Applied combinatorics, tucker, 6th

P1: FCH/FYX Frontmatter P2: FCH/FYX WB00623-Tucker QC: FCH/UKS T1: FCH November 1, 2011 12:27 This page is intentionally left blank www.itpub.net ii P1: FCH/FYX Frontmatter P2: FCH/FYX WB00623-Tucker QC: FCH/UKS November 1, 2011 T1: FCH 12:27 APPLIED COMBINATORICS i P1: FCH/FYX Frontmatter P2: FCH/FYX WB00623-Tucker QC: FCH/UKS T1: FCH November 1, 2011 12:27 This page is intentionally left blank www.itpub.net ii P1: FCH/FYX Frontmatter P2: FCH/FYX WB00623-Tucker QC: FCH/UKS T1: FCH November 28, 2011 8:0 APPLIED COMBINATORICS ALAN TUCKER SUNY Stony Brook John Wiley & Sons, Inc iii P1: FCH/FYX Frontmatter P2: FCH/FYX WB00623-Tucker QC: FCH/UKS T1: FCH December 1, 2011 16:7 VP AND PUBLISHER PROJECT EDITOR MARKETING MANAGER MARKETING ASSISTANT PRODUCTION MANAGER ASSISTANT PRODUCTION EDITOR COVER ILLUSTRATOR & DESIGNER Laurie Rosatone Shannon Corliss Debi Doyle Patrick Flatley Janis Soo Elaine S Chew Seng Ping Ngieng This book was set in Times by Aptara, Inc and printed and bound by Courier Westford, Inc The cover was printed by Courier Westford, Inc This book is printed on acid free paper ∞ Founded in 1807, John Wiley & Sons, Inc has been a valued source of knowledge and understanding for more than 200 years, helping people around the world meet their needs and fulfill their aspirations Our company is built on a foundation of principles that include responsibility to the communities we serve and where we live and work In 2008, we launched a Corporate Citizenship Initiative, a global effort to address the environmental, social, economic, and ethical challenges we face in our business Among the issues we are addressing are carbon impact, paper specifications and procurement, ethical conduct within our business and among our vendors, and community and charitable support For more information, please visit our website: www.wiley.com/go/citizenship Copyright C 2012, 2007, 2002, 1995 John Wiley & Sons, Inc All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc 222 Rosewood Drive, Danvers, MA 01923, website www.copyright.com Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030-5774, (201)748-6011, fax (201)748-6008, website http://www.wiley.com/go/permissions Evaluation copies are provided to qualified academics and professionals for review purposes only, for use in their courses during the next academic year These copies are licensed and may not be sold or transferred to a third party Upon completion of the review period, please return the evaluation copy to Wiley Return instructions and a free of charge return mailing label are available at www.wiley.com/go/returnlabel If you have chosen to adopt this textbook for use in your course, please accept this book as your complimentary desk copy Outside of the United States, please contact your local sales representative Library of Congress Cataloging-in-Publication Data Tucker, Alan, 1943 July 6Applied combinatorics / Alan Tucker — 6th ed p cm Includes bibliographical references and index ISBN 978-0-470-45838-9 (acid free paper) Combinatorial analysis Graph theory I Title QA164.T83 2012 511 6—dc23 2011044318 Printed in the United States of America 10 www.itpub.net iv P1: FCH/FYX Frontmatter P2: FCH/FYX WB00623-Tucker QC: FCH/UKS T1: FCH November 28, 2011 8:0 PREFACE Combinatorial reasoning underlies all analysis of computer systems It plays a similar role in discrete operations research problems and in finite probability Two of the most basic mathematical aspects of computer science concern the speed and logical structure of a computer program Speed involves enumeration of the number of times each step in a program can be performed Logical structure involves flow charts, a form of graphs Analysis of the speed and logical structure of operations research algorithms to optimize efficient manufacturing or garbage collection entails similar combinatorial mathematics Determining the probability that one of a certain subset of equally likely outcomes occurs requires counting the size of the subset Such combinatorial probability is the basis of many nonparametric statistical tests Thus, enumeration and graph theory are used pervasively throughout the mathematical sciences This book teaches students how to reason and model combinatorially It seeks to develop proficiency in basic discrete math problem solving in the way that a calculus textbook develops proficiency in basic analysis problem solving The three principal aspects of combinatorial reasoning emphasized in this book are the systematic analysis of different possibilities, the exploration of the logical structure of a problem (e.g., finding manageable subpieces or first solving the problem with three objects instead of n), and ingenuity Although important uses of combinatorics in computer science, operations research, and finite probability are mentioned, these applications are often used solely for motivation Numerical examples involving the same concepts use more interesting settings such as poker probabilities or logical games Theory is always first motivated by examples, and proofs are given only when their reasoning is needed to solve applied problems Elsewhere, results are stated without proof, such as the form of solutions to various recurrence relations, and then applied in problem solving Occasionally, a few theorems are stated simply to give students a flavor of what the theory in certain areas is like For decades, collegiate curriculum recommendations from the Mathematical Association of America have included combinatorial problem solving as an important component of training in the mathematical sciences Combinatorial problem solving underlies a wide spectrum of important subjects in the computer science curriculum Indeed, it is expected that most students in a course using this book will be computer science majors For both mathematics majors and computer science majors, v P1: FCH/FYX Frontmatter P2: FCH/FYX WB00623-Tucker QC: FCH/UKS November 1, 2011 T1: FCH 12:27 vi Preface this author believes that general reasoning skills stressed here are more important than mastering a variety of definitions and techniques This book is designed for use by students with a wide range of ability and maturity (sophomores through beginning graduate students) The stronger the students, the harder the exercises that can be assigned The book can be used for a one-quarter, two-quarter, or one-semester course depending on how much material is used It may also be used for a one-quarter course in applied graph theory or a one-semester or one-quarter course in enumerative combinatorics (starting from Chapter 5) A typical one-semester undergraduate discrete methods course should cover most of Chapters to and to 8, with selected topics from other chapters if time permits Instructors are strongly encouraged to obtain a copy of the instructor’s guide accompanying this book The guide has an extensive discussion of common student misconceptions about particular topics, hints about successful teaching styles for this course, and sample course outlines (weekly assignments, tests, etc.) The sixth edition of this book draws upon features from all the earlier editions For example, the game of Mastermind that appeared at the beginning of the first edition has been brought back, and a closing Postlude about cryptanalysis has been added The suggested solutions to selected enumeration exercises from the second and third editions have returned Of course, there are also new exercises Also, the numbers were changed in many of the old exercises in the counting chapters (to guard against student groups accumulating old solution sets) Many people gave useful comments about early drafts and the first edition of this text; Jim Frauenthal and Doug West were especially helpful The idea for this book is traceable to a combinatorics course taught by George Dantzig and George Polya at Stanford in 1969, a course for which I was the grader Many instructors who have used earlier editions of this book have supplied me with valuable feedback and suggestions that have, I hope, made this edition better I gratefully acknowledge my debt to them Ultimately, my interest in combinatorial mathematics and in its effective teaching rests squarely on the shoulders of my father, A W Tucker, who had long sought to give finite mathematics a greater role in mathematics as well as in the undergraduate mathematics curriculum Finally, special thanks go to former students of my combinatorial mathematics courses at Stony Brook It was they who taught me how to teach this subject Alan Tucker Stony Brook, New York www.itpub.net P1: FCH/FYX Frontmatter P2: FCH/FYX WB00623-Tucker QC: FCH/UKS T1: FCH November 28, 2011 8:0 CONTENTS PRELUDE xi PART ONE CHAPTER 1.1 1.2 1.3 1.4 1.5 GRAPH THEORY ELEMENTS OF GRAPH THEORY Graph Models Isomorphism 14 Edge Counting 24 Planar Graphs 31 Summary and References 44 Supplementary Exercises 45 CHAPTER 2.1 2.2 2.3 2.4 2.5 COVERING CIRCUITS AND GRAPH COLORING Euler Cycles 49 Hamilton Circuits 56 Graph Coloring 68 Coloring Theorems 77 Summary and References 49 86 Supplement: Graph Model for Instant Insanity 87 Supplement Exercises 92 CHAPTER 3.1 3.2 3.3 TREES AND SEARCHING 93 Properties of Trees 93 Search Trees and Spanning Trees 103 The Traveling Salesperson Problem 113 vii P1: FCH/FYX Frontmatter P2: FCH/FYX QC: FCH/UKS WB00623-Tucker T1: FCH November 28, 2011 8:0 viii Contents 3.4 3.5 Tree Analysis of Sorting Algorithms Summary and References 125 CHAPTER 4.1 4.2 4.3 4.4 4.5 4.6 NETWORK ALGORITHMS 127 Shortest Paths 127 Minimum Spanning Trees 131 Network Flows 135 Algorithmic Matching 153 The Transportation Problem 164 Summary and References 174 PART TWO CHAPTER 5.1 5.2 5.3 5.4 5.5 5.6 121 ENUMERATION 177 GENERAL COUNTING METHODS FOR ARRANGEMENTS AND SELECTIONS Two Basic Counting Principles 179 Simple Arrangements and Selections 189 Arrangements and Selections with Repetitions Distributions 214 Binomial Identities 226 Summary and References 236 179 206 Supplement: Selected Solutions to Problems in Chapter 237 CHAPTER 6.1 6.2 6.3 6.4 6.5 6.6 Generating Function Models 249 Calculating Coefficients of Generating Functions Partitions 266 Exponential Generating Functions 271 A Summation Method 277 Summary and References 281 CHAPTER 7.1 7.2 7.3 7.4 GENERATING FUNCTIONS 249 256 RECURRENCE RELATIONS Recurrence Relation Models 283 Divide-and-Conquer Relations 296 Solution of Linear Recurrence Relations 300 Solution of Inhomogeneous Recurrence Relations 283 304 www.itpub.net P1: JSN Solution WB00623-Tucker 466 October 29, 2011 10:43 Solutions to Odd-Numbered Problems 13 Similar to recurrence relations in Example except with 4n−1 replacing 3n−1 and a1 = instead of a1 = (still b1 = c1 = 1), an = 15 (4n − 1), n even, = n (4 − 4), n odd 15 15 (a) an = C(n − 1, k − 1)an−k , x (b) g(x) = ee −1 CHAPTER EIGHT SOLUTIONS Section 8.1 11 13 15 17 19 21 23 25 27 29 31 33 35 268 − 212 × 266 3n − × 2n−1 {C(52, 7) − C(13, 7) × C(4, 1)7 }/C(52, 7) × (94 − × × C(4, 3) 700 − 200 − 180 − 150 (a) 300 − 70 − 100 + 40 (b) 300 − 100 − 60 (142 + 500 − 71)/1000 (a) 200 − × 85 + × 30 − 15, (b) 85 − × 30 + 15 30 − 15 − 10 − + + + − 320 − × 220 + × 6!/2!3 − × 5!/2!2 + × 4!/2! − 3! C(37, 10) − [C(27, 10) + C(25, 10) + C(22, 10)] + [C(15, 10) + C(12, 10) + 1] C(52, 6) − × C(48, 6) + × C(44, 6) − C(40, 6) × C(6, 2) × 4! − {2 × C(6, 3) × 3! + 6!/2!2 } + C(6, 4) × 2! (15!/3!5 − × 5! × 10!/2!5 + × 5!3 − 5!3 )/(15!/3!5 ) 20 N (Y ) = N (Y − K ) + N (Y ∩ K ) = 50 + 20 25, Section 8.2 10m − × 9m + × 8m − 7m 13 × C(48, 5) − C(13, 2) × 44 (a) {C(52, 13) − C(4, 1) × C(39, 13) + C(4, 2) × C(26, 13) − C(4, 3) × C(13, 13)}/C(52, 13), (b) {C(4, 1) × C(39, 13) − C(4, 2) × C(26, 13) + C(4, 3) × C(13, 13)}/C(52, 13), (c) {C(52, 13) − C(4, 1) × C(48, 13) + C(4, 2) × C(44, 13) − C(4, 3) × C(40, 13) + C(36, 13)}/C(52, 13) 9!/3!3 − × 7!/3!2 + × 5!/3! − 3! 26! − {3 × 23! + 24!} + {2 × 20! + × 21!} − 18! www.itpub.net P1: JSN Solution WB00623-Tucker October 29, 2011 10:43 Solutions to Odd-Numbered Problems 467 11 C(26 + − 1, 26) − × C(19 + − 1, 19) + × C(12 + − 1, 12) − C(5 + − 1, 5) 13 C(10 + − 1, 10) − C(4, 1) × C(8 + − 1, 8) + C(4, 2) × − C(4, 3) × 15 27 17 ≈10!2 /e 19 10!/2!5 × {10!/2!5 − C(5, 1) × 8!/2!4 + C(5, 2) × 6!/2!3 − C(5, 3) × 4!/2!2 + C(5, 4) − C(5, 5)} 21 15!/3!5 − × × 12!/3!4 + C(5, 2) × × × 9!/3!3 − C(5, 3) × × × × 6!/3!2 + × 5! − 5! 23 (a) n − C(5, 1) × n + C(5, 2) × n − C(5, 3) × n + {C(5, 4) − C(5, 5)} × n; (b) n − C(5, 1) × n + C(5, 2) × n − {(C(5, 3) − 1) × n + n } + {(C(5, 4) − 2) × n + × n } − n (c) n − C(7, 1) × n + C(7, 2) × n − {C(7, 3) − 3) × n + × n } + {(C(7, 4) − 14) × n + 14 × n } − {(C(7, 5) − 2) × n + × n } + {C(7, 6) − C(7, 7)} × n 27 If 21 = C(2 + − 1, 2), then (−1)k × C(6, k) × (21 − k)n 29 5!n × {(2n − 1)! − C(n, k) × (2n − − k)!} 31 P(C(7, 3), 7) − × P(C(6, 3), 7) + C(7, 2) × P(C(5, 3), 7) 33 C(P(6, 3), 8) − × C(P(5, 3), 8) + C(6, 2) × C(P(4, 3), 8) 35 (−1)k × C(n, k) × (n − k)r /n! n−1 k+1 37 k=3 (−1)k+1 × C(k, 3) × C(n, k) × (n − k)r , n−1 × C(k − 1, 2) k=3 (−1) r × C(n, k) × (n − k) 39 [C(5, 2) × 9!/2!3 ] − [2 × C(5, 3) × 8!/2!2 ] + [3 × C(5, 4) × 7!/2!] − [4 × 6!] n j−k 47 nk=0 k × × C( j, k) × n!/j! j=k (−1) Section 8.3 × board with darkened squares on main diagonal 5! − × 4! + 20 × 3! − 16 × 2! + × 1! 7! − (9 × 6!) + (30 × 5!) − (46 × 4!) + (32 × 3!) − (8 × 2!) 5! − (7 × 4!) + (16 × 3!) − (13 × 2!) + (2 × 1!) (a) × board with darkened squares in positions just to right of main diagonal, (b) (x + 1)4 , (c) 5j=k (−1)k+ j × C( j, k) × C(n − 1, j) × (n − j)! 13 × array of darkened squares and “L” (a column of squares beside a single square both have + 4x + 2x ) 11 CHAPTER NINE SOLUTIONS Section 9.1 (a) Not symmetric, (b) Yes, (c) Not transitive, P1: JSN Solution WB00623-Tucker 468 October 29, 2011 10:43 Solutions to Odd-Numbered Problems (d) (e) (a) (b) (c) (a) Not transitive, Not transitive symmetries (as in Example 3), symmetries, symmetry All C j left fixed, (b) C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C1 C3 C4 C5 C2 C7 C8 C9 C6 C11 C10 C13 C14 C15 C12 C16 (c) C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C1 C3 C2 C5 C4 C6 C9 C8 C7 C11 C10 C15 C14 C13 C12 C16 (d) C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C1 C2 C5 C4 C3 C9 C8 C7 C6 C10 C11 C14 C13 C12 C15 C16 11 (a) a, b, c are rotations or 0◦ , 120◦ , and 240◦ , respectively d, e, f are flips around vertical axis, axis 30◦ clockwise of vertical, and axis 30◦ counterclockwise of vertical; row is first symmetry, column second symmetry: a b c d e f a b c d e f a b c d e f b c a e f d c a b f d e d f e a c b e d f b a c f e d c b a (b) Straightforward (c) Let a = a b c d 1234 1234 ,b = a b c d a b c d b a d c c d a b d c b a 1234 2143 ,c = 1234 3412 ,d= 1234 4321 , 13 Only left structure (right structure has isomers) 21 (b) {π1 , π3 , π5 , π6 } or {π1 , π3 , π7 , π8 }, (c) In addition to subgroups in (b) and G, G , G , other subgroups are {π1 πi }, for i = 3, 5, 6, Section 9.2 (a) 24, (b) 70 13 [315 + (2 × 35 )] 51 www.itpub.net P1: JSN Solution WB00623-Tucker October 29, 2011 10:43 Solutions to Odd-Numbered Problems 469 12 (5n + 5n/2 ) n even, and 12 [5n + (3 × 5(n−1)/2 )] n even (a) In cycle form: π1 = (1)(2)(3), π2 = (12)(3), π3 = (13)(2), π4 = (1)(23), π5 = (123), π6 = (132), (b) ␺(π1 ) = C(12 + − 1, 12) = 91, ␺ (π2 ) = ␺ (π3 ) = ␺ (π4 ) = 7, ␺ (π5 ) = ␺(π6 ) = 1, answer: 16 (91 + + + + + 1) = 19 11 ␺ (π1 ) = 18, ␺ (π3 ) = 6, ␺ (π7 ) = ␺ (π8 ) = 12, and other ␺ (πi ) = 0, answer (18 + + 12 + 12) = 6; 13 (b) {π1 , π7 }, (c) {π1 , π6 } Section 9.3 55 (a) 130, (b) 92, (c) Cyclic color sequence on hexagon of R-W-B-R-W-W and R-B-W-R-W-W (a) 16 (m + 2m + 3m ), (b) 18 (m 12 + 2m + 3m + 2m ), (c) 12 (m + m ), (d) 14 (m + 3m ), (m + 2m + 2m + 4m + 3m ), (e) 12 (f) Same as (b) (a) 16 (m + 2m + 3m ), (b) 18 (m 12 + 2m + 3m + 2m ), (c) 12 (m + m ), (d) 14 (m 10 + m + m + m ), (m 12 + 2m + 2m + m + 6m ), (e) 12 (f) (m 16 + 2m + m + 4m ) [7 + (3 × 72 )] = 637 11 (a) 12 (28 + 24 ) = 136, (b) 14 (28 + 24 + + 24 ) = 72 13 ␺(πi ) = number of cycles of length 15 (a) 1p [m p + ( p − 1) × m], (b) 21p {m p + [( p − 1) × m] + ( p × m ( p+1)/2 )} Section 9.4 b5 + b4 w + 2b3 w2 + 2b2 w2 + bw4 + w5 b4 + w4 + r + b3 w + b3r + bw3 + w3r + br + wr + 2b2 w2 + 2b2r + 2w2r + 2b2 wr + 2bw2r + 2bwr (a) 16 {(b + w)4 + 2(b3 + w3 )(b + w) + 3(b2 + w2 )(b + w)2 }, (b) 18 {(b + w)12 + 2(b4 + w4 )3 + 3(b2 + w2 )6 + 2(b2 + w2 )5 (b + w)2 }, (c) 12 {(b + w)5 + (b + w)(b2 + w2 )2 }, P1: JSN Solution WB00623-Tucker 470 October 29, 2011 10:43 Solutions to Odd-Numbered Problems (d) (e) {(b + w)8 + 3(b2 + w2 )4 }, {(b + w)7 + 2(b6 + w6 )(b + w) + 2(b3 12 2 + w3 )2 (b + w) + 4(b2 + w2 )3 (b + w) + 3(b + w ) (b + w) }, (f) Same as (b) (a) 16 {(b + w)6 + 2(b3 + w3 )2 + 3(b2 + w2 )2 (b + w)2 }, (b) 18 {(b + w)12 + 2(b4 + w4 )3 + 3(b2 + w2 )6 + 2(b2 + w2 )5 (b + w)2 }, (c) 12 {(b + w)6 + (b + w)2 (b2 + w2 )2 }, (d) 14 {(b + w)10 + (b2 + w2 )5 + (b + w)2 (b2 + w2 )4 + (b + w)4 (b2 + w2 )3 }, {(b + w)12 + 2(b6 + w6 )2 + 2(b3 + w3 )4 + (b2 + w2 )6 + 6(b2 + w2 )5 (e) 12 (b + w)2 }, (f) 18 {(b + w)16 + 2(b4 + w4 )4 + (b2 + w2 )8 + 4(b2 + w2 )7 (b + w)2 } {(b + w)4 + 8(b3 + w3 )(b + w) + 3(b2 + w2 )2 }, (a) 12 {(b + w)6 + 6(b4 + w4 )(b + w)2 + 3(b2 + w2 )2 (b + w)2 + 6(b2 + w2 )3 (b) 24 + 8(b3 + w3 )2 } 11 24 {(b + w)4 + 6(b4 + b4 ) + 8(b3 + w3 )(b + w) + 3(b2 + w2 )2 + 6(b2 + w2 ) (b + w)2 } 13 (a) If not a cyclic rotation of all corners, the length of the cycle would have to divide p—impossible, (b) C( p, k)/ p 15 (a) 36, (b) 216 CHAPTER TEN SOLUTIONS Section 10.1 (a) {a, c} or {b, d}, (b) f , (c) No kernel, consider directed 5-circuit b, a, d, g, h, b, a—if a is K (kernel), then d not in K , then g in K , then h not in K , then b in K —impossible since a in K ; similar sort of argument (also involving c, f, e) if a not in K {3, 4, 9, 11, 12, 16, 17, 21, 25, 26, 27, 31, 32, 36, over 40} A goes to 2, B must go to or else A will win Move to multiples of 5k + (initial position is win for second player) By symmetry assume g(a) = 0, then g(e) = 1, then g(d) = 0, then g(c) = 1, then g(b) = 0, but now two kernel vertices are adjacent 11 S is a kernel if and only if all vertices not in S have an edge to a vertex in S while no vertex in S has an edge to a vertex in S—that is, if and only if W (S) = S 13 Follows from parts (a) and (b) of Exercise 12 since g(x) = k means there is a path of length k starting at x while l(x) is length of longest path starting at x; longest path is at least length k (maybe there is another longer path starting at x) www.itpub.net P1: JSN Solution WB00623-Tucker October 29, 2011 10:43 Solutions to Odd-Numbered Problems 471 15 Suppose x and y are adjacent because there is an edge from x to y; then x must have a larger level number than y and a different Grundy value from its successor y 17 If there were an infinite number of vertices, then one of the finite number of starting vertices, call it x1 , must have an infinite number of vertices reachable from it, and one of the finite number of successors of x1 , call it x2 , must have an infinite number of vertices reachable from it, and one of the finite number of successors of x2 , call it x3 , must have an infinite number of vertices reachable from it, and so on, without end 19 (a) Let a = 0000, b = 000, c = 0− 00, d = 00− 0, e = 00, f = 0− 0, g = 0, h =− (win); s(a) = {b, c, d, e, f }, s(b) = {e, f, g}, s(c) = s(d) = {e, f, g}, s(e) = {g, h}, s( f ) = {g}, s(g) = h, s(h) = Ø; g( f ) = g(h) = 0, g(a) = g(g) = 1, g(e) = 2, g(b) = g(c) = g(d) = Section 10.2 (a) (b) (c) (d) (a) (b) (c) (d) (a) (b) (c) (d) (a) (b) 7, remove from 4th pile, 0, 4, remove from 2nd, 3rd, or 4th pile, 3, remove from 3rd pile or from 4th pile, 2, remove from 3rd or 4th pile, 0, 0, 0, 1, remove from 4th pile, 3, remove from 3rd pile 3, add nickel to 3rd pile, (0, 0), (0, 4), (0, 6), (0, 9), (1, 1), (2, 2), (2, 5), (2, 8), (3, 3), (3, 7), (4, 4), (4, 6), (5, 5) ˙ cj = c + ˙ d j ; if c + c j = c + ˙ d j , then c j and d j If c j = d j , then trivially c + must have 1s in the same positions in their binary representations—that is, they must be equal 11 Immediately the proposed strategy works 13 (a) Remove three balls along one of the three lines formed by ball on one of the three sides of the arrangement POSTLUDE SOLUTIONS THEORIES COMPLETE FAMILY UNIFORM VERTICES P1: JSN Solution WB00623-Tucker 472 October 29, 2011 10:43 Solutions to Odd-Numbered Problems APPENDIX SOLUTIONS Section A.1 (a) 12, 27 (b) 2, 3, 6, 7, 9, 12, 15, 17, 18, 21, 22, 24, 27, (c) 1, 4, 5, 8, 10, 11, 13, 14, 16, 19, 20, 23, 25, 26, 28, 29, (d) all ≤ k ≤ 29 except 12, 27 We are given N (R ∩ M) = as well as that N (M) = N (R) = N (M) = N (R) = 4; then N (R ∩ M) = N (M) − N (R ∩ M) = − = 2, N (R ∩ M) = N (R) − N (R ∩ M) = − = 2, and clearly N (R ∩ M) = (a) Impossible, (b) Yes, 20 − − = 4, (c) 20 − 15 = (a) (b) A B A B (c) (A ∪ B) ∩ (A ∩ B) = (A ∩ B), see Figure A1.3, (d) A − (B − A) = A; here all expressions involve ∩, so that ABC = A ∩ B ∩ C _ ABC _ ABC ABC _ ABC ABC _ _ ABC _ ABC _ ABC 17 (a) E = S ∪ H ∪ C, 523 − 393 , (b) E = (S ∪ H ∪ C) ∩ (S ∩ H ∩ C), 523 − 393 − 133 , (c) E = (S ∩ H ∩ C) ∪ (S ∩ H ∩ C) ∪ (S ∩ H ∩ C), × 132 × 39 Section A.2 23 One can prove by induction only a property that is a function of n—for example, one can prove that there are a finite number of binary sequences of length ≤n www.itpub.net P1: JSN Solution WB00623-Tucker October 29, 2011 10:43 Solutions to Odd-Numbered Problems 473 25 The initial step only assumes that n ≥ 1, not n ≥ 2, but for n = 1, a n−2 is undefined Section A.3 1/2 (a) 1/6, (b) 18/36 = 1/2, (c) 3/36 = 1/12 (a) 1/6, (b) 1/2 (a) 1/3 × 1/3 = 1/9, (b) × 1/3 × 2/3 = 4/9 (2 × × 2)/4! = 1/3 11 − [(50 + 40 − 20)/100] = 3/10 13 2/3 15 (a) All sequences with k tails, ≤ k ≤ 8, and one head followed by a head, (b) All positive integers, (c) All ordered pairs of positive integers, (d) All sequences of k black balls, k ≥ 0, followed by a red ball Section A.4 n + 15 Printer i is connected to computers i, i + 1, i + 2, i + 3, i + 4, i + P1: JSN Solution WB00623-Tucker October 29, 2011 10:43 This page is intentionally left blank www.itpub.net 474 P1: PBY Bindex WB00623-Tucker November 25, 2011 9:40 INDEX A AC Principle, 16 Addition Principle, 180 Adjacency matrix, 103 Adjacent vertices, Adler, I., 316, 317 Agnarsson, G., 439 Ahuja, R., 128, 174, 175 Aldous, J., 439 Allenby, R., 440 Ancestors in a tree, 94, 438 Andreescu, T., 440 Appel, K., 32, 69, 80, 86 Arrangement(s), 190, 435 with repetition, 206 Assignment problem, 121, 175 Art gallery problem, 79 Augmenting flow algorithm, 142 Ault, L., xiii, xv Automorphism of a graph, 48 B Backtracking in a graph, 104, 438 Balakrishnan, V., 439, 440 Balanced tree, 96, 438 Ball, W., 400 Barnette, D., 86 Berge, C., 400, 440 Berlekamp, E., 400 Bernoulli, Jacob, 237 Bernoulli, Jacques, 237, 425 Best, S., 439 Biggs, N., 44, 86, 440 Binomial coefficient, 190, 228, 435 Binomial identities, 230, 261, 291, 329 Binomial Theorem, 227 Bipartite graph, 5, 27, 67, 153, 435 deficiency, 163 Birthday paradox, 203 Blockwalking, 230 Boat crossing puzzles, 108 Bogart, K., 439 Bondy, J., 44, 87 Bono, M., 439 Boole, G., 418 Boolean algebra, 416 Bouton, C., 400 Boyer, C., 418 Branch-and-bound search, 113 Breadth-first search, 104, 438 Bridge in a graph, 46 Bridge probabilities, 215 Brook’s Theorem, 80 Brualdi, R., 439 Bubble sort, 122 Buck, R., 198, 237 Burnside’s Lemma, 365 Bussey, W., 421 C C(n,r), 190 Cameron, P., 281, 439 Capacity of a cut, 137 of an edge, 135 Capobianco, M., 440 Cardano, B., 425 Catalan number, 313 Cayley, A., 44, 99, 1125 Center of a tree, 101 Chain in a network, 142 Characteristic equation, 300, 310 Characteristic sequence of a tree, 112 Chartrand, G., 440 475 P1: PBY Bindex WB00623-Tucker 476 November 25, 2011 9:40 Index Chessboard, generalized, 342 Children in a tree, 94, 438 Chromatic number, 69, 80, 435 Chromatic polynomial, 81, 435 Chvatal’s theorem, V., 61 Circle graph, 51 Circle-chord method, 343 Circuit in a graph, 4, 435 length of a circuit, 27 Closure in a group, 359 Code text, 404 Coin balancing, 98 Color critical graph, 79, 85 Coloring a graph, 69, 334, 433 Combination(s), 190, 435 Complement: of a chessboard, 351 of a graph, 17, 85, 435 of a set, 416 Complete graph, 15, 25, 435 Component of a graph, 25, 436 Computational complexity, 430 Configuration in a graph, 35 Conjugate diagram, 268 Connected graph, 4, 435 Strongly connected, 55 Test for connectedness, 104 Conway, J., 400 Cook, S 431 Cormen, T., 298, 317 Coxeter, H., 400 Crossing number, 42 Cryptogram, 401 Cube, symmetries of, 379 Cut in network, 136 a-z cut, 136 Cut-set, 47, 151 Cycle, 49, 435 Cycle in a permutation, 359 Cycle index, 371 Cycle structure representation, 370 D David, F., 236, 237, 425 Deadheading edge, 52 Decomposition principle for Instant Insanity, 88 Deficiency of a bipartite graph, 163 Degree of a vertex, 6, 24, in-degree and out-degree, 18 Degree of a region, 38 De Carteblanche, F., 88 De Moivre, A., 281, 316, 351 De Morgan, A., 389, 417, 421 Depth-first search, 104, 438 Derangement, 333, 436 Descendants in a tree, 94, 438 Diameter of a graph, 67 Dictionary search, 5, 97 Difference equation, 290 Difference of sets, 416 Digraph, 401 Digital sum, 395 Dijkstra’s algorithm for shortest paths, 127 Dirac’s theorem, 61 Direct sum of graphs, 394 Directed graph, 3, 436 Dirichlet drawer principle, 427 Distribution, 215, 289, 436 Divide-and-conquer relations, 296 Dual graph, 32 Durfee square, 270 E Edge, 3, 24 directed edge, 3, 436 Edge chromatic number, 80 Edge coloring, 73 Edge cover, 8, 155, 436 Edge coloring, 73 Elements of a set, 415 Eliminating a team from contention, 157 Equivalence relation, 356 equivalence class, 356 Erickson, M., 439 Euler, L., 37, 44, 49, 51, 86, 281 Euler’s constant e, 198, 333 Euler cycle, 50, 112, 436 Euler’s formula for graphs, 37 www.itpub.net P1: PBY Bindex WB00623-Tucker November 10, 2011 10:22 Index 477 Euler trail, 53, 436 Event of outcomes, 423 compound event, 424 elementary event, 423 Expected value of a random variable, 265 Experiment, 423 Greenlaw, R., 440 Grinberg’s theorem, 61 Gross, J., 440 Group of symmetries, 360 Grundy, P., 390, 400 Grundy function, 390, 395 Guy, R., 400 F Factor in a graph, 89 labeled factor, 89 Family of elements, 415 Feil, T., 414 Feller, W., 281 Feng, Z., 440 Fermat, P., 236, 425 Ferrers diagram, 268 Fibonacci, 316 Fibonacci numbers, 281, 285, 302, 310, 316 Fibonacci relation, 285 Fisk’s theorem, 79 Fleury’s algorithm for Euler cycles, 55 Flow in network, 157 dynamic flow, 147 value of a flow, 137 Floyd’s algorithm for shortest paths, 129 Ford, L., 174, 175 Forest of trees, 101 Four-color problem, 32, 69, 80, 86 Fourier, J., 281 Fourier transform, 281 Fulkerson, D., 174, 175 H Haken, W., 32, 69, 80, 86 Hall, M., 440 Hall’s Marriage Theorem, 156 Halmos, P., 418 Hamilton, W., 86 Hamiltonian circuit and path, 56, 86, 119, 436 Harary, F., 383, 440 Harris, J., 440 Hartsfield, N., 440 Heap, 123 Heap sort, 123 Height of a tree, 96, 438 Hillier, F., 175 Hopcroft, J., 163 Hypercube, 63 G Gaines, H., 414 Galileo, 425 Garbage collection, 72 Generating functions, 249, 308, 436 exponential, 272 Generators of a group, 362 George, J., 440 Graham, R., 427, 439 Graph, 3, 436 Gray code, 62 I Identical Objects Rule, 425 Identity in a group, 359 Inclusion-exclusion formula, 328 Inclusion-exclusion principle, 320 Independent edges, 153 Independent set, 8, 69, 436 Induction, 420 Initial conditions, 283 Instant Insanity puzzle, 87 Integer solutions of an equation, 217, 250, 332 Interest problems, 287 Internal vertex in a tree, 95, 438 Intersection of sets, 416 Inverse of a symmetry, 359 Isolated vertices, 16 Isomers of organic compounds, 362 Isomorphism of graphs, 14, 436 P1: PBY Bindex WB00623-Tucker 478 November 10, 2011 10:22 Index K Kn (complete graph on n vertices), 15 Kan, A R., 126 Kayles, 393 Kernel of game, 387 Keys in Mastermind, xi Keyword, 405 Keyword transpose encoding, 405 Kiefer, J., 316, 317 Kirchhoff, G., 44, 125 Knight’s tour, 55, 67 Knuth, D., 440 Konig, D., 44 Konig-Egevary theorem, 156 Konigsberg bridges, 49 Kruse, R., 126 Kruskal’s algorithm for minimal spanning trees, 131 Kuratowski’s theorem, 35 L Laplace, S., 281, 423, 425 Laplace transform, 281 Lawler, E., 126 Leaves of a tree, 95, 438 Leibnitz, G., 237 Leiserson, C., 298, 317 Lenstra, J., 126 Lesniak, L., 440 Level in a game, 388 Level numbers in a tree, 94, 438 Letter frequencies, 401 Lewand, R., 414 Lieberman, G., 175 Line graph, 41, 48, 55, 68 Linear program, 175 Lloyd, E., 44, 86, 440 Lucas, E., 126 M MacMahon, P., 281 Magnanti, T., 175 Map coloring, 32, 437 Marcus, D., 440 Martin, G., 440 Mastermind, xi Matching in a graph, 4, 153, 437 maximal, 153 X-matching, 153 Matching network, 154 Maurolycus, 421 Max flow-min cut theorem, 145 Maximal planar graph, 42 Maze searching, 106 Mazur, D.,439 Member of a set, 415 Menage, 350 Merge sort, 122 Meriss, R., 440 Minimal spanning tree, 131 Minimum cost rule, 174 Missionaries-cannibals puzzle, 108 Molluzzo, J., 440 Moments of a random variable, 265 k-th moment, 277 Montmort, P., 351 Mountain climber’s puzzle, 25 Multigraph, 49, 437 Multiplication, fast, 298 Multiplication Principle, 180 Murty, U., 44, 87, 440 N Network, 127, 437 Network flow, 135, 437 Nim game, 393, 400 Northwest corner rule, 168 NP-completeness, 57, 69, 113, 431 Null set, 415 O Ore, O., 440 Orlin, J., 175 O’Rourke, J., 79, 87 P P(n,r ), 190 Palmer, E., 383 Parent in a tree, 94, 438 Parenthesization, 289, 312 www.itpub.net P1: PBY Bindex WB00623-Tucker November 25, 2011 9:40 Index Partitions, 266, 437 of an integer, 266 self-conjugate, 270 Pascal, B., 236, 421, 423, 425 Pascal’s triangle, 229, 236 Patashnik, O., 440 Path in a graph, 4, 437 directed path, 10 length of a path, 27 Pattern inventory, 353 Peacock, G.,417 Permutation, 190, 437 r -permutation, 190 Pigeonhole Principle, 427 Pisa, L., 316 Pitcher pouring puzzle, 106 Plain text, 404 Planar graph, 31, 80, 152, 437 Plane graph, 31 Platonic graph, 43 Poker probabilities, 192, 331 Polya, G., 229, 377, 382, 383, 440 Polya’s enumeration formula, 377 Polygon, 78 Power series, 249 Prim’s algorithm for minimal spanning tree, 131 Probability of an event, 423 Probability generating function, 265, 277 Progressively finite game, 385 Prufer sequence, 100 R Ramsey theory, 47, 427 Random variable, 265 Range graph, 25 Range in a matching, 156 Recurrence relation, 283, 437 homogeneous relation, 300 inhomogeneous relation, 304 systems of recurrence relations, 289 Reade, R., 383 Reflexivity in a relation, 356 Region, 37 479 Regular graph, 45 Rencontre, 339 Ringel, G., 440 Riordan, J., 351, 440 Rivest, R., 298, 317 Roberts, F., 439 Rook, non-capturing, 342 Rook polynomial, 343 Root of a tree, 93, 438 Rotation of a figure, 356 Rothschild, B., 427, 439 Round-robin tournament, 46, 73 Rubik’s cube, 68 Run in a sequence, 222 Ryser, H., 440 S Sample space, 424 Sandefur, J., 290, 317 Satisfiability problem, 431 Saturated and unsaturated edges, 139 Scheduling problems, 8, 29, 71, 72 Schwartz, B., 157, 175 Secret code xi Sedgewick, R., 126 Selection, 190, 437 Selection with repetition, 207 equivalent forms, 218, 223 Self-complementary graph, 48 Set, 415 Set Composition Principle, 194 Set of distinct representatives, 154 Shapley-Shubik index, 197 Shih-Chieh, C., 237 Shmoys, D., 126 Shortest path algorithms, 127 Sibling in a tree, 94, 438 Sink in a network, 135 Sinkov, A., 414 Slack in an edge, 139 Slomson, A., 440 Sorting algorithms, 121 Source in a network, 135 Spanning tree, 104, 131, 167 438 Spaziergangen problem, 49 P1: PBY Bindex WB00623-Tucker 480 November 10, 2011 10:22 Index Spencer, J., 427 Stanley, R., 440 Stanton, D and R., 440 Stirling number, 275 Stirling’s approximation for n!, 198 Street surveillance, Street sweeping, 52 Strongly connected graph, 45 Subdivided edge, 35 Subgraph, 15, 437 Subgroup, 362 Subset, 415 proper subset, 415 Subtree, 94, 438 Successor of a vertex, 388 Summation methods: using binomial identities, 230 using generating functions, 277 using recurrence relations, 305 Sylvester, J., 44, 351 Symmetry: of a geometric figure, 356 in a relation, 356 T Takeaway games, 385 Tetrahedron, symmetries of, 358 Tournaments, 44, 62, 73, 102, 297 Tower of Hanoi game, 286 Trail, 47, 49, 437 Transitive closure, 130 Transitivity in a relation, 356 Transportation problem, 164 Transportation tableau, 165 Transposition in a permutation, 363 Traveling salesperson problem, 113, 125, 431 Traversal of tree, 118 inorder, postorder, preorder, 118, 438 Tree, 93, 437 binary, 95, 438 m-ary, 95, 438 rooted, 93, 438 Tree sort, 125 Trial of experiment, 424 Triangulation of a graph, 78 Trigraph, 402 Trigraph frequency table, 402 Tripartite graph, 41 Tutte, W., 87 U Union of sets, 416 Unit-flow chain, 142 Unit-flow path, 139 Universal set, 415 V Value of a flow, 137 Vandermonde, A., 86 VanLint, J., 440 Venn diagram, 320, 417 Vertex, Vertex basis, 10 Vizing’s theorem, 80 Voter power, 197 W Wallis, W., 439 West, D., 44, 440 Wheel graph, 70 White, D., 440 Whitworth, W., 237 Wilson, R., 44, 86, 440 Winning position in a game, 386 Winning strategy in a game, 386 Winning vertex, 386 Woods, J., 440 Y Yellen, J., 440 www.itpub.net ... representative Library of Congress Cataloging-in-Publication Data Tucker, Alan, 1943 July 6Applied combinatorics / Alan Tucker — 6th ed p cm Includes bibliographical references and index ISBN 978-0-470-45838-9... reasoning is needed to solve applied problems Elsewhere, results are stated without proof, such as the form of solutions to various recurrence relations, and then applied in problem solving Occasionally,... ii P1: FCH/FYX Frontmatter P2: FCH/FYX WB00623-Tucker QC: FCH/UKS November 1, 2011 T1: FCH 12:27 APPLIED COMBINATORICS i P1: FCH/FYX Frontmatter P2: FCH/FYX WB00623-Tucker QC: FCH/UKS T1: FCH November