Math 582, Section F1 - Class Log
- 1. Wednesday, 1/20:
Handed out basic info about the class.
Introduction. Started counting spanning trees in multigraphs:
stated Cayley Formula and Prufer codes, an application of
Prufer codes.
Stated, gave examples and started proving the Matrix Tree Theorem.
- 2. Friday, 1/22:
An algebraic lemma. Graphs and polynomials.
Finished proof of the Matrix Tree Theorem.
- 3. Monday, 1/25:
Stated and
discussed (but did not prove) the Directed Matrix Tree Theorem.
Stated the Matrix Arborescence Theorem. Examples.
Showed how
Directed Matrix Tree Theorem follows from
the Matrix Arborescence Theorem. Started to prove the theorem.
- 4. Wednesday, 1/27:
Proved the Matrix Arborescence Theorem.
Connections between spanning in-trees and Eulerian circuits
in directed graphs.
- 5. Friday, 1/29:
BEST Theorem on the number of Eulerian
circuits in digraphs. Ringel's conjecture on tree-decompositions
of complete graphs. Defined graceful labelings of graphs.
Examples of graceful graphs. Discussion of
the Graceful Tree Conjecture.
On decompositions of 2m-regular graphs into m-edge trees.
- 6. Monday, 2/1:
A theorem on decompositions of 2m-regular graphs into m-edge trees.
Gyarfas-Lehel Conjecture.
Packing of graphs: definitions and examples.
- 7. Wednesday, 2/3:
Some problems and results
in the language of packing. Some extremal results and
conjectures on packing of graphs. Proved Sauer-Spencer
Theorems on packing of graphs with bounded products of
sizes and maximum degrees.
- 8 . Friday, 2/5:
Equitable colorings of graphs. Stated and half-proved
Hajnal-Szemeredi Theorem on
equitable (r+1)-coloring of graphs with maximum degree at most r.
- 9. Monday, 2/8:
Finished proof of Hajnal-Szemeredi Theorem on
equitable (r+1)-coloring of graphs with maximum degree at most r.
- 10. Wednesday, 2/10:
Short discussion of Chen-Lih-Wu Conjecture on equtable colorings of
graphs.
Stated Erdos-Gallai Theorem on graphic sequences. Stated and
proved Edmonds' Theorem on potentially k-edge-connected
sequences.
- 11. Friday, 2/12:
Vertex partitions.
Lovasz' Theorem on partitions with bounded maximum degree.
Theorem of Stiebitz on partitions with bounded minimum degree.
- 12. Monday, 02/15:
Introduction into the Reconstruction Problem. Definitions and
notation. Kelly's Lemma. Regular graphs are reconstructible,
disconnected graphs are reconstructible. Started discussion of
reconstructible trees.
- 13. Wednesday, 02/17:
Every tree with at least 3 vertices is reconstructible.
- 14. Friday, 02/19:
Tutte's Theorem: Proof and applications. Chromatic number
is reconstructible.
- 15. Monday, 02/22:
Started edge-reconstruction: definitions, edge-Kelly
Lemma. Proved Lovasz' Theorem. Stated
Nash-Williams' Theorem and proved the corollary
that each n-vertex graph with at least 1+ln(n!) edges
is edge-reconstructible.
- 16. Wednesday, 02/24:
Proved Nash-Williams' Theorem. Examples of digraphs
for which Reconstruction Conjecture fails.
Connectivity: definitions and first facts.
Started to prove a theorem on the
connectivity of the Cartesian product of connected graphs.
- 17. Friday, 02/26:
Finished the proof of a theorem on the
connectivity of the Cartesian product of connected graphs.
Menger's Theorem.
Started proof of Edmonds' Branching Theorem.
- 18. Monday, 03/01:
Finished Edmonds's Branching Theorem.
Corollaries. Dicuts in digraphs.
- 19. Wednesday, 03/03:
A discussion of dicuts in digraphs. Stated and proved Luccesi-Younger
Theorem modulo Lovasz' Lemma. Started proof of the lemma.
- 20. Friday, 03/05:
Proved Lovasz' Lemma. Introduced k-linked graphs.
An example of a 5-connected graph that is not 2-linked.
- 21. Monday, 3/08:
Every k-linked
graph is (2k-1)-connected. Stated and proved a lemma and a
theorem by Mader-Thomassen on
subdivisions of graphs.
- 22. Wednesday, 3/10:
Proved that each f(k)-connected graph is k-linked. Discussed
bounds on f(k) and on h(k) - the minimum t such that each
graph with minimum degree t has a subdivision of the complete
graph with k vertices. In particular, h(4)=3.
- 23. Friday, 3/12:
A digression on reconstruction of multigraphs.
H-linked graphs and their properties. Proved that every
graph with average degree at least 4k has a k-connected
subgraph.
- 24. Monday, 3/15:
Characterizations of 2-connected graphs. Expansion Lemma.
Vertex k-spits. Proved a lemma on 3-contractible edges in
3-connected graphs.
Tutte-Thomassen's
characterization of 3-connected graphs.
- 25. Wednesday, 3/17:
A lemma for Tutte's characterization of 3-connected graphs in terms
of disjoint 3-splits and edge-additions. D. McDonald started
describing his result on edge-reconstructibility of multigraphs
with at least one multiple edge.
- 26. Friday, 3/19:
Proved the lemmas for
Tutte's characterization of 3-connected graphs in terms
of disjoint 3-splits and edge-additions.
- 27. Monday, 3/29:
Proved Tutte's characterization of 3-connected graphs in terms
of disjoint 3-splits and edge-additions.
Stated Mader's Theorem on minimally k-connected graphs
and derived from it a bound on the number of vertices of degree k in a
minimally k-connected graph.
- 28. Wednesday, 03/31:
Proved Mader's Theorem on minimally k-connected graphs
and the main lemma for this theorem.
Started proving the orientation theorem.
- 29. Friday, 04/02:
Proved the orientation theorem and lemmas, in particular,
the shortcut lemma.
- 30. Monday, 04/05:
Stated and discussed the theorem by Gyori and Lovasz on
k-connected graphs. Planar graphs, Kuratowski's Theorem.
Cycle space and bond space of
a graph.
Stated and started to prove a planarity criterion due to MacLane:
a graph is planar if and only if its cycle space has a 2-basis.
- 31. Wednesday, 04/07:
Proved MacLane's criterion of planarity and a criterion due to
Whitney: a graph is planar if and only if it has an abstract dual.
Started Schnyder labelings: definitions and examples.
- 32. Monday, 04/12:
Simple properties of Schnyder labelings.
Extistence of Schnyder labelings. Every Schnyder labeling of
a triangulation is
obtained from a labeling of a smaller triangulation in a simple way.
Stated Uniform Angle Lemma.
- 33. Wednesday, 04/14:
Proved Uniform Angle Lemma.
Further properties of Schnyder labelings.
- 34. Friday, 04/16:
Straight-line embeddings of
planar graphs.
More economical straight-line embeddings of planar graphs.
The dimension of the vertex-edge containment order of a graph.
Stated and started to prove a criterion of planarity in terms of
the dimension of the vertex-edge containment order.
- 35. Monday, 04/19:
Finished the proof of the fact that the dimension of the
vertex-edge containment order
of every planar
graph is at most 3.
Started separators in planar graphs. Proved a lemma and
stated another for Planar Separator Theorem.
- 36. Wednesday, 04/21:
Proved Lipton-Tarjan Theorem on small separators in planar
graphs. Proved the corollary on (m,1/2)-separators in
planar graphs.
- 37. Friday, 04/23:
Proved that an algorithm using separators finds large
independent sets in planar graphs. Stated and started to
prove Borodin's Theorem on light triangles in normal
planar maps with minimum degree 5.
- 38. Monday, 04/26:
Proved Borodin's Theorem on light triangles in normal
planar maps with minimum degree 5. Stated and discussed
Grotschz's Theorem that every planar triangle-free graph is
3-colorable.
- 39. Wednesday, 04/28:
Proved Grotschz's Theorem modulo the main lemma.
- 40. Friday, 04/30:
Finished the proof of
Grotschz's Theorem. A discussion of Havel's examples. Steinberg's
Conjecture. Almost finished the proof of a theorem by Borodin, Sanders
and Zhao that every planar graph with no cycles of length
4, 5, 6, 7, 8, and 9 is 3-colorable.
- 41. Monday, 05/03:
Finished the proof of Borodin-Sanders-Zhao Theorem.
Nowhere-zero flows in graphs. Elementary properties of such flows.
Modular flows. A graph has a nowhere-zero 2-flow if and
only if it is even.
- 42. Wednesday, 05/05:
A cubic graph has a nowhere-zero 3-flow if and
only if it is bipartite. A cubic graph has a nowhere-zero 4-flow if and
only if it is edge-3-colorable. If a graph has a modular
nowhere-zero k-flow, then it has a
nowhere-zero k-flow. Flows in unions of graphs. Statements of Tutte's
5-Flow Conjecture and corresponding 8-Flow Theorem and 6-Flow Theorem.
Last changed on May 5, 2010.