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Diestel: Graph Theory
Reinhard Diestel
Graph Theory
Electronic Edition 2000
°
Springer-Verlag New York 1997, 2000
This is an electronic version of the second (2000) edition of the above
Springer book, from their series Graduate Texts in Mathematics , vol. 173.
The cross-references in the text and in the margins are active links: click
on them to be taken to the appropriate page.
The printed edition of this book can be ordered from your bookseller, or
electronically from Springer through the Web sites referred to below.
Softcover $34.95, ISBN 0-387-98976-5
Hardcover $69.95, ISBN 0-387-95014-1
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Preface
Almost two decades have passed since the appearance of those graph the-
ory texts that still set the agenda for most introductory courses taught
today. The canon created by those books has helped to identify some
main ¯elds of study and research, and will doubtless continue to in°uence
the development of the discipline for some time to come.
Yet much has happened in those 20 years, in graph theory no less
than elsewhere: deep new theorems have been found, seemingly disparate
methods and results have become interrelated, entire new branches have
arisen. To name just a few such developments, one may think of how
the new notion of list colouring has bridged the gulf between invari-
ants such as average degree and chromatic number, how probabilistic
methods and the regularity lemma have pervaded extremal graph theo-
ry and Ramsey theory, or how the entirely new ¯eld of graph minors and
tree-decompositions has brought standard methods of surface topology
to bear on long-standing algorithmic graph problems.
Clearly, then, the time has come for a reappraisal: what are, today,
the essential areas, methods and results that should form the centre of
an introductory graph theory course aiming to equip its audience for the
most likely developments ahead?
I have tried in this book to o®er material for such a course. In
view of the increasing complexity and maturity of the subject, I have
broken with the tradition of attempting to cover both theory and appli-
cations: this book o®ers an introduction to the theory of graphs as part
of (pure) mathematics; it contains neither explicit algorithms nor `real
world' applications. My hope is that the potential for depth gained by
this restriction in scope will serve students of computer science as much
as their peers in mathematics: assuming that they prefer algorithms but
will bene¯t from an encounter with pure mathematics of some kind, it
seems an ideal opportunity to look for this close to where their heart lies!
In the selection and presentation of material, I have tried to ac-
commodate two con°icting goals. On the one hand, I believe that an
viii
Preface
introductory text should be lean and concentrate on the essential, so as
to o®er guidance to those new to the ¯eld. As a graduate text, moreover,
it should get to the heart of the matter quickly: after all, the idea is to
convey at least an impression of the depth and methods of the subject.
On the other hand, it has been my particular concern to write with
su±cient detail to make the text enjoyable and easy to read: guiding
questions and ideas will be discussed explicitly, and all proofs presented
will be rigorous and complete.
A typical chapter, therefore, begins with a brief discussion of what
are the guiding questions in the area it covers, continues with a succinct
account of its classic results (often with simpli¯ed proofs), and then
presents one or two deeper theorems that bring out the full °avour of
that area. The proofs of these latter results are typically preceded by (or
interspersed with) an informal account of their main ideas, but are then
presented formally at the same level of detail as their simpler counter-
parts. I soon noticed that, as a consequence, some of those proofs came
out rather longer in print than seemed fair to their often beautifully
simple conception. I would hope, however, that even for the professional
reader the relatively detailed account of those proofs will at least help
to minimize reading time :::
If desired, this text can be used for a lecture course with little or
no further preparation. The simplest way to do this would be to follow
the order of presentation, chapter by chapter: apart from two clearly
marked exceptions, any results used in the proof of others precede them
in the text.
Alternatively, a lecturer may wish to divide the material into an easy
basic course for one semester, and a more challenging follow-up course
for another. To help with the preparation of courses deviating from the
order of presentation, I have listed in the margin next to each proof the
reference numbers of those results that are used in that proof. These
references are given in round brackets: for example, a reference (4.1.2)
in the margin next to the proof of Theorem 4.3.2 indicates that Lemma
4.1.2 will be used in this proof. Correspondingly, in the margin next to
Lemma 4.1.2 there is a reference [ 4.3.2 ] (in square brackets) informing
the reader that this lemma will be used in the proof of Theorem 4.3.2.
Note that this system applies between di®erent sections only (of the same
or of di®erent chapters): the sections themselves are written as units and
best read in their order of presentation.
The mathematical prerequisites for this book, as for most graph
theory texts, are minimal: a ¯rst grounding in linear algebra is assumed
for Chapter 1.9 and once in Chapter 5.5, some basic topological con-
cepts about the Euclidean plane and 3-space are used in Chapter 4, and
a previous ¯rst encounter with elementary probability will help with
Chapter 11. (Even here, all that is assumed formally is the knowledge
of basic de¯nitions: the few probabilistic tools used are developed in the
Preface
ix
text.) There are two areas of graph theory which I ¯nd both fascinat-
ing and important, especially from the perspective of pure mathematics
adopted here, but which are not covered in this book: these are algebraic
graph theory and in¯nite graphs.
At the end of each chapter, there is a section with exercises and
another with bibliographical and historical notes. Many of the exercises
were chosen to complement the main narrative of the text: they illus-
trate new concepts, show how a new invariant relates to earlier ones,
or indicate ways in which a result stated in the text is best possible.
Particularly easy exercises are identi¯ed by the superscript ¡ , the more
challenging ones carry a + . The notes are intended to guide the reader
on to further reading, in particular to any monographs or survey articles
on the theme of that chapter. They also o®er some historical and other
remarks on the material presented in the text.
Ends of proofs are marked by the symbol
¤
.
Almost every book contains errors, and this one will hardly be an
exception. I shall try to post on the Web any corrections that become
necessary. The relevant site may change in time, but will always be
accessible via the following two addresses:
http://www.springer.de/catalog/html-¯les/deutsch/math/3540609180.html
Please let me know about any errors you ¯nd.
Little in a textbook is truly original: even the style of writing and
of presentation will invariably be in°uenced by examples. The book that
no doubt in°uenced me most is the classic GTM graph theory text by
Bollobas: it was in the course recorded by this text that I learnt my ¯rst
graph theory as a student. Anyone who knows this book well will feel
its in°uence here, despite all di®erences in contents and presentation.
I should like to thank all who gave so generously of their time,
knowledge and advice in connection with this book. I have bene¯ted
particularly from the help of N. Alon, G. Brightwell, R. Gillett, R. Halin,
M. Hintz, A. Huck, I. Leader, T. ÃLuczak, W. Mader, V. Rodl, A.D. Scott,
P.D. Seymour, G. Simonyi, M. Skoviera, R. Thomas, C. Thomassen and
P. Valtr. I am particularly grateful also to Tommy R. Jensen, who taught
me much about colouring and all I know about k -°ows, and who invest-
ed immense amounts of diligence and energy in his proofreading of the
preliminary German version of this book.
March 1997
RD
. Where this symbol is
found directly below a formal assertion, it means that the proof should
be clear after what has been said|a claim waiting to be veri¯ed! There
are also some deeper theorems which are stated, without proof, as back-
ground information: these can be identi¯ed by the absence of both proof
and
¤
x
Preface
About the second edition
Naturally, I am delighted at having to write this addendum so soon after
this book came out in the summer of 1997. It is particularly gratifying
to hear that people are gradually adopting it not only for their personal
use but more and more also as a course text; this, after all, was my aim
when I wrote it, and my excuse for agonizing more over presentation
than I might otherwise have done.
There are two major changes. The last chapter on graph minors
now gives a complete proof of one of the major results of the Robertson-
Seymour theory, their theorem that excluding a graph as a minor bounds
the tree-width if and only if that graph is planar. This short proof did
not exist when I wrote the ¯rst edition, which is why I then included a
short proof of the next best thing, the analogous result for path-width.
That theorem has now been dropped from Chapter 12. Another addition
in this chapter is that the tree-width duality theorem, Theorem 12.3.9,
now comes with a (short) proof too.
The second major change is the addition of a complete set of hints
for the exercises. These are largely Tommy Jensen's work, and I am
grateful for the time he donated to this project. The aim of these hints
is to help those who use the book to study graph theory on their own,
but not to spoil the fun. The exercises, including hints, continue to be
intended for classroom use.
Apart from these two changes, there are a few additions. The most
noticable of these are the formal introduction of depth-¯rst search trees
in Section 1.5 (which has led to some simpli¯cations in later proofs) and
an ingenious new proof of Menger's theorem due to Bohme, Goring and
Harant (which has not otherwise been published).
Finally, there is a host of small simpli¯cations and clari¯cations
of arguments that I noticed as I taught from the book, or which were
pointed out to me by others. To all these I o®er my special thanks.
The Web site for the book has followed me to
I expect this address to be stable for some time.
Once more, my thanks go to all who contributed to this second
edition by commenting on the ¯rst|and I look forward to further com-
ments!
December 1999
RD

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