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Matroid theory. (English) Zbl 0784.05002
Oxford Graduate Texts in Mathematics 3. Oxford Science Publications. Oxford: Oxford University Press (ISBN 0-19-853563-5/hbk). xi, 532 p. £45.00 (1992).

There are two fundamental problems which the author had to face in writing Matroid Theory. The first was the decision of what material to leave out and the second was how to organize what was left. This could be said of almost any book, but with matroid theory both problems are quite difficult, particularly the second. There are over a half a dozen different ways to approach the subject. Of these, three lead to three fundamentally different ways of interpreting a matroid and correspond to three major early papers in the field: H. Whitney [On the abstract properties of linear dependence, Am. J. Math. 57, 509-533 (1935; Zbl 0012.00404)] (the defining paper); W. T. Tutte [A homotopy theorem for matrois.I,II. Trans. Am. Math. Soc. 88, 144-174 (1958; Zbl 0081.173)] and H. H. Crapo and G.-C. Rota [On the foundations of combinatorial theory: Combinatorial geometries, MIT Press (1970; Zbl 0216.021)].

Each of these three points of view motivates its own body of results and each has played a major role in the growth of matroid theory. In the first part of this text, Chapters 1-6, the author does an admirable job of balancing these three approaches in an excellent introduction to the basics of matroid theory. As the author states, these chapters with some deletion could be the basis of a one semester introduction to matroid theory. The remaining chapters are a selection of advanced topics from matroid theory. (No single text can cover all of matroid theory.) The titles of these chapters are: Chapter 7: Construction; Chapter 8: Higher connectivity, Chapter 9: Binary matroids, Chapter 10: Ternary matroids, Chapter 11: The splitter theorem, Chapter 12: Submodular functions and matroid union, Chapter 13: Regular matroids, Chapter 14: Unsolved problems.

These topics are admirably orchestrated so that the author has been able to achieve the following goal which he had set for himself: “One of the main tasks of the second half of the book is to present these proofs in reasonably full detail. I have never enjoyed reading proofs in which numerous intermediate steps are left to the reader, so I have tried to avoid writing such proofs.” The book includes a large variety of very good exercises (over 500) with a great variation in difficulty, and closes with a full chapter of the questions in fields which remain open.

The reviewer cannot as yet claim to have read the entire 500 pages of this text, but what he has read has been so well written, he is sure that within the next semester or so he will have read it from cover to cover.


MSC:
05-02Research monographs (combinatorics)
05B35Matroids, geometric lattices (combinatorics)