Objective:
MCS 423 is intended as a rigorous course that challenges students to think. Homework and tests require proofs. The material is interesting, accessible
and applicable; most students who stick with the course will give a fair amount
of time and thoughts.
Content:
Basic concepts of graph theory including Eulerian and hamiltonian cycles, trees, colorings, connectivity, shortest paths, minimum spanning trees, network flows,
bipartite matching, planar graphs.
Plan: Cover most sections in chapters 1-6, some sections in chapters 7-12. A more detailed schedule will be announced later. Note: You should read Appendix if you have forgotten basic knowledge in discrete mathematics. In fact, we covered (lecture #, content, ? means plan)1. introduction, 1.1 graph basics
2. 1.1 degree sequences, 1.2 special graphs
3. 1.4 walk, path, tour.
4. 1.5 bipartite graph theorem, Eulerian tour Thereom (Theorem 4.6.11)
5. 2.1 subgraphs, 2.2 isomorphism
6. 2.3-2.5 isomorphism test, reconstruction conjecture, matrix representations
7. 3.1 tree properties (omitted Proposition 3.1.9 and 3.1.10)
8. 3.1 and 4.4 Prüfer code and Cayley formula
9. 3.1 Theorem 3.1.12 and 4.5 minimal spanning tree (Prim and Kruskal)
10. 4.5 shortest path algorithm (Dijkstra), homework hints
11. 5.1 vertex and edge connectivity
12. 5.1 Whitney’s Theorem, 5.2 Harary graph,
13. 5.2 Whitney’s Synthesis, Harary graphs
14. 5.4 blocks, 6.1 directed Eulerian tour theorem
15. 6.2 De Brujin sequence
16. Review