Table Of ContentIntroduction to
Graph Theory
Solutions Manual
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Introduction to
Graph Theory
Solutions Manual
Koh Khee Meng
Notional University of Singapore, Singapore
Dong Fengming
Tay Eng Guan
Nanyang TKhnological Universily, Singapore
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World Scientific
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NEW JERSEY LONDON * SINGAPORE * BElJlNG SHANGHAI * HONG KONG * TAIPEI * CHENNAI
Published by
World Scientific Publishing Co. Pte. Ltd.
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A catalogue record for this book is available from the British Library.
INTRODUCTION TO GRAPH THEORY
Solutions Manual
Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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ISBN-13 978-981-277-175-9 (pbk)
ISBN-10 981-277-175-1 (pbk)
Printed in Singapore.
EH - Intro to Graph Theory (Solu Manual).pmd1 9/20/2007, 9:23 AM
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Preface
Discrete Mathematics is a branch of mathematics dealing with finite or
countable processes and elements. Graph theory is an area of Discrete
Mathematics which studies configurations (called graphs) consisting of a
set of nodes (called vertices) interconnecting by lines (called edges). From
humble beginnings and almost recreational type problems, Graph Theory
hasfounditscallinginthemodernworldofcomplexsystemsandespecially
of the computer. Graph Theory and its applications can be found not
only in other branches of mathematics, but also in scientific disciplines
such as engineering, computer science, operational research, management
sciences and the life sciences. Since computers require discrete formulation
of problems, Graph Theory has become an essential and powerful tool for
engineers and applied scientists, in particular, in the area of designing and
analyzing algorithms for various problems which range from designing the
itineraries for a shipping company to sequencing the human genome in life
sciences.
Graph Theory shows its versatility in the most surprising of areas. Re-
cently, the connectivity of the World Wide Web and the number of links
needed to move from one webpage to another has been remarkably mod-
eledwithgraphs,thusopeningthe realworldinternetconnectivitytomore
rigorous studies. These studies form part of research in the phenomena
of the property of a ‘small world’ even in huge systems such as the afore-
mentioned internet and global human relationships (in the so-called ‘Six
Degrees of Separation’).
This book is intended as a companion to our earlier book Introduction
to Graph Theory (World Scientific, 2006). Here, we present worked solu-
tions to all the exercise problems in the earlier book. Such a collection of
solutions is perhaps the first of its kind. We believe that the student who
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vi Introduction to Graph Theory, Solutions Manual
has worked on the problems himself will find the solutions presented here
useful as a check and as a model for rigorous mathematical writing. For
ease of reference, each chapter begins with a recapitulation of some of the
important concepts and/or formulae from the earlier book.
We would like to thank Prof. G.L. Chia, Ms Goh Chee Ying, Dr Jin
Xian’an, Dr Ng Kah Loon, Prof. Y.H. Peng, Dr Roger Poh, Ms Ren
Haizhen, Mr Soh Chin Ann, Dr Tan Ban Pin, Dr Tay Tiong Seng and
Dr K.L. Teo for reading the draft and for checking through the solutions -
any mistakes that remain are ours alone.
Koh Khee Meng
Dong Fengming
Tay Eng Guan
April 2007
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Notation
N= {1,2,3,···}
(cid:1)|S(cid:2)|= the number of elements in the finite set S
n = the number of r-element subsets of an n-element set = n!
r r!(n−r)!
B\A= {x∈B|x∈/ A}, where A and B are sets
(cid:3)
S = {x|x∈S for some i∈I}, where S is a set for each i∈I
i i i
i∈I
(⇒) proof of the implication “if P then Q” in the statement “P if and
only if Q”
(⇐) proof of the implication “if Q then P” in the statement “P if and
only if Q”
[Necessity] proof of the implication “if P then Q” in the statement “P if and
only if Q”
[Sufficiency] proof of the implication “if Q then P” in the statement “P if and
only if Q”
In what follows, G and H are multigraphs, and D is a digraph.
V(G): the vertex set of G
E(G): the edge set of G
v(G): the number of vertices in G or the order of G
e(G): the number of edges in G or the size of G
V(D): the vertex set of D
E(D): the arc set of D
v(D): the number of vertices in D or the order of D
e(D): the number of arcs in D
x→y : x is adjacent to y, where x,y are vertices in D
x(cid:5)→y : x is not adjacent to y, where x,y are vertices in D
∼
G=H : G is isomorphic to H
A(G): the adjacency matrix of G
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viii Introduction toGraph Theory, Solutions Manual
G: the complement of G
[A]: the subgraph of G induced by A, where A⊆V(G)
e(A,B): the number of edges in G having an end in A and the other in B,
where A,B ⊆V(G)
G−v : the subgraph of G obtained by removing v and all edges incident
with v from G, where v ∈V(G)
G−e: thesubgraphofGobtainedbyremovingefromG,wheree∈E(G)
G−F : the subgraph of G obtained by removing all edges in F from G,
where F ⊆E(G)
G−A: the subgraphof G obtained by removing each vertex in A together
withtheedgesincidentwithverticesinAfromG,whereA⊆V(G)
G+xy : the graph obtained by adding a new edge xy to G, where x,y ∈
V(G) and xy ∈/ E(G)
N (u): the set of vertices v such that uv ∈E(G)
G
N(u)= N (u)
(cid:3)G
N(S)= N(u), where S ⊆V(G)
u∈S
d(v)=d (v): the degree of v in G, where v ∈V(G)
G
id(v): the indegree of v in D, where v ∈V(D)
od(v): the outdegree of v in D, where v ∈V(D)
d(u,v): the distance between u and v in G, where u,v ∈V(G)
d(u,v): the distance from u to v in D, where u,v∈V(D)
c(G): the number of components in G
δ(G): the minimum degree of G
∆(G): the maximum degree of G
χ(G): the chromatic number of G
α(G): the independence number of G
G+H : the join of G and H
G∪H : the disjoint union of G and H
kG: the disjoint union of k copies of G
G(D): the underlying graph of D
n (H): the number of subgraphs in G which are isomorphic to H
G
C : the cycle of order n
n
K : the complete graph of order n
n
N : the null graph or empty graph of order n
n
P : the path of order n
n
Wn : the wheel of order n, Wn =Cn−1+K1
K(p,q): the complete bipartite graph with a bipartition (X,Y) such that
|X|=p and |Y|=q
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Contents
Preface v
Notation vii
1. Fundamental Concepts and Basic Results 1
Exercise 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Exercise 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Exercise 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2. Isomorphisms, Subgraphs and the Complement of a Graph 29
Exercise 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Exercise 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Exercise 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3. Bipartite Graphs and Trees 73
Exercise 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Exercise 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Exercise 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4. Vertex-colourings of Graphs 99
Exercise 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Exercise 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Exercise 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Exercise 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Exercise 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
ix
