Introduction to Graph Theory

1. Fundamental Concepts. Definitions and examples. Paths and proofs. Vertex degrees and counting. Degrees and algorithmic proof. 2. Trees and Distance. Basic properties. Spanning trees and enumeration. Optimization and trees. Eulerian graphs and digraphs. 3. Matchings and Factors. Matchings in bipartite graphs. Applications and algorithms. Matchings in general graphs. 4. Connectivity and Paths. Cuts and connectivity. k-connected graphs. Network flow problems. 5. Graph Coloring. Vertex colorings and upper bounds. Structure of k-chromatic graphs. Enumerative aspects. 6. Edges and Cycles. Line graphs and edge-coloring. Hamiltonian cycles. Complexity. 7. Planar Graphs. Embeddings and Eulers formula.…