Mathematics DISCRETE MATHEMATICS AND ITS APPLICATIONS DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN Series Editor KENNETH H. ROSEN D Discrete Discrete Structures i s c Interactions r and Their e t Structures e S Discover the Connections between Different Structures and Fields t r Discrete Structures and Their Interactions highlights the connec- u tions among various discrete structures, including graphs, directed c and Their t graphs, hypergraphs, partial orders, finite topologies, and simplicial u complexes. It also explores their relationships to classical areas of r e mathematics, such as linear and multilinear algebra, analysis, prob- Interactions s ability, logic, and topology. a The text introduces a number of discrete structures, such as n d hypergraphs, finite topologies, preorders, simplicial complexes, T and order ideals of monomials, that most graduate students in h e combinatorics, and even some researchers in the field, seldom i r experience. The author explains how these structures have important I Jason I. Brown n applications in many areas inside and outside of combinatorics. He t e also discusses how to recognize valuable research connections r through the structures. a c t Intended for graduate and upper-level undergraduate students in i o mathematics who have taken an initial course in discrete mathematics n or graph theory, this book shows how discrete structures offer new s insights into the classical fields of mathematics. It illustrates how to use discrete structures to represent the salient features and discover the underlying combinatorial principles of seemingly unrelated areas B r of mathematics. o w n K16844 K16844_Cover.indd 1 5/21/13 12:55 PM Discrete Structures and Their Interactions © 2013 by Taylor & Francis Group, LLC DISCRETE MATHEMATICS ITS APPLICATIONS Series Editor Kenneth H. Rosen, Ph.D. R. B. J. T. Allenby and Alan Slomson, How to Count: An Introduction to Combinatorics, Third Edition Craig P. Bauer, Secret History: The Story of Cryptology Juergen Bierbrauer, Introduction to Coding Theory Katalin Bimbó, Combinatory Logic: Pure, Applied and Typed Donald Bindner and Martin Erickson, A Student’s Guide to the Study, Practice, and Tools of Modern Mathematics Francine Blanchet-Sadri, Algorithmic Combinatorics on Partial Words Miklós Bóna, Combinatorics of Permutations, Second Edition Jason I. Brown, Discrete Structures and Their Interactions Richard A. Brualdi and Drago˘s Cvetkovi´c, A Combinatorial Approach to Matrix Theory and Its Applications Kun-Mao Chao and Bang Ye Wu, Spanning Trees and Optimization Problems Charalambos A. Charalambides, Enumerative Combinatorics Gary Chartrand and Ping Zhang, Chromatic Graph Theory Henri Cohen, Gerhard Frey, et al., Handbook of Elliptic and Hyperelliptic Curve Cryptography Charles J. Colbourn and Jeffrey H. Dinitz, Handbook of Combinatorial Designs, Second Edition Abhijit Das, Computational Number Theory Martin Erickson, Pearls of Discrete Mathematics Martin Erickson and Anthony Vazzana, Introduction to Number Theory Steven Furino, Ying Miao, and Jianxing Yin, Frames and Resolvable Designs: Uses, Constructions, and Existence Mark S. Gockenbach, Finite-Dimensional Linear Algebra Randy Goldberg and Lance Riek, A Practical Handbook of Speech Coders © 2013 by Taylor & Francis Group, LLC Titles (continued) Jacob E. Goodman and Joseph O’Rourke, Handbook of Discrete and Computational Geometry, Second Edition Jonathan L. Gross, Combinatorial Methods with Computer Applications Jonathan L. Gross and Jay Yellen, Graph Theory and Its Applications, Second Edition Jonathan L. Gross and Jay Yellen, Handbook of Graph Theory David S. 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Stitzinger, Applications of Abstract Algebra with Maple™ and MATLAB®, Second Edition Richard E. Klima and Neil P. Sigmon, Cryptology: Classical and Modern with Maplets Patrick Knupp and Kambiz Salari, Verification of Computer Codes in Computational Science and Engineering William Kocay and Donald L. Kreher, Graphs, Algorithms, and Optimization Donald L. Kreher and Douglas R. Stinson, Combinatorial Algorithms: Generation Enumeration and Search Hang T. Lau, A Java Library of Graph Algorithms and Optimization C. C. Lindner and C. A. Rodger, Design Theory, Second Edition San Ling, Huaxiong Wang, and Chaoping Xing, Algebraic Curves in Cryptography Nicholas A. Loehr, Bijective Combinatorics Toufik Mansour, Combinatorics of Set Partitions Alasdair McAndrew, Introduction to Cryptography with Open-Source Software Elliott Mendelson, Introduction to Mathematical Logic, Fifth Edition Alfred J. Menezes, Paul C. van Oorschot, and Scott A. 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Rosen, Handbook of Discrete and Combinatorial Mathematics Douglas R. Shier and K.T. Wallenius, Applied Mathematical Modeling: A Multidisciplinary Approach Alexander Stanoyevitch, Introduction to Cryptography with Mathematical Foundations and Computer Implementations Jörn Steuding, Diophantine Analysis Douglas R. Stinson, Cryptography: Theory and Practice, Third Edition Roberto Togneri and Christopher J. deSilva, Fundamentals of Information Theory and Coding Design W. D. Wallis, Introduction to Combinatorial Designs, Second Edition W. D. Wallis and J. C. George, Introduction to Combinatorics Jiacun Wang, Handbook of Finite State Based Models and Applications Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography, Second Edition © 2013 by Taylor & Francis Group, LLC DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN Discrete Structures and Their Interactions Jason I. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com © 2013 by Taylor & Francis Group, LLC Contents List of Figures xi Preface xv About the Author xvii 1 Introduction 1 1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Computational Complexity . . . . . . . . . . . . . . . . . . . 3 2 Discrete Structures - A Common Framework 5 2.1 Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Substructures . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Properties, Parametersand Operations . . . . . . . . . . . . 11 2.4 Representations and Models . . . . . . . . . . . . . . . . . . 12 2.4.1 Geometric Models . . . . . . . . . . . . . . . . . . . . 13 2.4.2 Algebraic Models . . . . . . . . . . . . . . . . . . . . . 13 2.4.3 Logical Models . . . . . . . . . . . . . . . . . . . . . . 16 2.4.4 Probabilistic Models . . . . . . . . . . . . . . . . . . . 17 3 Graphs and Directed Graphs 23 3.1 Graphs and Directed Graphs as Models . . . . . . . . . . . . 29 3.1.1 Graph Colourings . . . . . . . . . . . . . . . . . . . . 29 3.1.2 Reliability . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1.3 Proofs in Matrix Theory . . . . . . . . . . . . . . . . . 34 3.2 Graphs and Other Branches of Mathematics . . . . . . . . . 38 3.2.1 Graphs and Topology . . . . . . . . . . . . . . . . . . 38 3.2.2 Graphs and Algebra . . . . . . . . . . . . . . . . . . . 40 vii © 2013 by Taylor & Francis Group, LLC viii Contents 3.2.3 Graphs and Analysis . . . . . . . . . . . . . . . . . . . 45 3.2.4 Graphs and Logic . . . . . . . . . . . . . . . . . . . . 53 3.2.5 Graphs and Probability . . . . . . . . . . . . . . . . . 55 4 Preorders and Partial Orders 65 4.1 Finite Topologies and Preorders . . . . . . . . . . . . . . . . 69 4.1.1 The Correspondence . . . . . . . . . . . . . . . . . . . 69 4.1.2 Open Sets . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.1.3 The Lattice of All Topologies . . . . . . . . . . . . . . 72 4.1.4 Algorithmic Considerations . . . . . . . . . . . . . . . 72 4.2 Representing Preorders and Partial Orders . . . . . . . . . . 77 4.2.1 Random Preorders and Partial Orders . . . . . . . . . 77 4.2.2 Graphs for Preorders . . . . . . . . . . . . . . . . . . . 80 5 Hypergraphs 89 5.1 Applying Hypergraphs . . . . . . . . . . . . . . . . . . . . . 89 5.1.1 Hypergraphs and Graph Colourings . . . . . . . . . . 89 5.1.2 Hypergraphs and Generalized Ramsey Theory . . . . 93 5.1.3 Designs and Graphs . . . . . . . . . . . . . . . . . . . 94 5.1.4 Hypergraphs and Dimension of Partial Orders . . . . 99 5.2 Modeling Hypergraphs . . . . . . . . . . . . . . . . . . . . . 102 5.2.1 Criticality and Matrix Rank . . . . . . . . . . . . . . . 102 5.2.2 Criticality and Multilinear Algebra . . . . . . . . . . . 104 5.2.3 Finite Geometries and Orthogonality . . . . . . . . . . 106 5.2.4 Designs from Codes . . . . . . . . . . . . . . . . . . . 109 6 Complexes and Multicomplexes 117 6.1 Representations of Complexes and Multicomplexes . . . . . . 126 6.1.1 Topological Realizations of Complexes . . . . . . . . . 126 6.1.2 Connections to Commutative Algebra . . . . . . . . . 130 6.2 Applications of Complexes and Multicomplexes . . . . . . . 137 6.2.1 A “Complex” View of Partial Orders . . . . . . . . . . 137 6.2.2 Order Ideals of Monomials and Graph Colourings. . . 139 7 Research Problems 163 Selected Solutions 167 © 2013 by Taylor & Francis Group, LLC Contents ix AppendixA Set Theory 171 AppendixB Matrix Theory and Linear Algebra 173 AppendixC Abstract Algebra 175 AppendixD Probability 177 AppendixE Topology 179 AppendixF Logic 181 Bibliography 185 Index 195 © 2013 by Taylor & Francis Group, LLC