Introduction to Graph Theory (2nd) - West - Solution Manual

8/10/2019 Introduction to Graph Theory (2nd) - West - Solution Manual

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INTRODUCTION TO GRAPH THEORY

SECOND EDITION (2001)

SOLUTION MANUAL

SUMMER 2005 VERSION

c DOUGLAS B. WEST

MATHEMATICS DEPARTMENT

UNIVERSITY OF ILLINOIS

All rights reserved. No part of this work may be reproduced

or transmitted in any form without permission.

NOTICE

This is the Summer 2005 version of the Instructors Solution Manual for

Introduction to Graph Theory, by Douglas B. West. A few solutions have

been added or clarified since last years 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 17 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 7.3.31 in Chapters 17 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 introducethe 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

instructors. 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 [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://www.math.uiuc.edu/ west/igt/newprob.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 authorapologizes for any inconvenience caused by the absence of some solutions.

Douglas B. West


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1 Chapter 1: Fundamental Concepts Section 1.1: What Is a Graph? 2

1.FUNDAMENTAL CONCEPTS

1.1. WHAT IS A GRAPH?

1.1.1. Complete bipartite graphs and complete graphs. The complete bipar-

tite graphKm,nis a completegraphif and onlyifm = n = 1 or {m, n} = {1, 0}.1.1.2. Adjacency matrices and incidence matrices for a 3-vertex path.

0 1 1

1 0 0

1 0 0

0 1 0

1 0 1

0 1 0

0 0 1

0 0 1

1 1 0

1 1

1 0

0 1

1 1

0 1

1 0

1 0

1 1

0 1

0 1

1 1

1 0

1 0

0 1

1 1

0 1

1 0

1 1

Adjacency matrices for a path and a cycle with six vertices.

0 1 0 0 0 0

1 0 1 0 0 0

0 1 0 1 0 0

0 0 1 0 1 0

0 0 0 1 0 1

0 0 0 0 1 0

0 1 0 0 0 1

1 0 1 0 0 0

0 1 0 1 0 0

0 0 1 0 1 0

0 0 0 1 0 1

1 0 0 0 1 0

1.1.3. Adjacency matrix for Km,n.

m n

m 0 1

n 1 0

1.1.4. G = H if and only ifG = H. If fis an isomorphism from G to H,

then fis a vertex bijection preserving adjacency and nonadjacency, and

hence f preserves non-adjacency and adjacency in G and is an isomor-phism fromG to H. The same argument applies for the converse, since the

complement ofG is G .

1.1.5. If every vertex of a graphG has degree 2, then G is a cycleFALSE.

Such a graph can be a disconnected graph with each component a cycle. (If

infinite graphs are allowed, then the graph can be an infinite path.)

1.1.6. The graph below decomposes into copies of P4.

1.1.7. A graph with more than six vertices of odd degree cannot be decom-

posed into three paths. Every vertex of odd degree must be the endpoint

of some path in a decomposition into paths. Three paths have only six

endpoints.

1.1.8. Decompositions of a graph. The graph below decomposes into copies

of K1,3 with centers at the marked vertices. It decomposes into bold and

solid copies ofP4 as shown.

1.1.9. A graph and its complement. With vertices labeled as shown, two

vertices are adjacent in the graph on the right if and only if they are not

adjacent in the graph on the left.

a

b c

d

e

f g

h

a

f d

g

c

h b

e

1.1.10. The complement of a simple disconnected graph must be connected

TRUE. A disconnected graph G has vertices x,y that do not belong to apath. Hencexand y are adjacent inG . Furthermore, xand y have no com-

mon neighbor in G , since that would yield a path connecting them. Hence


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293 Chapter 5: Coloring of Graphs Section 5.3: Enumerative Aspects 294

currence for the chromatic polynomial, we begin with the compatible pairs

forG e and consider the effect of addinge. If(D, f) is a compatible pairfor G e such that f(u)= f(v), say f(u) < f(v), thene must be orientedfrom u to v to obtain an orientation ofG compatible with f. The result

is indeed acyclic, else it has a directed v, u-path along which the value f

must step downward at some point. Conversely, we can delete e from a

compatible pair for G with f(u)

= f(v) to obtain a compatible pair for

G e. Hence the compatible pairs with differing labels foru andv are in1-1 correspondence in G and G e.

Now consider pairs with f(u)= f(v). It suffices to show that eachsuch pair for G e becomes a compatible pair for G by addinge orientedin at least one way, and that for (G e, k) of these, both orientations ofeyield compatible pairs for G . For the first statement, consider an arbitrary

compatible pair (D, f) with f(u)= f(v) for Ge, and suppose neitherorientation for e yields a compatible pair for G This requires D to have


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orientation for e yields a compatible pair for G. This requires D to haveboth a u, v-path and a v , u-path, which cannot happen since D is acyclic.

For the second statement, suppose that (D, f) is a compatible pair for Gwith f(u)= f(v) and that the orientation obtained by reversinge is alsocompatible with f. Then D e has neither a u, v-path nor a v, u-path,and contractinge yields a compatible pair for Ge. Conversely, given acompatible pair for G e, we can split the contracted vertex to obtain acompatible pair for Ge with f(u)= f(v) so that orientinge in eitherdirection yields a compatible pair for G .


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339 Chapter 7: Edges and Cycles Section 7.1: Line Graphs and Edge Coloring 340

7.EDGES AND CYCLES

7.1. LINE GRAPHS & EDGE COLORING

7.1.1. Edge-chromatic number and line graph for the two graphs below.

The labelings are proper edge colorings, the number of colors used is the

maximum degree, so the colorings are optimal.

2

7.1.4. For every graph G, (G) e(G)/(G). In a proper edge-coloring,

each color class has at most (G)edges. The lower bound follows because

alle(G)edges must be colored.

7.1.5. The Petersen graph is the complement ofL (K5). The vertices ofL (K5)

are the edges in K5, which can be named as the 2-element subsets of [5].

Two such pairs are adjacent in the Petersen graph if and only if they are

disjoint, which is the condition for them being nonadjacent in L(K5).

7.1.6. The line graph of the Petersen graph has 10 triangles. For a simple

graphG, there is a triangle in L(G) for every set of three edges in G that

share one common endpoint and for every set of three edges that form

a triangle in G. The Petersen graph has no triangles, so the latter type

does not arise. However, the Petersen graph has 10 triples of edges with a

common endpoint, one for each of its vertices.

7.1.7. P5 is a line graph. The complement ofP5 is a 5-cycle with a chord.

It is the line graph of a 4-cycle with a pendant edge.


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1

2

3

0

3

0 1

0

2

0

1

2

0

1

2

0

1

2

7.1.2. (Qk) = (Qk), by explicit coloring. In the cube Qk, the edges

between vertices differing in coordinate j form a complete matching. Over

thekchoices of j , these partition the edges.

7.1.3. (Cn K2)= 3. The lower bound is given by the maximum degree.

For the upper bound, when n is even colors 0 and 1 can alternate alongthe two cycles, with color 2 appearing on the edges between the two copies

of the factor Cn. Whenn is odd, colors 0 and 1 can alternate in this way

except for the use of one 2. Color 2 appears on all cross edges except those

incident to edges on the cycles with color 2, as shown below.

2 2 0 1 2 2

0 1 2 0 1

0 1 2 0 1

HP5 = L(H)

7.1.8. The line graph ofKm,n is the cartesian product ofKm and Kn . For

each edge xiyj in Km,n , we have a vertex (i, j) in L(Km,n); these are also

the vertices ofKm Kn . Pairs(i, j)and (k, l)in V(Km Kn)are adjacent inKm Kn if and only ifi = kor j =l . This is the same as the condition for

adjacency inL (Km,n), becausexiyj andxkyl share an endpoint inKm,n if and

only ifi = kor j = l.

7.1.9. A set of vertices in the line graph of a simple graphG form a clique

if and only if the corresponding edges in G have one common endpoint orform a triangle. Let Sbe the corresponding set of edges in G, and choosee S. If all other elements of Sintersect e at the same endpoint ofe, we

have one common endpoint. Otherwise, we have edges f and g such that

f shares endpoint x with e and g shares endpoint y with e. Since f and

g must share an endpoint, they share their other endpoint z and complete

a triangle. Since no single vertex lies in all ofe, f, g, no additional edge of

the simple graphG can share a vertex with all of these.


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Introduction to Graph Theory (2nd) - West - Solution Manual