I Fundamentals. - 1. Definitions. - 2. Paths Cycles and Trees. - 3. Hamilton Cycles and Euler Circuits. - 4. Planar Graphs. - 5. An Application of Euler Trails to Algebra. - Exercises. - Notes. - II Electrical Networks. - 1. Graphs and Electrical Networks. - 2. Squaring the Square. - 3. Vector Spaces and Matrices Associated with Graphs. - Exercises. - Notes. - III Flows Connectivity and Matching. - 1. Flows in Directed Graphs. - 2. Connectivity and Menger's Theorem. - 3. Matching. - 4. Tutte's 1-Factor Theorem. - Exercises. - Notes. - IV Extremal Problems. - 1. Paths and Cycles. - 2. Complete Subgraphs. - 3. Hamilton Paths and Cycles. - 4. The Structure of Graphs. - Exercises. - Notes. - V Colouring. - 1. Vertex Colouring. - 2. Edge Colouring. - 3. Graphs on Surfaces. - Exercises. - Notes. - VI Ramsey Theory. - 1. The Fundamental Ramsey Theorems. - 2. Monochromatic Subgraphs. - 3. Ramsey Theorems in Algebra and Geometry. - 4. Subsequences. - Exercises. - Notes. - VII Random Graphs. - 1. Complete Subgraphs and Ramsey NumbersThe Use of the Expectation. - 2. Girth and Chromatic NumberAltering a Random Graph. - 3. Simple Properties of Almost All GraphsThe Basic Use of Probability. - 4. Almost Determined VariablesThe Use of the Variance. - 5. Hamilton CyclesThe Use of Graph Theoretic Tools. - Exercises. - Notes. - VIII Graphs and Groups. - 1. Cayley and Schreier Diagrams. - 2. Applications of the Adjacency Matrix. - 3. Enumeration and Pólya's Theorem. - Exercises. - Notes. - Index of Symbols. Language: English