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This book supplements the textbook of the authors" Lectures on Graph The ory" [6] by more than thousand exercises of varying complexity. The books match each other in their contents, notations, and terminology. The authors hope that both students and lecturers will find this book helpful for mastering and verifying the understanding of the peculiarities of graphs. The exercises are grouped into eleven chapters and numerous sections accord ing to the topics of graph theory: paths, cycles, components, subgraphs, re constructibility, operations on graphs, graphs and matrices, trees, independence, matchings, coverings, connectivity, matroids, planarity, Eulerian and Hamiltonian graphs, degree sequences, colorings, digraphs, hypergraphs. Each section starts with main definitions and brief theoretical discussions. They constitute a minimal background, just a reminder, for solving the exercises. the presented facts and a more extended exposition may be found in Proofs of the mentioned textbook of the authors, as well as in many other books in graph theory. Most exercises are supplied with answers and hints. In many cases complete solutions are given. At the end of the book you may find the index of terms and the glossary of notations. The "Bibliography" list refers only to the books used by the authors during the preparation of the exercisebook. Clearly, it mentions only a fraction of available books in graph theory. The invention of the authors was also driven by numerous journal articles, which are impossible to list here.
Table des matières
1 ABC of Graph Theory.- 2 Trees.- 3 Independence and Coverings.- 4 Connectivity.- 5 Matroids.- 6 Planarity.- 7 Graph Traversals.- 8 Degree Sequences.- 9 Graph Colorings.- 10 Directed Graphs.- 11 Hypergraphs.- Answers to Chapter 1: ABC of Graph Theory.- 1.1 Graphs: Basic Notions.- 1.2 Walks, Paths, Components.- 1.3 Subgraphs and Hereditary Properties of Graphs. Reconstructibility.- 1.4 Operations on Graphs.- 1.5 Matrices Associated with Graphs.- 1.6 Automorphism Group of Graph.- Answers to Chapter 2: Trees.- 2.1 Trees: Basic Notions.- 2.2 Skeletons and Spanning Trees.- Answers to Chapter 3: Independence and Coverings.- 3.1 Independent Vertex Sets and Cliques.- 3.2 Coverings.- 3.3 Dominating Sets.- 3.4 Matchings.- 3.5 Matchings in Bipartite Graphs.- Answers to Chapter 4: Connectivity.- 4.1 Biconnected Graphs and Biconnected Components.- 4.3 Cycles and Cuts.- Answers to Chapter 5: Matroids.- 5.1 Independence Systems.- 5.2 Matroids.- 5.3 Binary Matroids.- Answers to Chapter 6: Planarity.-6.1 Embeddings of Graphs. Euler Formula.- 6.2 Plane Triangulation.- 6.3 Planarity Criteria.- 6.4 Duality and Planarity.- 6.5 Measures of Displanarity.- Answers to Chapter 7: Graph Traversals.- 7.1 Eulerian Graphs.- 7.2 Hamiltonian Graphs.- Answers to Chapter 8: Degree Sequences.- 8.1 Graphical Sequences.- 8.3 Split and Threshold Graphs.- 8.4 Degree Sets and Arity Partitions.- Answers to Chapter 9: Graph Colorings.- 9.1 Vertex Coloring.- 9.2 Chromatic Polynomial.- 9.3 Edge Coloring.- 9.4 Colorings of Planar Graphs.- 9.5 Perfect Graphs.- Answers to Chapter 10: Directed Graphs.- 10.1 Directed Graphs: Basic Notions.- 10.2 Reachability and Components.- 10.3 Matrices Associated with Digraph.- 10.4 Tours and Paths.- 10.5 Tournaments.- 10.6 Base and Kernel.- Answers to Chapter 11: Hypergraphs.- 11.1 Hypergraphs: Basic Notions.- 11.2 Hypergraph Realizations.- Notations.
