A Source Book in Matroid Theory

400 Pages·1986·11.746 MB·English

by  Joseph P. S. Kung (auth.)

Joseph P.S. Kung
A Source Book in Matroid Theory
Springer Science+Business Media, LLC
Author's address
Joseph P.S. Kung,
Department of Mathematics,
North Texas State University,
Denton, TX 76203-5116 (USA)
Library of Congress Cataloging in Publication Data
Kung, Joseph P.S.
A source book in matroid theory.
Bibliography: p.
Includes index.
1. Matroids. I. Title.
QA166.6.K86 511'.6 84-24334
ISBN 978-0-8176-3173-4 ISBN 978-1-4684-9199-9 (eBook)
DOI 10.1007/978-1-4684-9199-9
CIP Kurztitelaufnahme der Deutschen Bibliothek
Kuug, Joseph P.S.:
A source book in matroid theory 1 Joseph P.S. Kung.
Boston; Basel; Stuttgart:
Birkhăuser, 1986
Ali rights reserved.
No part of this publication may be reproduced, stored in a retrieval system,
or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or
otherwise, without the prior permission of the copyright owner.
© 1986 Springer Science+Business Media New York
Originally published by Birhäuser Boston, Inc. 1986
Softcover reprint of the hardcover 1st edition 1986
Table of Contents
Foreword (by Gian-Carlo Rota) 9
Introduction . . . . 11
Acknowledgements . . . . . . . 14
Chapter I. Origins and basic concepts
Commentary
1. 1 The origins of matroid theory 15
1. 2 Geometric lattices . 24
1. 3 Orthogonal duality . . . . . . 25
1. 4 Exchange lattices . . . . . . . 28
1. 5 Survey of the early literature . 29
1. 6 Strong maps . . . . . . . . . 48
Reprints
1. Whitney, H.: On the abstract properties of linear dependence,
American Journal of Mathematics 57 (1935), 509-533 . . . . . 55
2. Birkhoff, G.: Abstract linear dependence in lattices, American
Journal of Mathematics 57 (1935), 800-804 . . . . . . . . . . . 81
3. Whitney, H.: Non-separable and planar graphs, Transactions of
the American Mathematical Society 34 (1932), 339-362 . . . . . 87
4. Mac Lane, S.: A lattice formulation for transcendence degrees
and p-bases, Duke Mathematical Journa/4 (1938), 455-468 . . 111
5. Higgs, D. A.: Strongmapsofgeometries,JournalofCombinato-
rial Theory 5 (1968), 185-191. (MR 38 #89) . . . . . . . . . . 125
5
Chapter D: Linear representations of matroids
Commentary
2. 1 Non-representable matroids 133
2. 2 Homotopy . . . . . . . . . 137
2. 3 Binary and regular matroids 140
2. 4 Ternary matroids . . . . . . 144
Reprints
1. Mac Lane, S.: Some interpretation of abstract linear dependence
in terms of projective geometry, . . . . . . . . . . . . . . 147
American Journal of Mathematics 58 (1936), 236-240.
2. Tutte, W.T.: A homotopy theorem for matroids, I and II, 153
Transactions of the American Mathematical Society 88 (1958),
144-174. (MR 21 =IF 336)
3. Seymour, P.D.: Matroid representation over GF(3), . . . . . 185
Journal of Combinatorial Theory Ser. B 26 (1979), 159-173. (MR
80k: 05031)
Chapter m. Enumeration in geometric lattices
Commentary
3. 1 Mobius functions . 201
3. 2 Homology . . . . 201
3. 3 Modular factorization 204
3. 4 Whitney numbers of the second kind 205
3. 5 The Spemer property . . . . . . . . 209
Reprints
1. Rota, G.-C.: On the foundations of combinatorial theory I.
Theory of Mobius functions, . . . . . . . . . . . . . . . . . . 213
Zeitschrift fiir Wahrscheinlichkeitstheorie und Verwandte Gebiete
2 (1964), 340-368. (MR 30 =IF 4688)
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2. Folkman, J.: The homology groups of a lattice, , . . . . . . . 243
Journal of Mathematics and Mechanics 15 (1966), 631-636. (MR
*
32 5557)
3. Stanley, R. P.: Modular elements in geometric lattices, . . . . 249
Algebra Universalis 1 (1971), 214-217. (MR 45 91= 5037)
4. Dilworth, R. P. and Greene, C.: A counterexample to the gene-
ralization of Sperner's theorem, . . . . . . . . . . . . . . . *. 253
Journal of Combinatorial Theory 10 (1971), 18-21. (MR 43
1893)
5. Dowling, T. A. and Wilson, R. M.: Whitney number inequalities
for geometric lattices, . . . . . . . . . . . . . . . . . . . . . 257
Proceedings of the American Mathematical Society 47 (1975),
504-512. (MR 50 91=6900)
Chapter IV: The Tutte decomposition
Commentary
4. 1 Contractions and deletions . 267
4. 2 The critical problem 273
4. 3 Cutting up space . . . . . . 278
Reprints
1. Tutte, W. T.: A ring in graph theory, . . . . . . . . . . . . . 283
Proceedings of the Cambridge Philosophical Society 43 (1947),
26-40. (MR 8, 284k)
2. Greene, C.: Weight enumeration and the geometry of linear
codes, . . . . . . . . . . . . . . . . . . . . . . . . . . . . *. 299
Studies in Applied Mathematics 55 (1976), 119-128. (MR 56
5335)
3. Zaslavsky, T.: Facing up to arrangements: Face-count formulas
for partition of space by hyperplanes, . . . . . . . . . . . . . 309
Memoirs of the American Mathematical Society Number 154,
American Mathematical Society, Providence, R. 1., 1975.
Extract. (MR 50 91= 9603)
7
Chapter V: Recent advances
Commentary
5. 1 Regular matroids . . . . . . . 333
5. 2 Hereditary classes of matroids 335
Reprints
1. P. D. Seymour, Decomposition of regular matroids, . . . . . 339
Journal of Combinatorial Theory Ser. B 28(1980), 305-359. (MR
82j: 05046)
2. J. Kahn and J.P.S. Kung, Varieties of combinatorial geome-
tries, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
Transactions of the American Mathematical Society 271(1982),
485-499. (MR 84j: 05043)
8
Foreword
by Gian-Carlo Rota
The subjects of mathematics, like the subjects of mankind, have finite
lifespans, which the historian will record as he freezes history at one instant
of time.
There are the old subjects, loaded with distinctions and honors. As their
problems are solved away and the applications reaped by engineers and other
moneymen, ponderous treatises gather dust in library basements, awaiting
the day when a generation as yet unborn will rediscover the lost paradise in
awe. Then there are the middle-aged subjects. You can tell which they are by
roaming the halls of Ivy League universities or the Institute for Advanced
Studies. Their high priests haughtily refuse fabulous offers from eager provin
cial universities while receiving special permission from the President of
France to lecture in English at the College de France. Little do they know that
the load of technicalities is already critical, about to crack and submerge their
theorems in the dust of oblivion that once enveloped the dinosaurs.
Finally, there are the young subjects-combinatorics, for instance. Wild
eyed individuals gingerly pick from a mountain of intractable problems, chil
dishly babbling the first words of what will soon be a new language. Child
hood will end with the first Seminaire Bourbaki.
It could be impossible to find a more fitting example than matroid theory
of a subject now in its infancy. The telltale signs, for an unfailing diagnosis, are
the abundance of deep theorems, going together with a paucity of theories.
Like many another great idea, matroid theory was invented by one of the
great American pioneers, Hassler Whitney. His paper, which is still today the
best entry to the subject, flagrantly reveals the unique peculiarity of this field,
namely, the exceptional variety of cryptomorphic definitions for a matroid,
embarassingly unrelated to each other and exhibiting wholly different mathe
matical pedigrees. It is as if one were to condense all trends of present day
mathematics onto a single finite structure, a feat that anyone would a priori
deem impossible, were it not for the mere fact that matroids do exist.
The original motivation, both in Whitney and the later papers of Tutte,
was graph theoretic. Matroids are objects that play the role of the dual graph
when the graph is not planar. Almost every fact about graphs that can be
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formulated without using the term "vertex" has a matroidal analogue. The
deepest insight obtained from this yoga is Tutte's homotopy theorem, truly a
great combinatorial feat.
But one of the axioms for matroids, as was first recognized by Mac Lane,
is Steinitz' axiom for linear independence of vectors. From this follows a
second yoga: almost every fact about linear independence that can be stated
without reference to the underlying field gives a theorem about matroids. For
example, Higgs was able to develop a matroidal analogue of linear transfor
mation.
The suspicion quickly followed that matroids could be represented by
points in projective space, a suspicion quickly quashed by Mac Lane. In fact,
the condition for representability of a matroid in a given projective space can
be stated in terms of the absence of "obstructions" or forbidden configura
tions, as in the archetypal theorem of Kuratowski on planar graphs. Such
theorems are difficult to come by, witness Seymour's theorem for GF(3).
Next, we find the lattice theorists-Birkhoff, Crapo, Rota-whose yoga
is to see a matroid as its lattice of flats. Many deep enumerative properties,
often associated with homological properties (Folkman), hold for matroids.
The Whitney numbers of a matroid (a term introduced by Harper and Rota)
exhibit some of the properties of binomial coefficients, as Dowling and
Wilson proved, though not all, as the counterexample of Dilworth and
Greene showed.
As if three yogas were not enough, there followed the Tutte-Grothen
dieck decomposition theory (developed most energetically by Brylawski),
which displayed an astonishing analogy with K-theory. Zaslavsky's solution
of Steiner's problem, which gives an explicit formula for the number of
regions into which space is subdivided by a set of hyperplanes, is the finest
application to date of this yoga.
What next? Kahn and Kung have just come along with a new yoga: a
varietal theory of matroids, that brings universal algebra into an already
crowded game. Similarly, the intricate decomposition theory of Seymour
should lead to yet another yoga.
All these yogas lead to a deep suspicion. Anyone who has worked with
matroids has come away with the conviction that the notion of a matroid is
one of the richest and most useful concepts of our day. Yet, we long, as we
always do, for one idea that will allow us to see through the plethora of dispa
rate points of view. Whether this idea will ever come along will depend
largely on who reads the essays collected in this fine volume.
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