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Graph theory: a problem oriented approach

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1. Verfasser:	Marcus, Daniel A. (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Washington, DC Math. Assoc. of America 2008
Schriftenreihe:	MAA textbooks
Schlagworte:	Graphentheorie
Online-Zugang:	Inhaltsverzeichnis
Beschreibung:	XVI, 205 S. graph. Darst.
ISBN:	9780883857533

Internformat

MARC


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Datensatz im Suchindex

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adam_text	Contents Preface ix Introduction: Problems of Graph Theory . 1 Path Problems .................................. 1 Coloring Problems ................................ 2 Isomorphic Graphs ................................ 4 Planar Graphs ................................... 4 DisjointPaths ................................... 5 Shortest Paths ................................... 6 ... and More .................................... 7 A Basic Concepts 9 Equivalent Graphs ................................ 9 Multigraphs .................................... 10 Directed Graphs and Mixed Graphs ....................... 11 Complete Graphs ................................. 11 Cycle Graphs ................................... 11 Paths in a Graph ................................. 11 Open and Closed Paths; Cycles ......................... 12 Subgraphs ..................................... 13 The Complement of a Graph ........................... 13 Degrees of Vertices ................................ 13 The Degree Sequence of a Graph ......................... 14 Regular Graphs .................................. 15 Connected and Disconnected Graphs ...................... 15 Components of a Graph ............................. 15 More Problems .................................. 16 В Isomorphic Graphs 21 More Problems ................................... 23 xiii XIV Graph Theory С Bipartite Graphs 25 Complete Bipartite Graphs ............................ 26 Bipartite Graphs and Matrices .......................... 26 Cycles in a Bipartite Graph ............................ 27 Cycle Theorem for Bipartite Graphs ....................... 27 Proof of the Cycle Theorem ........................... 28 More Problems .................................. 29 D Trees and Forests 31 Pruning a Tree .................................. 32 Directed Trees .................................. 33 Spanning Trees .................................. 34 Counting Spanning Trees ............................. 34 Codewords for Trees: Prufer s Method ...................... 35 More Problems .................................. 36 E Spanning Tree Algorithms 41 Constructing Spanning Trees ........................... 41 Weighted Graphs ................................. 41 Minimal Spanning Trees ............................. 42 Prim s Algorithm ................................. 42 Tables for Prim s Algorithm ........................... 43 The Reduction Algorithm ............................ 43 Spanning Trees and Shortest Paths ........................ 44 Minimal Paths in a Weighted Graph ....................... 45 Minimal Path Algorithm, first attempt ..................... . 45 Minimal Path Algorithm, revised ......................... 46 Tables for Dijkstra s Algorithm ......................... 46 Minimal Paths in a Directed Graph ....................... 48 Negative Weights ................................. 48 More Problems .................................. 51 F Euler Paths 57 The Königsberg Bridge Problem ......................... 57 Euler Paths in Directed Graphs and Directed Multigraphs ............ 59 Application of Euler Paths: State diagrams, DeBruijn sequences, and rotating wheels ................. 60 More Problems .................................. 61 G Hamilton Paths and Cycles 65 Some Negative Tests ............................... 65 Positive Tests for Hamilton Cycles ........................ 67 Some Proofs ................................... 70 More Problems .................................. 73 H Planar Graphs 77 Regions Formed by a Plane Diagram ...................... 78 Proof that K5 is Non-Planar, Using Euler s Formula .............. 80 Contents xv Non-Planar Graphs and Kuratowski s Theorem ................. 81 More Problems . . . ,.............................. 83 I Independence and Covering 85 The Independence Numbers of a Graph ..................... 85 A Graph Game .................................. 88 Covering Sets and Covering Numbers ...................... 89 More Problems .................................. 91 J Connections and Obstructions 95 Internally Disjoint Paths ............................. 95 Edge-Disjoint Paths ............................... 95 Path Connection Numbers ............................ 96 Blocking Sets ................................... 96 ^-Connected Graphs ............................... 98 Vertex Cut Sets and Vertex Cut Numbers .................... 99 More Problems .................................. 100 К Vertex Coloring 103 The Vertex Coloring Number of a Graph .................... 103 Vertex Coloring Theorems ............................ 104 Map Coloring ................................... 110 More Problems .................................. Ill L Edge Coloring 119 The Edge Coloring Number of a Graph ..................... 119 Edge Coloring of Complete Graphs ....................... 120 Edge Coloring of Bipartite Graphs ........................ 122 Edge Color Switch ................................ 122 Proof of Edge Coloring Theorem #3....................... 123 Application of Edge Coloring: the Scheduling Problem ............. 124 More Problems .................................. 124 M Matching Theory for Bipartite Graphs 131 The Max/Min Principle .............................. 132 Proof of the König-Egervary Theorem ..................... 133 The Colored Digraph Construction ........................ 133 Matching Extension Algorithm ......................... 135 Proof of the Colored Digraph Theorem ..................... 135 Matrix Interpretation of the König-Egervary Theorem ............. 136 Hall s Theorem and Its Consequences ...................... 138 More Problems .................................. 139 N Applications of Matching Theory 143 Sets and Representatives ............................. 143 Latin Squares ................................... 144 Permutation Matrices ............................... 145 xvi Graph Theory The Optimal Assignment Problem ........................ 146 More Problems .................................. 149 О Cycle-Free Digraphs 153 Chains and Antichains .............................. 153 Chain Decompositions .............................. 154 Proof of Dilworth s Theorem ........................... 156 More Problems .................................. 158 Ρ Network Flow Theory 161 Flows in a Network ................................ 161 Cuts and Capacities ................................ 169 More Problems .................................. 172 Q Flow Problems with Lower Bounds 175 The Supply and Demand Problem ........................ 175 More Problems .................................. 184 Answers to Selected Problems 187 Index 201 About the Author 205
adam_txt	Contents Preface ix Introduction: Problems of Graph Theory . 1 Path Problems . 1 Coloring Problems . 2 Isomorphic Graphs . 4 Planar Graphs . 4 DisjointPaths . 5 Shortest Paths . 6 . and More . 7 A Basic Concepts 9 Equivalent Graphs . 9 Multigraphs . 10 Directed Graphs and Mixed Graphs . 11 Complete Graphs . 11 Cycle Graphs . 11 Paths in a Graph . 11 Open and Closed Paths; Cycles . 12 Subgraphs . 13 The Complement of a Graph . 13 Degrees of Vertices . 13 The Degree Sequence of a Graph . 14 Regular Graphs . 15 Connected and Disconnected Graphs . 15 Components of a Graph . 15 More Problems . 16 В Isomorphic Graphs 21 More Problems . 23 xiii XIV Graph Theory С Bipartite Graphs 25 Complete Bipartite Graphs . 26 Bipartite Graphs and Matrices . 26 Cycles in a Bipartite Graph . 27 Cycle Theorem for Bipartite Graphs . 27 Proof of the Cycle Theorem . 28 More Problems . 29 D Trees and Forests 31 Pruning a Tree . 32 Directed Trees . 33 Spanning Trees . 34 Counting Spanning Trees . 34 Codewords for Trees: Prufer's Method . 35 More Problems . 36 E Spanning Tree Algorithms 41 Constructing Spanning Trees . 41 Weighted Graphs . 41 Minimal Spanning Trees . 42 Prim's Algorithm . 42 Tables for Prim's Algorithm . 43 The Reduction Algorithm . 43 Spanning Trees and Shortest Paths . 44 Minimal Paths in a Weighted Graph . 45 Minimal Path Algorithm, first attempt . . 45 Minimal Path Algorithm, revised . 46 Tables for Dijkstra's Algorithm . 46 Minimal Paths in a Directed Graph . 48 Negative Weights . 48 More Problems . 51 F Euler Paths 57 The Königsberg Bridge Problem . 57 Euler Paths in Directed Graphs and Directed Multigraphs . 59 Application of Euler Paths: State diagrams, DeBruijn sequences, and rotating wheels . 60 More Problems . 61 G Hamilton Paths and Cycles 65 Some Negative Tests . 65 Positive Tests for Hamilton Cycles . 67 Some Proofs . 70 More Problems . 73 H Planar Graphs 77 Regions Formed by a Plane Diagram . 78 Proof that K5 is Non-Planar, Using Euler's Formula . 80 Contents xv Non-Planar Graphs and Kuratowski's Theorem . 81 More Problems . . . ,. 83 I Independence and Covering 85 The Independence Numbers of a Graph . 85 A Graph Game . 88 Covering Sets and Covering Numbers . 89 More Problems . 91 J Connections and Obstructions 95 Internally Disjoint Paths . 95 Edge-Disjoint Paths . 95 Path Connection Numbers . 96 Blocking Sets . 96 ^-Connected Graphs . 98 Vertex Cut Sets and Vertex Cut Numbers . 99 More Problems . 100 К Vertex Coloring 103 The Vertex Coloring Number of a Graph . 103 Vertex Coloring Theorems . 104 Map Coloring . 110 More Problems . Ill L Edge Coloring 119 The Edge Coloring Number of a Graph . 119 Edge Coloring of Complete Graphs . 120 Edge Coloring of Bipartite Graphs . 122 Edge Color Switch . 122 Proof of Edge Coloring Theorem #3. 123 Application of Edge Coloring: the Scheduling Problem . 124 More Problems . 124 M Matching Theory for Bipartite Graphs 131 The Max/Min Principle . 132 Proof of the König-Egervary Theorem . 133 The Colored Digraph Construction . 133 Matching Extension Algorithm . 135 Proof of the Colored Digraph Theorem . 135 Matrix Interpretation of the König-Egervary Theorem . 136 Hall's Theorem and Its Consequences . 138 More Problems . 139 N Applications of Matching Theory 143 Sets and Representatives . 143 Latin Squares . 144 Permutation Matrices . 145 xvi Graph Theory The Optimal Assignment Problem . 146 More Problems . 149 О Cycle-Free Digraphs 153 Chains and Antichains . 153 Chain Decompositions . 154 Proof of Dilworth's Theorem . 156 More Problems . 158 Ρ Network Flow Theory 161 Flows in a Network . 161 Cuts and Capacities . 169 More Problems . 172 Q Flow Problems with Lower Bounds 175 The Supply and Demand Problem . 175 More Problems . 184 Answers to Selected Problems 187 Index 201 About the Author 205
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title	Graph theory a problem oriented approach
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title_full	Graph theory a problem oriented approach Daniel A. Marcus
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