pearls in graph theory: a comprehensive introduction.by n. hartsfield; g. ringel

Pearls in Graph Theory: A Comprehensive Introduction. by N. Hartsfield; G. RingelReview by: John S. MaybeeSIAM Review, Vol. 33, No. 4 (Dec., 1991), pp. 664-665Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2031030 .

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664 BOOK REVIEWS

general reader had translations been provided for the quotations in French, German, Italian, and Latin that are scattered throughout the text.

Coolidge concluded his own 1949 Introduc- tion with the remark that he would be "most happy [if] the number of men included [were] doubled or trebled." In his 1951 review of this book in the Mathematical Monthly [58, No. 2 (1951), pp. 118-119], F. B. Wiley reiterates the hope that someone might undertake the task of continuing this type of study and writ- ing. In this tradition, let me add my own hope that the new 1990 edition of Coolidge will in- spire others to make additional episodes in the history of mathematics so accessible to as wide a readership.

PETER BRAUNFELD University of Illinois at Urbana-Champaign

An Introduction to Dynamical Systems. By D. K. Arrowsmith and C. M. Place. Cambridge University Press, New York, 1990. iv + 423 pp. $79.50, cloth; $29.95, paper. ISBN 0-52 1- 30362- 1.

There are many books with the word "in- troduction" in the title, but few are truly com- prehensible to a novice. In order to test the level of this book I lent my review copy to two graduate students, one in mathematics and one in engineering, who were taking my course on dynamical systems. Both of them came back with a rave review. This book is a true intro- duction to the modem theory of dynamical systems. They found the text interesting and easy to read. It has many clear figures and it has copious exercises which were readily solv- able and instructive.

This book tries to cover a vast amount of material, as can be seen from the summary of the chapters given below. Therefore, it does not present the proofs of the more difficult re- sults in the theory. This makes the book more accessible to the engineer or scientist, but less useful for the research mathematician in train- ing. Also, by omitting the more difficult proofs, the level of difficulty is reduced considerably, thus justifying the word "introduction" in the title. It is clear that the authors learned the material from secondary sources, since many major theorems are credited to expositors and not the discoverers.

With these caveats, I recommend this text. The following is an outline of the chapters.

Chapter 1: Diffeomorphisms andflows. The definitions of manifold, diffeomorphism, flow, invariant set, conjugacy, Poincare map, and suspension are given and illustrated. A brief discussion of Hamiltonian systems is given also.

Chapter 2: Local properties offlows and dif- feomorphisms. Hyperbolic critical and fixed points are introduced and the stable manifold and Hartman-Grobman theorems are ex- plained but not proven. Normal forms for nonhyperbolic points are discussed in several cases. The center manifold and simple blowing- up techniques are explained.

Chapter 3: Structural stability, hyperbolicity, and homoclinic points. This chapter discusses a number of classics such as Piexoto's work on the structural stability of flows on a two man- ifold, Anosov's work on the structural stability of Anosov diffeomorphisms, Smale's horse- shoe, the Melnikov method, etc. Here, as stated above, the text is easy to read, the figures are clear, the topics are interesting, but the tough proofs are missing.

Chapter 4: Local bifurcations I: planar vector fields and diffeomorphisms on R. The saddle- node, Hopf, and Takens-Bogdanov bifurca- tions in two-dimensional flows are treated along with similar bifurcations of one-dimen- sional diffeomorphisms. The chapter ends with a discussion of the logistic map.

Chapter 5: Local bifurcations II: Diffeo- morphisms of R2. Here we find Arnol'd tongues and the unfolding of various reso- nances.

Chapter 6: Area-preserving maps and their perturbations. The book finishes with a quick introduction to many important ideas in Hamiltonian systems. Birkhoff's normal form, the Poincare-Birkhoff theorem, generic bifur- cation of periodic points, KAM theory and the Aubry-Mather theory, and Chenciner bifur- cations are all touched on.

KENNETH R. MEYER

University of Cincinnati

Pearls in Graph Theory: A Comprehensive In- troduction. By N. Hartsfield and G. Ringel. Academic Press, San Diego, 1990. ix + 246 pp. $29.95. ISBN 0-12-328552-6.

The reviewer has ambivalent feelings about this little book on graph theory. This ambiv- alence is induced both by the phrase "A Com- prehensive Introduction" in the title as well as

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BOOK REVIEWS 665

the statement on the back cover that "The se- lection of material . . . includes all major top- ics and theorems ...." It was indeed disap- pointing to find that several important topics are omitted and a number of basic concepts of graph theory are not even mentioned. To be specific, although the authors introduce the concept of a bridge, the related concepts of cutpoints and blocks are never mentioned. Thus, there is no discussion anywhere of bi- connectedness or of connectivity in general. Several classes of graphs which are important in current research are also not mentioned. These include line graphs, two-step graphs, in- terval graphs, and unit interval graphs, al- though, interestingly, certain analogous graphs in two dimensions to the interval graphs and unit interval graphs are introduced briefly and called, respectively, coin graphs and penny graphs. While the text contains many inter- esting results about complete graphs, the con- cept of a clique is never introduced; hence there is no mention whatsoever of covering problems which have an honorable history beginning with Erdos and are the subject of a great deal of current research in graph theory. Perhaps the most serious omission in the reviewer's opinion is that no matrices are introduced. Thus, the student learning graph theory from this text would never realize that there is a symbiotic relationship between graph theory and matrix theory. In view of the rather intense current interest in both algebraic graph theory and in combinatorial matrix analysis, this omission seems the most serious of all.

Having pointed out that the claims of com- prehensiveness for this introduction to graph theory are somewhat dubious, let me turn to the positive features of the text. First of all, this little book does indeed contain many pearls. It is the authors' intention, as stated in the foreword, to maximize the pleasure for both the teacher and the student in learning graph theory and to stimulate the student with a new view of mathematics. The authors accomplish this very well. The topics covered in this text are presented beautifully with great care and attention to detail. The examples used always illustrate the ideas accurately and enhance the readers' understanding of the material pre- sented. The exercises are extensive and chosen so that they provide the student with an op- portunity both to apply the ideas introduced in the text and, in some cases, to be motivated to further study. For each topic covered the

authors either show how to solve the problem or carry the reader to the point where inter- esting unsolved problems occur naturally. Thus, I found the book a joy to read; and, since it covers a variety of interesting subjects, some in greater depth than can be found in most other books, many graph theorists will wel- come it to their library.

The selection of topics covered reflects the main interests of the senior author, Gerhard Ringel. Thus, four chapters out of ten are de- voted to coloring problems, planarity, and graphs on surfaces, with the result that the text goes far more deeply and carefully into these topics than other introductory texts do. There is a chapter devoted to extremal problems which presents Turan's Theorem, discusses cages, and introduces the student to Ramsey theory. Still another chapter on counting in- troduces the problem of determining how many 1-factors are in Kn n and similar prob- lems about counting subgraphs as well as de- rangements, Cayley's spanning tree formula, and related ideas. There is also a chapter on labeling graphs. Here the student learns about magic graphs, graceful trees, and antimagic graphs. The authors also present a very brief introduction to directed graphs in this chapter, taking the opportunity to discuss the notion of conservative graphs and Kirchoff's current law. Thus they come remarkably close to the con- cepts revolving around flows in networks without actually touching upon this important topic at all. Finally, there is a chapter on ap- plications and algorithms. Here spanning tree algorithms-including the algorithm of Krus- kal for finding a minimum weight spanning tree-are presented. Matching algorithms, in particular, the Hungarian algorithm, are pre- sented for bipartite graphs. Here also is found some further discussion of directed graphs, which leads to a brief treatment of binary trees and prefix codes and an illustration of how to use Huffman's algorithm.

Those who belong to the school of thought which believes that in an introduction to a subject as comprehensive as graph theory it is more important to stimulate the student's in- terest and arouse his curiosity than it is to pro- vide him with background in a specific range of topics will find this an excellent introductory text.

JOHN S. MAYBEE University of Colorado

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