Graphs and Applications An Introductory Approach

Discrete Mathematics is one of the fastest growing areas in mathematics today with an ever-increasing number of courses in schools and universities. Graphs and Applications is based on a highly successful Open University course and the authors have paid particular attention to the presentation, clarity and arrangement of the material, making it ideally suited for independent study and classroom use. An important part of learning graph theory is problem solving; for this reason large numbers of examples, problems (with full solutions) and exercises (without solutions) are included.

1 Introduction.- 1.1 Graphs, Digraphs and Networks.- 1.2 Classifying Problems.- 1.3 Seeking Solutions.- 2 Graphs.- 2.1 Graphs and Subgraphs.- 2.2 Vertex Degrees.- 2.3 Paths and Cycles.- 2.4 Regular and Bipartite Graphs.- 2.5 Case Studies.- Exercises 2.- 3 Eulerian and Hamiltonian Graphs.- 3.1 Exploring and Travelling.- 3.2 Eulerian Graphs.- 3.3 Hamiltonian Graphs.- 3.4 Case Studies.- Exercises 3.- 4 Digraphs.- 4.1 Digraphs and Subdigraphs.- 4.2 Vertex Degrees.- 4.3 Paths and Cycles.- 4.4 Eulerian and Hamiltonian Digraphs.- 4.5 Case Studies.- Exercises 4.- 5 Matrix Representations.- 5.1 Adjacency Matrices.- 5.2 Walks in Graphs and Digraphs.- 5.3 Incidence Matrices.- 5.4 Case Studies.- Exercises 5.- 6 Tree Structures.- 6.1 Mathematical Properties of Trees.- 6.2 Spanning Trees.- 6.3 Rooted Trees.- 6.4 Case Study.- Exercises 6.- 7 Counting Trees.- 7.1 Counting Labelled Trees.- 7.2 Counting Binary Trees.- 7.3 Counting Chemical Trees.- Exercises 7.- 8 Greedy Algorithms.- 8.1 Minimum Connector Problem.- 8.2 Travelling Salesman Problem.- Exercises 8.- 9 Path Algorithms.- 9.1 Fleury’s Algorithm.- 9.2 Shortest Path Algorithm.- 9.3 Case Study.- Exercises 9.- 10 Paths and Connectivity.- 10.1 Connected Graphs and Digraphs.- 10.2 Menger’s Theorem for Graphs.- 10.3 Some Analogues of Menger’s Theorem.- 10.4 Case Study.- Exercises 10.- 11 Planarity.- 11.1 Planar Graphs.- 11.2 Euler’s Formula.- 11.3 Cycle Method for Planarity Testing.- 11.4 Kuratowski’s Theorem.- 11.5 Duality.- 11.6 Convex Polyhedra.- Exercises 11.- 12 Vertex Colourings and Decompositions.- 12.1 Vertex Colourings.- 12.2 Algorithm for Vertex Colouring.- 12.3 Vertex Decompositions.- Exercises 12.- 13 Edge Colourings and Decompositions.- 13.1 Edge Colourings.- 13.2 Algorithm for Edge Colouring.- 13.3 EdgeDecompositions.- Exercises 13.- 14 Conclusion.- 14.1 Classification of Problems.- 14.2 Efficiency of Algorithms.- 14.3 Another Classification of Problems.- Suggestions for Further Reading.- Appendix: Methods of Proof.- Computing Notes.- Solutions to Computer Activities.- Solutions to Problems in the Text.