Class Log
- 1. Wednesday, 01/21:
Handing out and discussing basic info about the class. Basic definitions.
Examples and more definitions (paths, cycles, complete graphs, stars,
Petersen graph, complements of simple graphs,
Konigsberg bridges).
- 2. Friday, 01/23:
Job Assignment Problem. Definitions: bipartite graphs,
complete bipartite graphs. subgraphs.
Incidence and adjacency matrices, lists
of neighbors. Isomorphism of graphs.
- 3. Monday, 01/26:
More on isomorphism of graphs. Clique number and independence number
of graphs.
Walks, trails, paths, and cycles.
Connected
graphs and connected components, isolated vertices.
Lemma 1.2.5 on walks and paths.
- 4. Wednesday, 01/28:
Lemma 1.2.15 on odd closed walks.
A characterization of bipartite graphs (Theorem 1.2.18).
Lemma on cycles in graphs and simple graphs with minimum
degree k. Notions of Eulerian circuits and trails.
Stated and started to prove Euler's theorem.
- 5. Friday, 01/30:
Proved Problem 1 in HW1.
Proof of Euler's Theorem, Eulerian trails. Degree Sum Formula,
k-regular graphs, maximum and minimum degrees of graphs.
Hypercubes.
Stated and started to prove
Mantel's Theorem on maximum number of edges in
triangle-free graphs.
- 6. Monday, 02/02:
Proved Mantel's Theorem.
Degree sequences of graphs, graphic sequences.
Havel-Hakimi Theorem on graphic sequences.
- 7. Wednesday, 02/04:
Defined directed graphs, degrees, outdegrees and indegrees of their
vertices. Adjacency and
incidence matrices. Weakly connected and strongly connected
directed graphs. Degree Sum Formula. De Bruijn graphs. Tournaments.
Quiz 1.
- 8. Monday, 02/09:
Eulerian circuits in directed graphs.
and their properties.
Special cyclic arrangements of 0s and 1s and de Bruijn graphs.
Kings in tournaments.
A theorem on the existence of kings. Main results in Chapter 1.
Trees: basic definitions (tree, forest, leaf) and a lemma.
- 9. Wednesday, 02/11:
Proved a lemma on trees and a characterization theorem for trees.
Distance in graphs: eccentricity, diameter, radius.
- 10. Friday, 02/13:
Proved Problem 5 in HW2. Centers of graphs.
Jordan's theorem on centers of trees.
Spanning trees, Prufer Code, Cayley Formula.
Proved half of the properties of Prufer Code.
- 11. Monday, 02/16:
Finished part of the proof of properties of Prufer Code.
Spanning trees in general graphs.
A recurrence for the number of spanning trees in a graph.
Matrix Tree Theorem: the statement and an example.
A short discussion of the Minimum Spanning Tree Problem.
- 12. Wednesday, 02/18:
Stated a basic lemma on minimum spanning trees. Prim's and Kruskal's algorithms
for finding minimum spanning trees in graphs.
Maximum spanning trees. Quiz 2.
- 13. Friday, 02/20:
Solved Problem 6 in HW4.
Proved the basic lemma on minimum spanning trees.
Dijkstra's algorithm for
finding shortest paths.
- 14. Monday, 02/23:
Proved the properties of Dijkstra's algorithm for
finding shortest paths.
Main results in Chapter 2.
Matchings in graphs: basic definitions.
Augmenting paths. Proved Berge's theorem on augmenting paths.
Matchings in bipartite graphs. Stated
Hall's Theorem.
- 15. Wednesday, 02/25:
Proved Hall's Theorem and the Marriage Theorem.
Systems of distinct representatives.
The notion of vertex cover and its relation to that
of independent set.
- 16. Wednesday, 02/25:
Test 1 at 6pm in 145 AH.
- 17. Friday, 02/27:
An extended comment on Test 1.
Relations between independence number, the size of a
maximum matching, and the size of a minimum
vertex cover. Proved Konig-Egervary Theorem.
Stated Gallai's Theorem.
- 18. Monday, 03/02:
Proved Gallai's Theorem.
The Augmenting
Path Algorithm for finding maximum matchings in bipartite
graphs.
- 19. Wednesday, 03/04:
Stable matchings, the Gale-Shapley Algorithm.
Matchings in general graphs. Stated
Tutte's Theorem on perfect matchings in graphs. Quiz 3.
- 20. Friday, 03/06 :
Solutions of Problems 2 and 6 in HW6.
Derived from Tutte's Theorem a
corollary for graphs with even number of vertices.
Graphs as models of special polynomials. Petersen graph
does not decompose into 3 perfect matchings.
Proved Petersen's theorem on 1-factors in 3-regular graphs.
- 21. Monday, 03/09:
Proved Tutte's Theorem on perfect
matchings in graphs and
Petersen's Theorem on
2-factors in 2k-regular graphs.
An example of a 3-regular graph without a perfect metching.
- 22. Wednesday, 03/11:
Berge-Tutte Formula and its proof. The six important theorems in Chapter 3.
First definitions for connectivity. Quiz 4.
- 23. Friday, 03/13:
A discussion of Quiz 4. Proved Problem 1 from HW7.
Definitions in connectivity of graphs, examples.
Edge cuts. Stated the theorem on relations between
connectivity, edge-connectivity and minimum degree of a graph.
- 24. Monday, 03/16:
Proved the theorem on relations between
connectivity, edge-connectivity and minimum degree of a graph.
Connectivity of a 3-regular simple graph equals its edge-connectivity.
Blocks; no two blocks share more than one vertex.
- 25. Wednesday, 03/18:
Solved Problems 1 and 2 in HW8. Whitney's Theorem for 2-connected graphs.
Expansion lemma.
- 26. Wednesday, 03/18:
Test 2 at 6pm in 145 AH.
- 27. Monday, 03/30:
A discussion of Test 2.
Equivalent conditions for 2-connectdness.
Subdivisions of graphs.
Ear decomposition. Characterization of 2-connected graphs in
terms of ear decompositions (proof not finished).
- 28. Wednesday, 04/01:
Finished characterization of 2-connected graphs in
terms of ear decompositions.
Notion of x,y-cuts and related characterstics.
Stated and half-proved Menger's Theorem
for vertex connectivity. Quiz 5.
- 29. Friday, 04/03:
Proved
Menger's Theorem for vertex connectivity.
Menger's criterion for a graph to be
k-connected. The Fan Lemma.
- 30. Monday, 04/06:
Reproved a part of Fan Lemma.
Notion of connectivity of digraphs.
Stated Menger's Theorem for digraphs.
Definitions for flows in networks.
Started a theorem on the structure of flows (see a handout on the web).
- 31. Wednesday, 04/08:
Proved a theorem on the structure of flows (see handout on the web).
Max Flow Problem for networks, s,t-cuts in networks.
Ford-Fulkerson's algorithm for finding maximum flow in a network.
Stated Max Flow-Min Cut Theorem.
- 32. Wednesday, 04/08:
Test 3 at 6pm in 145 AH.
- 33. Friday, 04/10:
A discussion of Test 3: Problems 1, 2, and 5.
Proved Max Flow-Min Cut Theorem.
Computational issues of the Ford-Fulkerson's algorithm for
the Max-Flow Problem. An example.
- 34. Monday, 04/13:
Applications of the Max Flow-Min Cut Theorem. Finding maximum matchings in
a bipartite graph is a Max Flow Problem. Edge version of Menger's Theorem
for digraphs using the Max Flow-Min Cut Theorem.
Main results in Chapter 4.
Definitions and examples of planar and plane graphs.
- 35. Wednesday, 04/15:
More definitions and examples for planar and plane graphs.
Faces and their length.
Dual graphs. Proved Euler's Formula and stated two
consequences. Quiz 6.
- 36. Friday, 04/17:
A short discussion of Quiz 6. Consequences of Euler's Formula:
K_5 and K_{3,3} are non-planar. Discussion of Kuratowski's Theorem
and Wagner's Theorem. Derived Wagner's Theorem from Kuratowski's Theorem.
Petersen's Graph is nonplanar. Outerplanar graphs.
- 37. Monday, 04/20:
Maximal planar graphs and triangulations.
The 4-Color Problem.
Graph colorings, definitions and examples.
Simple properties of chromatic number.
Greedy coloring.
- 38. Wednesday, 04/22:
Greedy coloring and d-degenerate graphs. Every d-degenerate graph
is (d+1)-colorable. Every planar graph is 6-colorable.
Stated
Brooks' Theorem (without proof).
Notion of color-critical graphs.
Quiz 7.
- 39. Friday, 04/24:
A comment on Quiz 7. Properties of k-critical
and vertex-k-critical graphs.
An example of a 2-connected k-critical graph.
Every k-critical graph is (k-1)-edge connected.
Proof of 5-Color Theorem.
Edge coloring: definitions.
- 40. Monday, 04/27:
Edge coloring: definitions and properties.
Edge coloring of regular graphs with a cut edge.
Edge coloring of bipartite graphs. Petersen Graph is not
3-edge-colorable.
Line graphs.
Stated Shannon's Theorem and Vizing's Theorem
on edge coloring. Stated
Tait's Theorem on edge coloring of plane graphs.
Quiz 8.
- 41. Wednesday, 04/29:
Discussed and half-proved Tait's Theorem. Hamiltionian cycles.
Hamiltonian properties of 3-regular 3-connected planar graphs,
Tutte's example. A smaller example.
Proved Dirac's Theorem on
hamiltonian cycles.
- 42. Wednesday, 04/29:
Test 4 at 6pm in 145 Altgeld Hall.
- 43. Friday, 05/01:
A discussion of Test 4. 10 most important theorems on graphs.
Review of the course,
topics for the final.
Last changed on May 1, 2009.