Solution Manual for Introduction to Graph Theory, Second Edition

521 Pages·2001·6.84 MB·English

by  Douglas B. West

INTRODUCTION TO GRAPH THEORY
SECOND EDITION (2001)
SOLUTION MANUAL
SUMMER 2005 VERSION
® DOUGLAS B. WEST
MATHEMATICS DEPARTMENT
UNIVERSITY OF ILLINOIS
Allrights reserved. No part of this work may be reproduced
or transmitted in any form without permission.
11
NOTICE
This is the Summer 2005 version of the Instructor's Solution Manual for
Introduction to Graph Theory, by Douglas B. West. A few solutions have
been added or clarified since last year's version.
Also present is a (slightly edited) annotated syllabus for the one-
semester course taught from this book at the University of Illinois.
This version of the Solution Manual contains solutions for 99.4% of
the problems in Chapters 1—7 and 93% of the problems in Chapter 8. The
author believes that only Problems 4.2.10, 7.1.36, 7.1.37, 7.2.39, 7.2.47,
and 7.3.3 1 in Chapters 1—7 are lacking solutions here. There problems are
too long or difficult for this text or use concepts not covered in the text; they
will be deleted in the third edition.
The positions of solutions that have not yet been written into the files
are occupied by the statements of the corresponding problems. These prob-
lems retain the (—), (!), (+), (*) indicators. Also (.) is added to introduce
the statements of problems without other indicators. Thus every problem
whose solution is not included is marked by one of the indicators, for ease
of identification.
The author hopes that the solutions contained herein will be useful to
instnictors. The level of detail in solutions varies. Instructors should feel
free to write up solutions with more or less detail according to the needs of
the class. Please do not leave solutions posted on the web.
Due to time limitations, the solutions have not been proofread or edited
as carefully as the text, especially in Chapter 8. Please send corrections to
[email protected]. The author thanks Fred Galvin in particular for con-
tributing improvements or alternative solutions for many of the problems
in the earlier chapters.
This will be the last version of the Solution Manual for the second
edition of the text, The third edition will have many new problems, such
as those posted at http:/Iwww.math.uiuc.edul westligtlnewprob.html. The
effort to include all solutions will resume for the third edition. Possibly
other pedagogical features may also be added later.
Inquiries may be sent to [email protected]. Meanwhile, the author
apologizes for any inconvenience caused by the absence of some solutions.
Douglas B. West
iii Solutions
Mathematics Department - University of Illinois
MATH 412
SYLLABUS FOR INSTRUCTORS
Text: West, Introduction to Graph Theory, second edition,
Prentice Hall, 2001.
Many students in this course see graph algorithms repeatedly in
courses in computer science. Hence this course aims primarily to improve
students' writing of proofs in discrete mathematics while learning about
the structure of graphs. Some algorithms are presented along the way, and
many of the proofs are constructive. The aspect of algorithms emphasized
in CS courses is running time; in a mathematics course in graph theory
from this book the algorithmic focus is on proving that the algorithms work.
Math 412 is intended as a rigorous course that challenges students to
think. Homework and tests should require proofs, and most of the exercises
in the text do so. The material is interesting, accessible, and applicable;
most students who stick with the course will give it a fair amount of time
and thought.
An important aspect of the course is the clear presentation of solutions,
which involves careful writing. Many of the problems in the text have hints,
either where the problem is posed or in Appendix C (or both). Producing
a solution involves two main steps: finding a proof and properly writing
it. It is generally beneficial to the learning process to provide "collabora-
tive study sessions" in which students can discuss homework problems in
small groups and an instructor or teaching assistant is available to answer
questions and provide direction. Students should then write up clear and
complete solutions on their own.
This course works best when students have had prior exposure to
writing proofs, as in a "transition" course. Some students may need fur-
ther explicit discussions of the structure of proofs. Such discussion appear
in many texts, such as
D'Angelo snd West, Mathematical Thinking: Problem-Solving and Proofs;
Eisenberg, The Mathematical Method: A Transition to Advanced Mathematics;
Fletcher/Patty, Foundations of Higher Mathematics;
Galovich, Introduction to Mathematical Structures;
Galovich, Doing Mathematics: An Introduction to Proofs and Problem Solving;
Solow, How to Read and Do Proofs.
Preface iv
Suggested Schedule
The subject matter for the course is the first seven chapters of the text,
skipping most optional materiaL Modifications to this are discussed below.
The 22 sections are allotted an average of slightly under two lectures each.
In the exercises, problems designated by (—)are easier or shorter than
most, often good for tests or for "warmup" before doing homework problems.
Problems designated by (-F-) are harder than most. Those designated by (!)
are particularly instructive, entertaining, or important. Those designated
by (*) make use of optional material.
The semester at the University of Illinois has 43 fifty-minute lectures.
The final two lectures are for optional topics, usually chosen by the students
from topics in Chapter 8.
Chapter 1 Fundamental Concepts 8
Chapter 2 Trees and Distance 5.5
Chapter 3 Matchings and Factors 5.5
Chapter 4 Connectivity and Paths 6
Chapter 5 Graph Coloring 6
Chapter 6 Planar Graphs 5
Chapter 7 Edges and Cycles 5
* Total 41
Optional Material
No later material requires material marked optional. The "optional"
marking also suggests to students that the final examination will not cover
that material.
The optional subsections on Disjoint Spanning Trees (Bridg-It) in Sec-
tion 2.1 and Stable Matchings in Section 3.2 are a'ways quite popthar with
the students, The planarity algorithm (without proof) in 6.2 is appealing
to students, as is the notion of embedding graphs on the torus through
Example 6.3.21. Our course usually includes these four items.
The discussion of f-factors in Section 3.3 is also very instructive and
is covered when the class is proceeding on schedule. Other potential addi-
tions include the applications of Menger's Theorem at 4.2.24 or 4.2.25.
Other items marked optional generally should not be covered in class.
Additional text items not marked optional that can be skipped when
behind schedule:
1.1: 31, 35 1.2: 16, 21—23 1.3: 24, 31—32 1.4: 1, 4, 7, 25—26
2.1: 8, 14—16 2.2: 13—19 2.3: 7—8 3.2: 4
4,1: 4—6 4,2: 20—21 5,1: 11, 22(proof) 5.3: 10—11, 16(proof)
6.1: 18—20, 28 6.3: 9—10, 13—15 7.2: 17
v Solutions
Comments
There are several underlying themes in the course, and mentioning
these at appropriate moments helps establish continuity. These include
1) TONCAS (The Obvious Necessary Condition(s) are Also Sufficient).
2) Weak duality in dual maximation and minimization problems.
3) Proof techniques such as the use of extremality, the paradigm for induc-
tive proofs of conditional statements, and the technique of transforming a
problem into a previously solved problem.
Students sometimes find it strange that so many exercises concern
the Petersen graph. This is not so much because of the importance of the
Petersen graph itself, but rather because it is a small graph and yet has
complex enough structure to permit many interesting exercises to be asked.
Chapter 1. In recent years, most students enter the course having
been exposed to proof techniques, so reviewing these in the first five sec-
tions has become less necessary; remarks in class can emphasis techniques
as reminders. To minimize confusion, digraphs should not be mentioned
until section 1.4; students absorb the additional model more easily after
becoming comfortable with the first.
1.1: p3-6 contaln motivational examples as an overview of the course;
this discussion should not extend past the first day no matter where it
ends (the definitions are later repeated where needed). The material on
the Petersen graph estabhshes its basic properties for use in later examples
and exercises.
1.2: The definitions of path and cycle are intended to be intuitive; one
shouldn't dwell on the heaviness of the notation for walks.
1.3: Although characterization of graphic sequences is a classical topic,
some reviewers have questioned its importance. Nevertheless, here is a
computation that students enjoy and can perform.
1,4: The examples are presented to motivate the model; these can be
skipped to save time. The de Bruijn graph is a classical apphcation. It is
desirable to present it, but it takes a while to explaln.
Chapter 2.
2,1: Characterization of trees is a good place to ask for input from the
class, both in listing properties and in proving equivalence.
2,2: The inductive proof for the Prufer correspondence seems to be
easier for most students to grasp than the full bijective it also illus-
trates the usual type of induction with trees. Most students in the class
have seen determinants, but most have considerable difficulty understand-
ing the proof of the Matrix Tree Theorem; given the time involved, it is best
Preface vi
just to state the result and give an example (the next edition will include
a purely inductive proof that uses only determinant expansion, not the
Cauchy-Binet Formula). Students find the material on graceful labelings
enjoyable and illuminating; it doesn't take long, but also it isn't required.
The material on branchings should certaily be skipped in this course.
2.3: Many students have seen rooted trees in computer science and
find ordinary trees unnatural; Kruskal's algorithm should clarify the dis-
tinction. Many CS courses now cover the algorithms of Kruskal, Dijkstra,
and Huffrnan; here cover Kruskal and perhaps Dijkstra (many students
have seen the algorithm but not a proof of correctness), and skip Huffman,
Chapter 3.
3.1: Skip "Dominating Sets", but present the rest of the section.
3.2: Students find the Hungarian algorithm difficult; explicit examples
need to be worked along with the theoretical discussion of the equality
subgraph. "Stable Matchings" is very popular with students and should be
presented unless far behind in schedule. Skip "Faster Bipartite Matching".
3.3: Present all of the subsection on Tutte's 1-factor Theorem. The
material on f-factors is intellectually beautiful and leads to one proof of
the Erdös-Gallai conditions, but it is not used again in the course and is an
"aside". Skip everything on Edmonds' Blossom Algorithm: matching algo-
rithms in general graphs are important algorithmically but would require
too much time in this course; this is "recommended reading".
Chapter 4.
4.1: Students have trouble distinguishing "k-connected" from "connec-
tivity k" and "bonds" from "edge cuts". Bonds are dual to cycles in the
matroidal sense; there are hints of this in exercises and in Chapter 7, but
the full duality cannot be explored before Chapter 8.
4.2: Students find this section a bit difficult. The proof of 4.2.10 is
similar to that of 4.2.7, making it omittable, but the application in 4.2.14
is appealing. The details of 4.2.20-21 can be skipped or treated lightiy,
since the main issue is the local version of Menger's theorem. 4.2.24-25 are
appealing appllcations that can be added; 4.2.5 (CSDR) is a fundamental
result but takes a fair amount of effort.
4.3: The material on network flow is quite easy but can take a long
time to present due to the overhead of defining new concepts. The basic
idea of 4.3.13-15 should be presented without belaboring the details too
much, 4.3.16 is a more appealing application that perhaps makes the point
more effectively. Skip "Supplies and Demands".
vii Solutions
Chapter 5.
5,1: If time is short, the proof of 5.1.22 (Brooks' Theorem) can be merely
sketched.
5.2: Be sure to cover Turán's Theorem. Presentation of Dirac's The-
orem in 5 +2+20 is valuable as an application of the Fan Lemma (Menger's
Theorem). The subsequent material has limited appeal to undergraduates.
5.3: The inclusion-exclusion formula for the chromatic polynomial is
derived here (5+3+10) without using inclusion-exclusion, making it accessi-
ble to this class without prerequisite. However, this proof is difficult for
students to follow in favor of the simple inclusion-exclusion proof, which
will be optional since that formula is not prerequisite for the course. Hence
this formula should be omitted unless students know inclusion-exclusion.
Chordal graphs and perfect graphs are more important, but these can also
be treated lightly if short of time, Skip Acyclic Orientations".
Chapter 6.
6+1: The technical definitions of objects in the plane should be treated
very lightly. It is better to be informal here, without writing out formal
definitions unless explicitly requested by students. Outerplanar graphs
are useful as a much easier class on which to solve problems (exercises!)
like those on planar graphs; 6.18-20 are fundamental observations about
outerplanar graphs, but other items are more important if time is short,
6.1.28 (polyhedra) is an appeahng application but can be skipped.
6.2: The preparatory material 6.2.4-7 on Kuratowski's Theorem can be
presented lightly, leaving the annoying details as reading; the subsequent
material on convex embedding of 3-connected graphs is much more inter-
esting. Presentation of the planarity algorithm is appealing but optional;
skip the proof that it works.
6.3: The four color problem is popular; for undergraduates, show the
flaw in Kempe's proof (p271), but don't present the discharging rule un-
less ahead of schedule, Students find the concept of crossing number easy
to grasp, but the results are fairly difficult; try to go as far as the recur-
sive quartic lower bound for the complete graph+ The edge bound and its
geometric application are impressive but take too much time for under-
graduates. The idea of embeddings on surfaces can be conveyed through
the examples in 6.3+2 1 on the torus. Interested students can be advised to
read the rest of this section.
Chapter 7.
7.1: The proof of Vizing's Theorem is one of the more difficult in the
course, but undergraduates can gain follow it when it is presented with
sufficient colored chalk. The proof can be skipped if short of time. Skip
Preface viii
"Characterization of Line Graphs", although if time and interest is plenti-
ftil the necessity of Krausz's condition can be explained.
7.2: Chvátal's theorem (7.2.13) is not as hard to present as it looks if
the instructor has the statement and proof clearly in mind. Nevertheless,
the proof is somewhat technical and can be skipped (the same can be said
of 7.2.17). More appealing is the Chvátal—Erdôs Theorem (7.2.19), which
certainly should be presented. Skip "Cycles in Directed Graphs".
7.3: The theorems of Tait and Grinberg make a nice culmination to
the required material of the course. Skip "Snarks" and "Flows and Cycle
Covers". Nevertheless, these are lively topics that can be recommended for
advanced students.
Chapter 8. If time permits, material from the first part of sections of
Chapter 8 can be presented to give the students a glimpse of other topics.
The best choices for conveying some understanding in a brief treatment are
Section 8.3 (Ramsey Theory or Sperner's Lemma) and Section 8.5 (Random
Graphs). Also possible are the Gossip Problem (or other items) from Sec-
tion 8.4 and some of the optional material from earlier chapters. The first
part of Section 8.1 (Perfect Graphs) may also be usable for this purpose if
perfect graphs have been discussed in Section 5.3. Sections 8.2 and 8.6 re-
quire more investment in preliminary material and thus are less suitable
for giving a "glimpse".
1 Chapter1: Fundamental Concepts
I.FUNDAMENTAL CONCEPTS
1.1. WHAT IS A GRAPH?
1.1.1. Completebipartitegraphsandcompletegraphs. Thecompletebipar
titegraphKm,n isacompletegraphifandonlyifm = n = lor{m, n} = {I, O}.
1.1.2. Adjacencymatrices and incidence matrices for a 3-vertexpath.
(: ~ ~) (1 ~ 1) (~ ~ l)
(l ~) (~ l) (l :) (: l) (~ :) (: ~)
Adjacency matrices for a path and a cycle with sixvertices.
(~ ~ ~ ~ ~ ~l (~~ ~ ~ ~ ~l
010100 010100
001010 001010
000101 000101
000010 100010
1.1.3. Adjacencymatrix for Km,n'
m n
:[ffi]
1.1.4. G;:; H ifand only ifG ;:; H. Iff is an isomorphismfrom G to H,
then f is a vertex bijection preserving adjacency and nonadjacency, and
hence f preserves non-adjacency and adjacency in G and is an isomor
phismfrom Gto H. Thesameargumentappliesforthe converse,sincethe
complement ofG is G.