Discrete Mathematics (離散數學)
Content for the PhD qualifying examination on Discrete Mathematics Department of Mathematics, National Taiwan University (2007-06-11)
Reference books:
1. P. J. Cameron, Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press, 1994.
2. D. B. West, Introduction to Graph Theory, Second Edition, Prentice-Hall, 2001.
Contents:
1. Recurrence relations and generating functions.
2. The principle of inclusion and exclusion.
3. Design---Latin squares, Steiner triple systems, finite geometies, block designs.
4. Extremal sets---intersecting families, Sperner families, de Drujin-Erdos theorem.
5. Ramsey theory---pigeonhole principle, Ramsey’s theorem, bounds, applications.
6. Posets, lattices, matroids---Dilworth’s theorem, Mobius function, rank function.
7. Trees and distance---distance in graphs, spanning trees, minimum spanning trees.
8. Matchings---Hall’s theorem, Tutte’s 1-factor theorem, matching algorithms.
9. Connectivity---edge-connectivity, blocks, Menger’s theorems, network flows.
10. Colorings---Brooks’ theorem, Turan’s theorem, chordal graphs, orientations.
11. Planar graphs---dual graphs, Euler’s formula, Kuratowski’s theorem, crossing numbers, coloring and Hamiltonicity in planar graphs.
12. Special topics---perfect graphs, matroids, Ramsey theory, extremal graph theory, Random graphs, eigenvalues of graphs.
