Graph Theory and Interconnection Networks

Transcription

1 Graph Theory and Interconnection Networks Lih-Hsing Hsu and Cheng-Kuan CRC Press Taylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business

2 Contents Preface Authors xi xiii Chapter 1: Fundamental Concepts Graphs and Simple Graphs Matrices and Isomorphisms Paths and Cycles Vertex Degrees Graph Operations Some Basic Techniques Degree Sequences 18 Chapter 2: Applications on Graph Isomorphisms Generalized Honeycomb Tori Isomorphism between Cyclic-Cubes and Wrapped Butterfly Networks Edge Fault-Tolerant Design for Meshes Faithful 1-Edge Fault-Tolerant Graphs Ä-Edge Fault-Tolerant Designs for Hypercubes 40 Chapter 3: Distance and Diameter Introduction Diameter for Some Interconnection Networks Shuffle-Cubes Routel(u, v) Moore Bound Star Graphs and Pancake Graphs Edge Congestion and Bisection Width Transmitting Problem 57 Chapter 4: Trees Basic Properties Breadth-First Search and Depth-First Search Rooted Trees Counting Trees Counting Binary Trees Number of Spanning Trees Contains a Certain Edge Embedding Problem 76 v

3 vi Contents Chapter 5: Eulerian Graphs and Digraphs Eulerian Graphs Eulerian Digraphs Applications Chinese Postman Problem Street-Sweeping Problem Drum Design Problem Functional Cell Layout Problem 87 Chapter 6: Matchings and Factors Matchings Bipartite Matching Edge Cover Perfect Matching Factors 103 Chapter 7: Connectivity Cut and Connectivity Connected Graphs Menger Theorem Ill 7.4 An Application Making a Road System One-Way Connectivity of Some Interconnection Networks Wide Diameters and Fault Diameters Superconnectivity and Super-Edge-Connectivity 120 Chapter 8: Graph Coloring Vertex Colorings and Bounds Properties of ^-Critical Graphs Bound for Chromatic Numbers Girth and Chromatic Number Hajos' Conjecture Enumerative Aspects Homomorphism Functions An Application Testing on Printed Circuit Boards Edge-Colorings 138 Chapter 9: Hamiltonian Cycles Hamiltonian Graphs Necessary Conditions Sufficient Conditions Hamiltonian-Connected Mutually Independent Hamiltonian Paths Diameter for Generalized Shuffle-Cubes Cycles in Directed Graphs 158

4 Contents vii Chapter 10: Planar Graphs Planar Embeddings Euler's Formula Characterization of Planar Graphs Coloring of Planar Graphs 167 Chapter 11: Optimal &-Fault-Tolerant Hamiltonian Graphs Introduction Node Expansion Other Construction Methods Fault-Tolerant Hamiltonicity and Fault-Tolerant Hamiltonian Connectivity of the Folded Petersen Cube Networks Fault-Tolerant Hamiltonicity and Fault-Tolerant Hamiltonian Connectivity of the Pancake Graphs Fault-Tolerant Hamiltonicity and Fault-Tolerant Hamiltonian Connectivity of Augmented Cubes Fault-Tolerant Hamiltonicity and Fault-Tolerant Hamiltonian Connectivity of the WK-Recursive Networks Fault-Tolerant Hamiltonicity of the Fully Connected Cubic Networks 221 Chapter 12: Optimal 1-Fault-Tolerant Hamiltonian Graphs Introduction Join Cycle Extension Cells for Optimal 1 -Hamiltonian Regular Graphs Generalized Petersen Graphs Honeycomb Rectangular Disks Properties with Respect to the 3-Join Examples of Various Cubic Planar Hamiltonian Graphs А П В П С АП В ПС Л П В П С АПВПС АПВПС АПВПС А П ПС А~П~ВП С Hamiltonian Properties of Double Loop Networks 275 Chapter 13: Optimal &-Fault-Tolerant Hamiltonian-Laceable Graphs Introduction Super Fault-Tolerant Hamiltonian Laceability of Hypercubes Super Fault-Tolerant Hamiltonian Laceability of Star Graphs 289

5 viii Contents 13.4 Construction Schemes Cubic Hamiltonian-Laceable Graphs lp-fault-tolerant Hamiltonian Graphs Spider-Web Networks Brother Trees Hamiltonian Laceability of Faulty Hypercubes Conditional Fault Hamiltonicity and Conditional Fault Hamiltonian Laceability of the Star Graphs 327 Chapter 14: Spanning Connectivity Introduction Spanning Connectivity of General Graphs Spanning Connectivity and Spanning Laceability of the Hypercube-Like Networks Spanning Connectivity of Crossed Cubes Spanning Connectivity and Spanning Laceability of the Enhanced Hypercube Networks Spanning Connectivity of the Pancake Graphs Spanning Laceability of the Star Graphs Spanning Fan-Connectivity and Spanning Pipe-Connectivity of Graphs 410 Chapter 15: Cubic 3*-Connected Graphs and Cubic 3*-Laceable Graphs Properties of Cubic 3*-Connected Graphs Examples of Cubic Super 3*-Connected Graphs Counterexamples of 3*-Connected Graphs Properties of Cubic 3*-Laceable Graphs Examples of Cubic Hyper 3*-Laceable Graphs Counterexamples of 3*-Laceable Graphs 447 Chapter 16: Spanning Diameter Introduction Spanning Diameter for the Star Graphs Spanning Diameter of Hypercubes Spanning Diameter for Some (n, )-Star Graphs 474 Chapter 17: Pancyclic and Panconnected Property Introduction Bipanconnected and Bipancyclic Properties of Hypercubes Edge Fault-Tolerant Bipancyclic Properties of Hypercubes Panconnected and Pancyclic Properties of Augmented Cubes Comparison between Panconnected and Panpositionable Hamiltonian Bipanpositionable Bipancyclic Property of Hypercube 504

6 Contents ix Chapter 18: Mutually Independent Hamiltonian Cycles Introduction Mutually Independent Hamiltonian Cycles on Some Graphs Mutually Independent Hamiltonian Cycles of Hypercubes Mutually Independent Hamiltonian Cycles of Pancake Graphs Mutually Independent Hamiltonian Cycles of Star Graphs Fault-Free Mutually Independent Hamiltonian Cycles in a Faulty Hypercube Fault-Free Mutually Independent Hamiltonian Cycles in Faulty Star Graphs Orthogonality for Sets of Mutually Independent Hamiltonian Cycles 543 Chapter 19: Mutually Independent Hamiltonian Paths Introduction Mutually Independent Hamiltonian Laceability for Hypercubes Mutually Independent Hamiltonian Laceability for Star Graphs Mutually Independent Hamiltonian Connectivity for (n, &)-Star Graphs Cubic 2-Independent Hamiltonian-Connected Graphs Examples of Cubic 2-Independent Hamiltonian-Connected Graphs That Are Super 3*-Connected Examples of Super 3*-Connected Graphs That Are Not Cubic 2-Independent Hamiltonian-Connected Example of a Cubic 2-Independent Hamiltonian- Connected Graph That Is Not Super 3*-Connected Example of a Cubic 1-Fault-Tolerant Hamiltonian Graph That Is Hamiltonian-Connected but Not Cubic 2-Independent Hamiltonian-Connected or Super 3*-Connected 581 Chapter 20: Topological Properties of Butterfly Graphs Introduction Cycle Embedding in Faulty Butterfly Graphs Spanning Connectivity for Butterfly Graphs Mutually Independent Hamiltonicity for Butterfly Graphs 616 Chapter 21: Diagnosis of Multiprocessor Systems Introduction Diagnosis Models PMC Model Comparison Model Diagnosability of the Matching Composition Networks Diagnosability of the Matching Composition Networks under the PMC Model 632

7 x Contents Diagnosability of the Matching Composition Networks under the Comparison Model Diagnosability of Cartesian Product Networks, Diagnosability of Cartesian Product Networks under the PMC Model Diagnosability of Cartesian Product Networks under the Comparison Model Diagnosability of f-connected Networks Diagnosability of Homogeneous Product Networks Diagnosability of Heterogeneous Product Networks Strongly?-Diagnosable Systems Strongly г-diagnosable Systems in the Matching Composition Networks Conditional Diagnosability Conditional Diagnosability of Q n under the PMC Model Conditional Diagnosability of Q under the Comparison Model Conditional Diagnosability of Cayley Graphs Generated by Transposition Trees under the Comparison Diagnosis Model Cayley Graphs Generated by Transposition Trees Conditional Diagnosability Local Diagnosability Strongly Local-Diagnosable Property Conditional Fault Local Diagnosability 679 References 687 Index 703