11 of 12 people found the following review helpful:
3.0 out of 5 stars
Positives and Negatives, December 16, 2008
This book was assigned for a class in applied combinatorics, and in many instances I had to ask "why?"
The Positives: This book has the simplest introduction to building generating function models that I've ever come across, in regards to ordinary generating functions. The examples in this section really shine, and if you spend time on this section (Chapter 6.1) you shouldn't have too much trouble at all as you progress through the rest of the chapter. I cannot stress that enough! Master 6.1; the rest falls into place.
The intro to graph theory is just that: Tucker doesn't spend too much time on any major results, in fact treats Euler's polyhedral formula almost as an afterthought. I mean, I realize this is enumeration, but the fact that Euler's proof was really combinatorial in the first place is an excellent place to tie in a branch of mathematical study. The emphasis though are on graph problems and it gives an excellent study of two algorithms for solving the traveling salesman problem.
Binomial identities (the ultimate goal of chapter 5) aren't covered quite as comprehensively as I would have hoped. (see the book "Art of Combinatorial Proof") But the writing here is good and I only had to consult outside material on some of the problems.
The Negatives:
The treatment of recurrence relations are a joke. I mean, seriously! It starts out assuming you've spent time on Diff EQ (which I hadn't) and uses language such as 'obviously,' a damn dangerous word to anyone who studies math seriously!
In Homgeneous recurrence relations they introduce the general solution buried within the discussion and make the additional mistake of using the * symbol in the notation. Organization is my biggest gripe here, they should treat this solution like a theorem or do SOMETHING to make it stand out. I finally just complied my own chart for my notes because I got tired of flipping pages back and forth and scanning for what I wanted. Tucker also misses the fact that an explicit treatment of notation when multiple or complex roots should be in order.
For solving recurrence relations I went back to my trusty "Discrete and Combinatorial Mathematics" text from Grimaldi. Real theory, and plainly spoken. For an undergrad text I expect my hand to be held at least a little, and Tucker obviously thinks recurrence relations are no big deal.
They aren't once you know them. I'm in a class of mostly math majors and you should have seen their faces when in an Inhomogeneous solution the teacher finished with "...can be solved with partial fraction decomposition."
The final gripe is the lack of any solutions manual. Maybe that's the point. I'm supposed to look at a solution and reverse engineer it on my own; its just that many times I'm not getting the point in the first place and I need to see more solutions to different kinds of problems.
In conclusion, I will keep this book on my shelf. The problems are incredibly challenging, and once you can solve them with one method, you can go back and solve them with another, that's the fun part about combinatorics.
For other clear treatments, Grimaldi, Harris, and Bona.
Harris's book is at the upper end of the undergrad spectrum, but its brevity on some topics is especially excellent as its derivation of generating function models sped up my understanding of this crucial area.
Docked 1 star for retail price for such a thin tome, and 1 star for assorted problems.
For self-study, I'd recommend the Grimaldi text before you tackle this one, but this one is good if you just want a source of problems. Many of these books I suggest should be available in your school's library, to avoid racking up high book costs.
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11 of 13 people found the following review helpful:
4.0 out of 5 stars
Excellent for applications, June 28, 2004
This review is from: Applied Combinatorics (Hardcover)
The book covers the fundamentals of graph theory and combinatorics (enumeration) and is designed for first courses for undergraduates.
The material is presented in a clear, friendly manner. The sections are short and specific and the emphasis is on problem-solving. Many examples are provided and constitute the majority of the book's volume. Each section ends with 20-30 exercises with answers (not full solutions) at the end of the book.
The book is excellent for computer science and applied math majors looking for a clear, application-based introduction to combinatorics and graph theory. It is also excellent for self-study.
The book's main flaw is that the proofs are not rigorous and are sometimes more intuitive than mathematical. For pure math students looking to explore graph theory and combinatorics in a more rigorous manner, other books (e.g. Diestel, "Graph Theory") will serve that purpose better.
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2 of 2 people found the following review helpful:
2.0 out of 5 stars
Meandering Approach Leaves Me Frustrated, October 27, 2008
Very briefly, Tucker loses you through examples rather than developing his theoretical discussions.
I've noticed that some people have said that this text is clearly written, to me, nothing could be further from the truth. He seems to have a great knack for over-complicating simple ideas. This may be a personality thing -- some people really identify with his approach, I do not. I think he gets bogged down in smoothing out the details of his examples and definitions and ends up obfuscating points where simple brevity not only would have sufficed, but would have been more illuminating.
Also, except by inference on the reader's part, it can be difficult to distinguish the important from the trivial . . . this is a very poor text for self-study. I think I'm going to check out what selections Springer offers covering these topics, I've had pretty good luck finding well-written texts with their UTM series.
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