Topics in Graph Theory

Main subject categories: • Graph theory • CombinatoricsThe interplay continues to grow between graph theory and a wide variety of models and applications in mathematics, computer science, operations research, and the natural and social sciences.Topics in Graph Theory is geared toward the more mathematically mature student. The first three chapters provide the basic definitions and theorems of graph theory and the remaining chapters introduce a variety of topics and directions for research. These topics draw on numerous areas of theoretical and applied mathematics, including combinatorics, probability, linear algebra, group theory, topology, operations research, and computer science. This makes the book appropriate for a first course at the graduate level or as a second course at the undergraduate level.The authors build upon material previously published in Graph Theory and Its Applications, Third Edition, by the same authors. That text covers material for both an undergraduate and graduate course, while this book builds on and expands the graduate-level material.Features• Extensive exercises and applications.• Flexibility: appropriate for either a first course at the graduate level or an advanced course at the undergraduate level.• Opens avenues to a variety of research areas in graph theory.• Emphasis on topological and algebraic graph theory. Cover Half Title Series Page Title Page Copyright Page CONTENTS PREFACE 1. FOUNDATIONS 1.1. Basic Definitions and Terminology 1.2. Walks and Connectivity 1.3. Subgraphs 1.4. Graph Operations 1.5. Directed Graphs 1.6. Formal Specifications for Graphs and Digraphs Glossary 2. ISOMORPHISMS AND SYMMETRY 2.1. Graph Homomorphisms and Isomorphisms 2.2. Automorphisms and Symmetry 2.3. Tests for Non-Isomorphism Glossary 3. TREES AND CONNECTIVITY 3.1. Characterizations and Properties of Trees 3.2. Cycles, Edge-Cuts, and Spanning Trees 3.3. Graphs and Vector Spaces 3.4. Vertex- and Edge-Connectivity 3.5. Max-Min Duality and Menger's Theorems 3.6. Block Decompositions Glossary 4. PLANARITY AND KURATOWSKI'S THEOREM 4.1. Planar Drawings and Some Basic Surfaces 4.2. Subdivision and Homeomorphism 4.3. Extending Planar Drawings 4.4. Kuratowski's Theorem 4.5. Algebraic Tests for Planarity 4.6. Planarity Algorithm 4.7. Crossing Numbers and Thickness Glossary 5. DRAWING GRAPHS AND MAPS 5.1. Topology of Low Dimensions 5.2. Higher-Order Surfaces 5.3. Mathematical Model for Drawing Graphs 5.4. Regular Maps on a Sphere 5.5. Imbeddings on Higher-Order Surfaces 5.6. Geometric Drawings of Graphs Glossary 6. GRAPH COLORINGS 6.1. Vertex-Colorings 6.2. Local Recolorings 6.3. Map-Colorings 6.4. Edge-Colorings 6.5. Factorization Glossary 7. MEASUREMENT AND MAPPINGS 7.1. Distance in Graphs 7.2. Domination in Graphs 7.3. Bandwidth 7.4. Intersection Graphs 7.5. Graph Homomorphisms 7.6. Modeling Network Emulation Glossary 8. ANALYTIC GRAPH THEORY 8.1. Ramsey Theory 8.2. Extremal Graph Theory 8.3. Random Graphs Glossary 9. GRAPH COLORINGS AND SYMMETRY 9.1. Automorphisms of Simple Graphs 9.2. Equivalence Classes of Colorings 9.3. Burnside's Lemma 9.4. Cycle-Index Polynomial of a Permutation Group 9.5. More Counting, Including Simple Graphs 9.6. Polya-Burnside Enumeration Glossary 10. ALGEBRAIC SPECIFICATION OF GRAPHS 10.1. Cyclic Voltages 10.2. Specifying Connected Graphs 10.3. Zn-Voltage Graphs and Graph Colorings 10.4. General Voltage Graphs 10.5. Permutation Voltages 10.6. Symmetric Graphs and Parallel Architectures Glossary 11. NON-PLANAR LAYOUTS 11.1. Representing Imbeddings by Rotations 11.2. Genus Distribution of a Graph 11.3. Voltage-Graph Specification of Graph Layouts 11.4. Non-KVL-Imbedded Voltage Graphs 11.5. Heawood Map-Coloring Problem Glossary A: APPENDIX A.1. Logic Fundamentals A.2. Relations and Functions A.3. Some Basic Combinatorics A.4. Algebraic Structures A.5. Algorithmic Complexity A.6. Supplementary Reading BIBLIOGRAPHY A.1. General Reading A.2. References INDEX The interplay continues to grow between graph theory and a wide variety of models and applications in mathematics, computer science, operations research, and the natural and social sciences. Topics in Graph Theory is geared toward the more mathematically mature student. The first three chapters provide the basic definitions and theorems of graph theory and the remaining chapters introduce a variety of topics and directions for research. These topics draw on numerous areas of theoretical and applied mathematics, including combinatorics, probability, linear algebra, group theory, topology, operations research, and computer science. This makes the book appropriate for a first course at the graduate level or as a second course at the undergraduate level. The authors build upon material previously published in Graph Theory and Its Applications, Third Edition, by the same authors. That text covers material for both an undergraduate and graduate course, while this book builds on and expands the graduate-level material. Features Extensive exercises and applications. Flexibility: appropriate for either a first course at the graduate level or an advanced course at the undergraduate level. Opens avenues to a variety of research areas in graph theory. Emphasis on topological and algebraic graph theory.