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This page intentionally left blank Copyright © 2007, New Age International (P) Ltd., Publishers Published by New Age International (P) Ltd., Publishers All rights reserved. No part of this ebook may be reproduced in any form, by photostat, microfilm, xerography, or any other means, or incorporated into any information retrieval system, electronic or mechanical, without the written permission of the publisher. All inquiries should be emailed to [email protected] ISBN (13) : 978-81-224-2890-2 PUBLISHING FOR ONE WORLD NEW AGE INTERNATIONAL (P) LIMITED, PUBLISHERS 4835/24, Ansari Road, Daryaganj, New Delhi - 110002 Visit us at www.newagepublishers.com IIIII DDDDDeeeeedddddiiiiicccccaaaaattttteeeeeddddd ttttthhhhhiiiiisssss bbbbbooooooooookkkkk tttttooooo ‘‘‘‘‘OOOOOMMMMM’’’’’ ‘‘‘‘‘MMMMMaaaaahhhhhaaaaa KKKKKaaaaaaaaaallllliiiii MMMMMaaaaaaaaaatttttaaaaa’’’’’ &&&&& mmmmmyyyyy ‘‘‘‘‘‘‘‘‘‘PPPPPaaaaarrrrreeeeennnnntttttsssss’’’’’’’’’’ (v) This page intentionally left blank PREFACE This text has been carefully designed for flexible use. It is primarily designed to provide an introduction to some fundamental concepts in COMBINATORICS AND GRAPH THEORY for post- graduate (Master of Computer Applications – M.C.A.) students. Each topic is divided into sections of approximately the same length, and each section is divided into subsections that form natural blocks of material for teaching. Instructors can easily pace their lectures using these blocks. All definitions and theorems in this text are stated extremely carefully so that students will ap- preciate the precision of language and rigor needed in mathematical sciences. Proofs are motivated and developed slowly, their steps are all carefully justified. Recursive definitions are explained and used extensively. The writing style in this book is direct and pragmatic. Precise mathematical language is used without excessive formalism and abstraction. Care has been taken to balance the mix of notation and words in mathematical statements. Over 1000 problems are used to illustrate concept, related to different topics, and introduce applications. In most examples, a question is first posed, then its solution is presented with appropriate details. The applications included in this text demonstrate the utility of combinatorics and Graph Theory in the solution of real world problem. This text includes applications to wide-variety of areas, including computer science and engineering. There are over 900 exercises in the text with many different types of questions posed. There is an ample supply of straight forward exercises that develop basic skills, a large number of intermediate exercises and many challenging (problem sets) exercise sets. Problem sets are stated clearly and unambiguously, and all are carefully graded for various levels of difficulty. It will be honest on my part to accept that it is not possible to include every thing in one book. Many people contributed directly or indirectly to the completion of this book. Thanks are due to my friends who were able to convince me that I should write this book. I am grateful to my students, who always encouraged me and many times thanked me for writing this book. Special thanks to my teachers, who made me realize that I can indeed write a book on “Combinatorial Mathematics”. Some pulled me down, some encouraged me and some gave me constructive suggestions. I am grateful to all of them. (vii) I specially thank to Mr. K. Krishna (president); Mr. R. Rajagopal (secretary); Mr. Lokanath (Treasurer); Dr. Rukhmangada (Chairman); Mr. Kuppaswamy (Convener); Mr. Subramanyam (Manager); K.S. Institute of Technology, Bangalore; who always encouraged me and gave me constructive suggestions. I am grateful to all of them. I an greateful to my parents; Grandmother; maternal uncle; elder sister, elder brothers; who tolerated me all along while I devoted my time to completing this book. I express my sincere thanks to the chairman Mr. R.K. Gupta, the Managing Director Mr. Saumya Gupta, the Marketing Manager Mr. V.R. Babu and Mr. R. Srinath and Lucknow Branch Manager Mr. L.N. Mishra of M/s New Age International (P) Limited, publishers, New Delhi, for their responsible work-done at every level in the publication of the book with high production standards. Healthy criticism and suggestions to improve the quality and standards of the text are most welcome. Bangalore, March 2007 C. Vasudev (viii) This page intentionally left blank