A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory (Second Edition)

It goes without saying that the text covers the classic areas, i.e. combinatorial choice problems and graph theory. What is unusual, for an undergraduate textbook, is that the author has included a number of more elaborate concepts, such as Ramsey theory, the probabilistic method and - probably the first of its kind - pattern avoidance. While the reader can only skim the surface of these areas, the author believes that they are interesting enough to catch the attention of some students. As the goal of the book is to encourage students to learn more combinatorics, every effort has been made to provide them with a not only useful, but also enjoyable and engaging reading.

The book is very exciting to read, and the author has a wonderful sense of humor: in Chapter 3 he introduces the idea of a permutation by the example of n people arriving at a dentist's office at the same time. They must decide the order in which they will be served. How many orders are possible?

The problems are a great strength of this text. Each chapter ends with a set of exercises with solutions. These tend to be very interesting and often quite challenging. A set of supplementary exercises follows. These tend to be a little easier, though not always, and make good homework assignments. The supplementary exercises do not have solutions, but a solutions manual is available to instructors.

The book walks through four parts: I. Basic Methods; II. Enumerative Combinatorics; III. Graph Theory; IV. Horizons. I particularly like the fourth part which includes Ramsey Theory, subsequence conditions on permutations, the probabilistic method, and partial orders and lattices. A glimpse of these subjects can whet the walker's appetite for more challenging terrain.

I would have liked to give this book 5 stars, but it suffers from a lack of clarity in some places. For example, the discussion of example 2.2 in Chapter 2 on induction just does not read clearly or make sense as it is written. Though an instructor can figure out what is missing, it would be much harder for a student to do so. And figure 13.1 on the colors of the edge of a triangle in Chapter 13 on Ramsey Theory is mislabeled. Again, this could steer an unwary student off the path of understanding. But these defects are minor compared to the riches contained in this text. The author has chosen his subjects carefully, illustrated them well and provided a wealth of wonderful exercises. And he has given the reader a glimpse of some of the less traditional and newer areas of combinatorics at the end of the book.


Well structured book   December 3, 2005
Rui Jiang (BELLEVUE, WA USA)
8 out of 8 found this review helpful

The best thing I like about this book, is that it has carefully selected subjects and rich set of exercises with detailed solutions. For the first several chapters, there are even more pages devoted to exercises+answers than the text. I think it is better to learn math by doing exercises than memorizing lots of theorems.

I would have given it 5 stars if there were not so many typos. It is annoying because a lot of times when I puzzled about something, it turns out be a typo. There are several versions of errata online, and none of them is complete. :) You can find them here:

http://www-math.mit.edu/~apost/courses/18.314/

I hope the author will correct all those typos then this would be the very best textbook!



A good book on combinatorics   July 9, 2008
Chee Lim Cheung
This is a good book on combinatorics. The problems constitute the real meat of this text. They range from the elementary to the very challenging. The first set of exercises has solutions. A set of supplementary exercises (without solutions) follows. This is a suitable book for self-study. Another combinatorics book worth looking at is Martin's Counting:The Art of Enumerative Combinatorics.