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Introduction to graph theory: solutions manual PDF

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Introduction to Graph Theory Solutions Manual TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk Introduction to Graph Theory Solutions Manual Koh Khee Meng Notional University of Singapore, Singapore Dong Fengming Tay Eng Guan Nanyang TKhnological Universily, Singapore w\ World Scientific - NEW JERSEY LONDON * SINGAPORE * BElJlNG SHANGHAI * HONG KONG * TAIPEI * CHENNAI Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data 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, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. 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 September7,2007 11:6 WorldScientificBook-9inx6in graph-theory-solu 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 v September7,2007 11:6 WorldScientificBook-9inx6in graph-theory-solu 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 September7,2007 11:6 WorldScientificBook-9inx6in graph-theory-solu 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 vii September7,2007 11:6 WorldScientificBook-9inx6in graph-theory-solu 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 September7,2007 11:6 WorldScientificBook-9inx6in graph-theory-solu 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

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