Table Of ContentDISCRETE MATHEMATICS AND ITS APPLICATIONS
Series Editor KENNETH H. ROSEN
GRAPH THEORY AND
ITS APPLICATIONS
SECOND EDITION
DISCRETE
MATHEMATICS
ITS APPLICATIONS
Series Editor
Kenneth H. Rosen, Ph.D.
Juergen Bierbrauer, Introduction to Coding Theory
Kun-Mao Chao and Bang Ye Wu, Spanning Trees and Optimization Problems
Charalambos A. Charalambides, Enumerative Combinatorics
Henri Cohen, Gerhard Frey, et at., Handbook of Elliptic and Hyperelliptic Curve Cryptography
Charles J. Co/bourn and Jeffrey H. Dinitz, The CRC Handbook of Combinatorial Designs
Steven Furino, Ying Miao, and Jianxing Yin, Frames and Resolvable Designs: Uses,
Constructions, and Existence
Randy Goldberg and Lance Riek, A Practical Handbook of Speech Coders
Jacob Goodman and Joseph O'Rourke, Handbook of Discrete and Computational Geometry,
SecoE.n d Edition
Jonathan L. Gross and Jay Yellen, Graph Theory and Its Applications, Second Edition
Jonathan L. Gross and Jay Yellen, Handbook of Graph Theory
Darrel R. Hankerson, Greg A. Harris, and Peter D. Johnson, Introduction to Information
Theory and Data Compression, Second Edition
Daryl D. Harms, Miroslav Kraetzl, Charles J. Co/bourn, and John S. Devitt, Network Reliability:
Experiments with a Symbolic Algebra Environment
Derek Holt with Bettina Eick and Eamonn A. O'Brien, Handbook of Computational Group Theory
F.
David M. Jackson and Terry Visentin, An Atlas of Smaller Maps in Orientable and
Nonorientable Surfaces I.
Richard E. Klima, Ernest Stitzinger, and Neil P. Sigmon, Abstract Algebra Applications
with Maple
Patrick Knupp and Kambiz Safari, Verification of Computer Codes in Computational Science
and Engineering
William Kocay and Donald L. Kreher, Graphs, Algorithms, and Optimization
Donald L. Kreher and Douglas R. Stinson, Combinatorial Algorithms: Generation Enumeration
and Search
Charles C. Lindner and Christopher A. Rodgers, Design Theory
Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone, Handbook of Applied
Cryptography
Continued Titles
Richard A. Mol/in, Algebraic Number Theory
Richard A. Mol/in, Codes: The Guide to Secrecy from Ancient to Modern Times
Richard A. Mol/in, Fundamental Number Theory with Applications
Richard A. Mol/in, An Introduction to Cryptography
Richard A. Mol/in, Quadratics
Richard A. Mol/in, RSA and Public-Key Cryptography
Kenneth H_ Rosen, Handbook of Discrete and Combinatorial Mathematics
Douglas R_ Shier and K. Wallen ius, Applied Mathematical Modeling: A Multidisciplinary
Approach T.
Steuding, Diophantine Analysis
Jam
Douglas R_ Stinson, Cryptography: Theory and Practice, Second Edition
Roberto Togneri and Christopher deSilva, Fundamentals of Information Theory and
Coding Design J.
Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography
DISCRETE MATHEMATICS AND ITS APPLICATIONS
Series Editor KENNETH H. ROSEN
GRAPH THEORY AND
ITS APPLICATIONS
SECOND EDITION
JONATHAN L. GROSS
JAY YELLEN
Boca Raton London New York
Published iu 2006 by
Chapman Hall/CRC
Taylor F&ra ncis Group
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Boca Raton, FL 33487-2742
2006 by
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PREFACE
Interest in graphs and their applications continues to grow rapidly, largely due to
the usefulness of graphs as models for computation and optimization. This text targets
the need for a comprehensive approach to the theory, integrating a careful exposition of
classical developments with emerging methods, models, and practical needs. It is suit
able for classroom presentation at the introductory graduate or advanced undergraduate
level, or for self-study and reference by working professionals.
Graph theory has evolved as a collection of seemingly disparate topics. The intent
of the authors is to present this material in a more cohesive framework, characteristic
of mathematical areas with longer traditions, such as linear algebra and group theory.
In the process, important techniques and analytic tools are transformed into a unified
mathematical methodology.
Emphasis throughout is conceptual, with more than 800 graph drawings included
to strengthen intuition and more than 2000 exercises ranging from routine drill to more
challenging problem solving. Applications and concrete examples are designed to stim
ulate interest and to demonstrate the relevance of new concepts.
Algorithms are presented in a concise format, shorn of the details of computer
implementation. Computer science students will find numerous projects inviting them
to convert algorithms to computer programs. Software design issues are addressed
throughout the book in the form of computational notes, which can be skipped by
those not interested in implementation. These design issues include the efficient use of
computational resources, software reusability, and the user interface.
Summary of Contents
Chapters 1 through 6 concentrate on graph representation, basic properties, model
ing, and applications. Graphical constructions here are concrete and primarily spatial.
When necessary, we introduce abstractions in a supportive role.
Chapters 7 and 8 present some of the underpinnings of topological graph theory.
Chapter 7 is devoted to planarity and Kuratowski's theorem, and in Chapter 8, the scope
of graph drawings is expanded in multiple directions, including the topological model
for drawings, drawings on higher-order surfaces, and computer drawings. Chapter 9
is about graph colorings, including vertex- and edge-colorings, map-colorings, and the
related topics of cliques, independence numbers, and graph factorization.
Chapter 10 discusses graph measurement, including measurements of graph map
pings. Chapter 11 provides a brief introduction to analytic graph theory, which com
prises three of the most extensively developed branches of graph theory-Ramsey theory,
extremal graphs, and random graphs. The material in Chapters 12 and 13, special di
graph models and network flows, overlaps with various areas in operations research and
computer science.
Chapters 14 through 16, the most algebraic chapters, concern enumeration, speci
fication by voltage graphs, and constructing nonplanar layouts.
Most of the material in Chapters 1 through 6 assumes no prerequisite. However,
some familiarity with topics typically found in an undergraduate course in discrete
mathematics would be useful, and those topics appear briefly in the appendix (along
with some linear algebra, which is used in §4.6). The appendix also provides a quick
review (including permutation groups) for some of the more advanced topics in the later
chapters.
To the Instructor
The book has ample material for a two-semester course, and a variety of one
semester courses can be designed with different slants using various combinations of
chapters. An introductory one-semester general course in graph theory would typically
include most of the topics covered in the first nine chapters, except possibly parts of
Chapter 8. However, some instructors of a fast-paced one-semester course might consider
leaving various sections of earlier chapters for self-study by students. This would allow
more time for selections from later chapters.
The definitions and results from all of Chapter 1, most of Chapter 2, and from §3.1
are used throughout the text, and we recommend that they be covered in any course
that uses the book. The remaining chapters are largely independent of each other, with
several notable exceptions, as follows:
§3.2 is used in §4.1.
•
§4.1 and §4.2 are used in §5.4 and §12.5.
•
§4.5 is referred to in §5.3 and §6.1.
•
Parts of Chapter 5 are used in §13.3.
•
Chapter 8 is used in §16.1 and §16.2.
•
§15.1, §15.2, and §15.3 are used in §16.3, §16.4, and §16.5.
•
operations research/optimization
A course oriented toward should include most or
all of the material in Chapters 4 (spanning trees), 5 (connectivity), 6 (traversability),
9 (colorings), 12 (digraph models), and 13 (flows), along with various other sections
of the instructor's choice, depending on the time available. A course emphasizing the
data structures and algorithms
role of might add to the above topics more material
from Chapter 3 (trees). A more algebraic and topological course in graph theory might
replace some of these selections with parts or all of Chapter 7 (planarity), Chapter 8
(graph drawings), Chapter 14 (enumeration), Chapter 15 (voltage graphs), and Chapter
16 (graphs on general surfaces).
New Features in the Second Edition
Solutions and Hints. Each exercise marked with a superscripts has a solution or
• hint appearing in the back of the book.
Supplementary Exercises. In addition to the section exercises, each chapter now
• concludes with a section of supplementary exercises, which are intended to develop
the problem-solving skills of students and test whether they can go beyond what
has been explicitly taught. Most of these exercises were designed as examination
questions for students at Columbia University.
Foreshadowing. The first three chapters now preview a number of concepts, mostly
• via the exercises, that are more fully developed in later chapters. This makes it
easier to encourage students to take earlier excursions in research areas that may
be of particular interest to the instructor.
New Material in the Second Edition
We were gratified to see the first edition of this book used successfully at many col
leges and universities in North America, Europe, Asia, and Oceania. Suggestions from
instructors at these institutions, and from their students, led to several improvements
in this second edition, including the addition of much new material. The inclusion of
other new material was inspired by contributions to our Handbook of Graph Theory.
Almost nothing has been deleted from the first edition, and well over 100 pages of new
or expanded material have been added to the second edition. The following 9 sections
are all new:
§8.6 Geometric Drawings
§9.4 Factorization
§10.1 Distance in Graphs
§10.2 Domination in Graphs
§10.3 Bandwidth
§10.4 Intersection Graphs
§11.1 Ramsey Graph Theory
§11.2 Extremal Graph Theory
§11.3 Random Graphs
Highlights of Changes in the Second Edition
Now, at the start of Chapter 1, we introduce the notation for an edge between
uv
vertices and of a simple graph. Our move to this more convenient notation was
u v
motivated by comments from our colleagues who work primarily in simple graphs. Of
course, we continue to use explicit edge names in contexts where multiple edges may be
present. We have expanded Chapter 2 with increased discussion of isomorphism testing,
automorphisms and symmetry, vertex and edge orbits, graph reconstruction, and some
graph operations that appeared much later in the first edition.
We revised and reorganized some of the material in Chapters 3 and 4 to create a
smoother flow and to make the unifying treatment of the tree-growing algorithms more
transparent. The main change in Chapter 5 is the inclusion in §5.2 of a short proof of
the Whitney-Robbins characterization of 2-edge-connected graphs.
With the new Chapter 7, Kuratowski's Theorem and its proof (which were in
Chapter 9 of the first edition) can now be studied without having to first cover the
more demanding topology material in Chapter 8. Chapter 7 now also includes crossing
numbers and thickness (from Chapter 15 of the first edition), which are more closely
related to planarity than to higher order surfaces. Most of the content of the old Chapter
7 (in the first edition) was redistributed to other chapters. Chapter 9 on graph colorings
is essentially the old Chapter 10 together with the new section on factorization (§9.4).
Chapters 10 and 11 are entirely new, except for the two sections on graph mappings
(§10.5 and §10.6), which migrated from the old Chapter 7. Except for some reorganiza
tion within each chapter, Chapter 12 is the old Chapter 11, and Chapter 13 is the old
Chapter 12.
The new Chapters 14, 15, and 16 were the old Chapters 13, 14, and 15, respectively.
Chapter 14 (graphical enumeration) has several improved proofs, especially of Burnside's
Lemma. Chapter 16 (non-planar layouts) no longer includes the sections on crossing
numbers and thickness, now in Chapter 7, or the section on generalizing planar drawings
to higher-order surfaces, now in Chapter 8.
