer Monographs in Mathematics
Price: $149.00
ISBN: 9781848009974
Category: Monograph
Topics: Graph Theory, Algorithms
[Reviewed by Miklós Bóna, on 02/19/2009]
This is a very comprehensive volume on directed graphs and related topics. It consists of 18 chapters (compared to 12 for the first edition) and almost 800 pages. This reviewer will not attempt to list the contents — prospective readers can find that in the link above. Instead, he will discuss a few characteristics of this impressive, well-conceived, and well-written book, and its possible uses.
First, the book is an excellent reference. The authors understand that even in 800 pages it would be impossible to cover everything about digraphs in a self-contained fashion, so they were not afraid to make some choices. They chose to be comprehensive, at the expense of being self-contained. This means that there is an unusually high number of results in the book that are not proved; but they are there, their place in the theory is discussed, and interested readers are given references for their proofs. You can be reasonably certain that if the result you are looking for is not proved in the book, the book will at least help you find it.
Second, for the reason given in the previous paragraph, this reviewer would not use the book for a regular, lecture-based, course, but rather for a series of seminars, where each student would be responsible for presenting a part of the material, including proofs that are in the book and proofs that are not in the book.
Third, unlike in a large number of research monographs, the authors do an excellent job putting their topic into context. Hence the book will be useful not just for researchers focusing on digraphs, but also for everyone else for whom digraphs are tools, and not goals. The applications discussed in the book are not limited to topics where digraphs are a necessity, such as networks. They include problems like Konig's theorem and Philip Hall's theorem, that may be proved by using digraphs, but may also be proved without using digraphs. This commendable approach will extend the reach of the book.
Last, but not least, an extensive list of conjectures and open questions is included in every chapter. These are typically very well presented and easy to understand. I am not taking a big risk when I predict that with the help of books like this, the area will continue to progress at a fast pace.
Miklós Bóna is associate professor of mathematics at the University of Florida.