Math 412
Introduction to Graph Theory
Sections F13 and F14

Instructor: Alexandr Kostochka
Office: 234 Illini Hall
Phone: (217) 265-8037 (office)
Fax: (217) 333-9576
E-mail: kostochk@math.uiuc.edu
Time and place: 2pm - 2:50 pm MWF, 245 Altgeld Hall
Final exam: 7:00-10:00 p.m., Friday, December 14.
Office hours: 3-4 pm MWF and by appointment



Class Announcements





  • We will have the make-up Quiz 5 on Monday, December 10. Topics for Quiz 5 are: 1. Matchings in bipartite graphs. Hall's Theorem. 2. Matchings and covers. Konig-Egervary Theorem. 3. Matchings in general graphs. Tutte's Theorem. 4. Berge-Tutte Formula and theorems of Petersen. 5. Matrix Tree Theorem. 6. A characterization of 2-connected graphs. 7. Expansion Lemma, Fan Lemma. 8. Menger's Theorems.(!) 9. Flows in networks. Decomposition of every flow into flows along cycles and s,t-paths. 10. Ford-Fulkerson Algorithm. Max Flow--Min Cut Theorem. 11. Integral flows (matchings in bipartite graphs, edge connectivity). 12. Planar and plane graphs. Dual graphs. 13. Euler's Formula and its corollaries. 14. Outerplanar graphs, maximal planar graphs. 15. Kuratowski's Theorem and Wagner's Theorem. 16. Colorings. Greedy algorithm for coloring. A bad example for a tree. 17. Color-critical graphs and their properties ((k-1)-edge-connected etc.). 18. Brooks' Theorem. 19. Turan's Theorem.


  • Final Exam Topics: 1. Bipartite graphs. A characterization of bipartite graphs. 2. Isomorphism of graphs. 3. Representations of graphs: adjacency and incidence matrices. 4. Eulerian circuits and trails. Euler's Theorem for graphs and digraphs. 5. Extremal problems on graphs. Mantel's Theorem. 6. Graphic sequences. Havel-Hakimi Theorem on such sequences. 7. Directed graphs: degrees, connectivity, Eulerian circuits, de Bruijn graphs. 8. Tournaments, kings in tournaments. 9. Trees, characterizations of trees. Centers of trees. 10. Counting spanning trees in graphs. Prufer codes. Matrix Tree Theorem. 11. Minimum spanning trees. Algorithms of Kruskal and Prim. 12. Matchings in bipartite graphs. Hall's Theorem. 13. Matchings and covers. Konig-Egervary Theorem. 14. Stable matchings. 15. Matchings in general graphs. Tutte's 1-factor Theorem. 16. Berge-Tutte Formula and theorems of Petersen. 17. Connectivity and edge connectivity. A characterization of 2-connected graphs. 18. Ear decomposition. 19. Menger's Theorems. 20. Flows in networks. Decomposition of a flow into flows along cycles and s,t-paths. 21. Ford-Fulkerson Algorithm. Max Flow--Min Cut Theorem. 22. Integral flows (matchings in bipartite graphs, edge connectivity). 23. Planar and plane graphs. 24. Euler's Formula. K_5 and K_{3,3} are not planar. Number of edges in planar graphs. 25. Outerplanar graphs. Triangulations. 26. Kuratowski's Theorem and Wagner's Theorem. 27. Colorings. Greedy algorithm for coloring; d-degenerate graphs. 28. Properties of k-critical graphs. 29. Mycielski's Construction. Brooks' Theorem. 30. 6-Color Theorem for planar graphs. Statement of 4-Color Theorem. 31. Extremal problems on graphs. Turan's Theorem. Ershov-Kozhuhin Theorem. 32. Edge colorings. Shannon's Theorem and statement of Vizing's Theorem. Petersen Graph is not 3-edge-colorable. Edge colorings of regular simple graphs with cut edges. 33. Hamiltonian cycles in graphs. Dirac's Theorem and its sharpness examples. Tutte's example of a 3-connected 3-regular planar simple nonhamiltonian graph.
    Graphs to remember: a) Complete and complete bipartite graphs, b) The k-dimensional cube Q_k, c) Petersen graph, d) de Bruijn graph, e) the Turan graph T(n,k), f) Mycielski's Graph for k=4. g) Shannon's Triangle.
    One of the problems on the final will be a proof of one of the following 7 theorems: 1. Konig's Theorem on bipartite graphs (Theorem 1.2.8 in the book), 2. Mantel's Theorem on the number of edges in triangle-free graphs(Theorem 1.2.8 in the book), 3. Theorem on the existence of kings in a tournament (Prop. 1.4.30 in the book), 4. Gallai's Theorem edge cover number versus matching number (Theorem 3.1.22 in the book), 5. Berge's Theorem on M-augmenting paths (Theorem 3.1.10 in the book), 6. Petersen's Theorem on 2-factors in 2k-regular graphs (Cor. 3.3.10 in the book), 7. Whitney's Theorem on internally disjoint paths in 2-connected graphs (Theorem 4.2.2 in the book).
  • We will have help sessions for the final on December 13 and 14 from Noon to 1 pm in our classroom.

  • To look at your grades, go to ``www.math.uiuc.edu'', then ``courses'', then ``scores reports''.


    You may send comments to: kostochk@math.uiuc.edu

    Last changed on December 07, 2018.