Math 582, Section F1 - Class Log

• 1. Monday, 01/14: Handed out basic info about the class. Introduction. Started counting spanning trees in complete graphs. Examples. Proved a generalization of Cayley's Formula (Theorem 1.1, Theorem 6.1.18 in the book). Introduced and gave an example of Prufer codes. Stated the claim on Prufer codes. Stated the Matrix Tree Theorem (Theorem 1.2, Theorem 6.1.26 in the book).
• 2. Wednesday, 01/16: Restated, gave an example and almost proved the Matrix Tree Theorem (Theorem 1.2, Theorem 6.1.26 in the book). What remains is to prove is Lemma 1.5 on minors of incidence matrices of loopless multigraphs.
• 3. Friday, 01/18: Proved Lemma 1.5. Stated and discussed (but did not prove) the Directed Matrix Tree Theorem (Theorem 1.6, Theorem 6.1.28 in the book). Examples. Stated the Matrix Arborescence Theorem (Theorem 1.7, Theorem 6.1.30 in the book). Showed how Directed Matrix Tree Theorem follows from the Matrix Arborescence Theorem. An example.
• 4. Wednesday, 01/23: Proved the Matrix Arborescence Theorem (Theorem 1.7, Theorem 6.1.30 in the book). Counting Eulerian circuits in directed graphs, an example. A lemma (Lemma 1.8, Lemma 6.1.33 in the book) on last edges exiting vertices in Eulerian circuits. Connections between spanning in-trees and Eulerian circuits in directed graphs. Stated BEST Theorem (Theorem 1.9, Theorem 6.1.36 in the book) on the number of Eulerian circuits in digraphs and spanning in-trees.
• 5. Friday, 01/25: A lemma (Lemma 1.10) for BEST Theorem (Theorem 1.9), and a proof of the theorem. Ringel's conjecture on tree-decompositions of complete graphs. Defined graceful labelings of graphs. Examples of graceful graphs. A discussion of the Graceful Tree Conjecture. Proved Theorem 1.11 on decompositions of complete graphs into copies of graphs with graceful labelings. Caterpillars have graceful labelings.
• 6. Monday, 01/28: Stated Wilson's Theorem (Theorem 1.12) on decompositions of large complete graphs into copies of the same graph. On decompositions of 2m-regular graphs into m-edge trees. Graham-Haggkvist Conjectures. Stated and proved Theorem 1.13 on decompositions of 2m-regular graphs into m-edge trees. Gyarfas-Lehel Conjecture on decomposition of the complete n-vertex into given set of trees of different sizes. Packing of graphs: definitions and examples.
• 7. Wednesday, 01/30: Snow day.
• 8. Friday, 02/01: Some graph-theoretic problems and results in the language of packing. Proved Sauer-Spencer Theorem (Theorem 1.14) on packing of graphs with bounded product of sizes. Proved Sauer-Spencer Theorem (Theorem 1.15) on packing of graphs with bounded products of maximum degrees. Some extremal results and conjectures on packing of graphs. A short discussion of Bollobas-Eldridge Conjecture of graph packing.
• 9. Monday, 02/04: Equitable colorings of graphs: properties and examples. Discussed the Chen-Lih-Wu Conjecture on equtable colorings of graphs. Stated, discussed and half-proved the Hajnal-Szemeredi Theorem on equitable (r+1)-coloring of graphs with maximum degree at most r.
• 10. Wednesday, 02/06 (at 10am): Guest lecture by Professor Pfender (U. Colorado, Denver) on 1-2-3-Conjecture.
• 11. Wednesday, 02/06: Finished the proof of the Hajnal-Szemeredi Theorem on equitable (r+1)-coloring of graphs with maximum degree at most r.
• 12. Friday, 02/08: Started Section 2 on vertex degrees. Stated Erdos-Gallai Theorem (Theorem 2.1) on graphic sequences. Stated and proved the Havel-Hakimi Theorem (Theorem 2.2) on such sequences. Discussed graphs with the same degree sequence. Stated and half-proved Edmonds' Theorem on potentially k-edge-connected sequences (Theorem 2.3).
• 13. Monday, 02/11: Proved Edmonds' Theorem on potentially k-edge-connected sequences (Theorem 2.3). Vertex partitions. Stated and proved Lovasz' Theorem on partitions with bounded maximum degree (Theorem 2.4). A conjecture on the topic by Correa, Havet and Sereni. Stated and a Theorem of Stiebitz on partitions with bounded minimum degree (Theorem 2.5). First words on the Reconstruction Problem.
• 14. Wednesday, 02/13: Introduction into the Reconstruction Problem. Definitions and notation. An example. Kelly's Lemma (Lemma 3.1). Regular graphs are reconstructible, Proved Theorem 3.2 that disconnected graphs with at least 3 vertices are reconstructible and started to prove Theorem 3.4 that every tree with at least 3 vertices is reconstructible.
• 15. Friday, 02/15: Proved Lemma 3.3 that in a tree the distance from a vertex to a closest vertex of degree at least 3 is reconstructible. Proved Theorem 3.4 that every tree with at least 3 vertices is reconstructible. Stated Tutte's Theorem (Theorem 3.5) on reconstructibility of the number of some spanning subgraphs of a graph.
• 16. Monday, 02/18: Proved Tutte's Theorem (Theorem 3.5) on reconstructibility of the number of some spanning subgraphs of a graph. Applications: the number of spanning trees and the number of hamiltonian cycles in a graph are reconstructible. Started edge-reconstruction: definitions, stated Harary's Edge-Reconstruction Conjecture. Stated and hinted the proof of edge-Kelly Lemma (Lemma 3.6). Stated Lovasz' Theorem (Theorem 3.7) on edge-reconstruction of graphs with many edges.
• 17. Wednesday, 02/20 at 10am in 341 AH (make-up class): Guest lecture by Professor West on graph reconstruction from the deck of k-vertex induced subgaphs.
• 18. Wednesday, 02/20: Proved Lovasz' Theorem (Theorem 3.7) on edge-reconstruction of graphs with many edges. Stated Nash-Williams' Theorem (Theorem 3.8) and derived from it the Theorem 3.9 by Muller that each n-vertex graph with at least 1+log(n!) edges is edge-reconstructible. Proved Nash-Williams' Theorem. Mentioned digraphs for which the Reconstruction Conjecture fails.
• 19. Friday, 02/22: Connectivity: initial discussion. Stated definitions and Menger's Theorem(s). Stated and proved Edmonds' Branching Theorem (Theorem 4.1). Also proved Theorem 4.2 that Theorem 4.1 implies the edge version of Menger's Theorem for digraphs.
• 20. Monday, 02/25: Stated and proved Corollary 4.3 of the Edmonds' Branching Theorem. A discussion of dicuts in digraphs. Stated the Lucchesi-Younger Theorem (Theorem 4.4) and Lovasz' Lemma (Lemma 4.5) on dicuts covering each edge at most twice. Almost proved Theorem 4.4 modulo Lemma 4.5.
• 21. Wednesday, 02/27: Proved Lovasz' Lemma (Lemma 4.5) on dicuts covering each edge at most twice. A short discussion of 2-linked graphs. An example of a 5-connected graph that is not 2-linked.
• 22. Friday, 03/01: Defined k-linked graphs and presented an example of a (2k-2)-connected graph that is not k-linked. Stated and proved a lemma (Lemma 4.6) by Mader-Thomassen on special partitions of graphs with a given minimum degree. Proved a theorem (Theorem 4.7) by Mader-Thomassen on subdivisions of graphs with high minimum degree. Discussed bounds on h(k) - the minimum t such that each graph with minimum degree t has a subdivision of the complete graph with k vertices. Stated Theorem 4.8 by Jung and Larman-Many that graphs with high connectivity are k-linked.
• 23. Monday, 03/04: Proved Theorem 4.8 by Jung and Larman-Many that graphs with high connectivity are k-linked. Every k-linked graph is (2k-1)-connected. There are (3k-3)-connected graphs that are not k-linked. A discussion of H-linked graphs and their properties. Stated and started to prove Theorem 4.9 that every graph with average degree at least 4k has a k-connected subgraph.
• 24. Wednesday, 03/06: Proved Theorem 4.9 that every graph with average degree at least 4k has a k-connected subgraph. Stated and discussed Mader's Conjecture on the topic. Recalled ear decomposition of graphs. Restated Theorem 5.1 that a graph is 2-connected if and only if it has an ear decomposition. Stated and proved Expansion Lemma (Lemma 5.2). Vertex k-splits. Proved Lemma 5.3 that vertex k-splits of k-connected graphs are k-connected. Stated Lemma 5.4 on 3-contractible edges in 3-connected graphs and Tutte-Thomassen's characterization of 3-connected graphs (Theorem 5.5).
• 25. Friday, 03/08: Proved Lemma 5.4 on 3-contractible edges in 3-connected graphs and Tutte-Thomassen's characterization of 3-connected graphs (Theorem 5.5). A discussion and examples of minimally k-connected graphs. Stated Mader's Theorem on minimally k-connected graphs (Theorem 5.6). Proved Lemma 5.8 on connectivity of graphs after deleting or contracting edges. Stated two more lemmas and Tutte's characterization of 3-connected graphs in terms of disjoint 3-splits and edge-additions (Theorem 5.7). Outlined the idea of the proof.
• 26. Monday, 03/11: Proved two lemmas and almost proved Tutte's characterization of 3-connected graphs in terms of disjoint 3-splits and edge-additions (Theorem 5.7).
• 27. Wednesday, 03/13: Finished proof of Theorem 5.7. Proved Mader's Theorem on minimally k-connected graphs (Theorem 5.6) modulo special lemma (Lemma 5.11). Derived from Mader's Theorem a bound (by Bollobas) on the number of vertices of degree k in a minimally k-connected graph with n vertices (Theorem 5.12). Proved Lemma 5.11.
• 28. Friday, 03/15: Proved Mader's theorem on vertices of degree k in minimally-k-edge-connected multigraphs (Theorem 5.13). Define relative connectivity and shortcuts. Stated the Shortcut Lemma (Lemma 5.14). Stated and proved the Orientation Theorem of Nash-Willians (Theorem 5.15) modulo the Shortcut Lemma (Lemma 5.12). Stated and briefly discussed the theorem by Gyori and Lovasz (Theorem 5.16) on k-connected graphs.
• 29. Monday, 03/25: Proved the Shortcut Lemma (Lemma 5.14). Graph drawings, plane and planar graphs. Euler's Formula. Graphs K_5 and K_{3,3} are not planar.
• 30. Wednesday, 03/27: For proving Kuratowski's Theorem (Theorem 6.1), proved the following facts: a) Lemma 6.2 that contractions of graphs not containing Kuratowski subgraphs also do not contain Kuratowski subgraphs, b) Lemma 6.3 that a vertex minimal counter-example to Theorem 6.1 must be 3-connected and c) Tutte's Theorem that every 3-connected graph with no Kuratowski subgraph has a convex embedding in the plane such that no three vertices belong to a line (Theorem 6.4). Derived from Kuratowski's Theorem and commented Wagner's Theorem (Theorem 6.5).
• 31. Friday, 03/29: The cycle space and the bond space of a graph; their properties (Theorem 6.6). Stated and proved the planarity criterion due to MacLane: a graph is planar if and only if its cycle space has a 2-basis (Theorem 6.7).
• 32. Monday, 04/01: Defined and discussed dual multigraphs to plane multigraphs. Abstract dual to a graph. Stated and proved a criterion due to Whitney: a 2-connected graph is planar if and only if it has an abstract dual (Theorem 6.8). Started Schnyder labelings: definitions and examples. Simple properties of Schnyder labelings. Lemma 6.9 on labelings of the external vertices in triangulations. Lemma 6.10: If a is an external vertex in a triangulation with at least 4 vertices, then a has an internal neighbor u such that the edge au is contractible. Stated and proved Theorem 6.11 on the existence of Schnyder labelings.
• 33. Wednesday, 04/03: Every Schnyder labeling of a triangulation is obtained from a labeling of a smaller triangulation in a simple way (Lemma 6.12). Stated and proved Uniform Angle Lemma (Theorem 6.13). Further properties of Schnyder labelings. Partitions of edges of planar graphs into 3 forests (Theorem 6.14). A lemma on regions R_i(v) (Lemma 6.15).
• 34. Friday, 04/05: Barycentric representations of graphs. Straight-line embeddings of planar graphs into grids of reasonable size using Schnyder labelings. Started separators in planar graphs. Stated the Lipton-Tarjan Theorem on small separators in planar graphs (Theorem 7.1).
• 35. Monday, 04/08: Proved the Lipton-Tarjan Theorem on small separators in planar graphs (a proof by Alon-Seymour-Thomas).
• 36. Wednesday, 04/10: A discussion of the Four Color Theorem. An example of discharging. A short discussion of light triangles in planar graphs. Stated and half-proved Borodin's Theorem (Theorem 8.1) on light triangles in normal planar maps with minimum degree 5.
• 37. Friday, 04/12: Finished the proof of Borodin's Theorem (Theorem 8.1) on light triangles in normal planar maps with minimum degree 5. Stated and discussed Grotschz's Theorem (Theorem 8.2) that every planar triangle-free graph is 3-colorable. Gave a simple proof of it using a theorem from Math 581. Steinberg's Conjecture. Stated a lemma (Lemma 8.4) on structure of plane graphs. Using this lemma, proved a theorem by Borodin, Sanders and Zhao (Theorem 8.3) that every planar graph with no cycles of length 4, 5, 6, 7, 8, and 9 is 3-colorable. Started a proof of Lemma 8.4.
• 38. Monday, 04/15: Proved Lemma 8.4 by Borodin, Sanders and Zhao. Started hamiltonian and longest cycles. Gave two proofs of the theorem by Dirac (Theorem 9.1). Proved a theorem by Ore (Theorem 9.2) and a theorem by Posa (Theorem 9.3) on the subject.
• 39. Wednesday, 04/17 at 10am in 341 AH (make-up class): Guest lecture by Professor Stiebitz (Germany) on a proof of his theorem in the course.
• 40. Wednesday, 04/17: Derived Corollary 9.4 on adding an edge xy when d(x)+d(y)>n-1. Proved theorems by Chvatal (Theorem 9.5) and Chvatal-Erdos (Theorem 9.6) on Hamiltonian cycles in graphs. Showed an extremal example for Chvatal Theorem. Started longest cycles. Stated Erdos-Gallai Theorems (Theorems 9.7 and 9.7') on long paths and cycles in graphs with given number of vertices and edges. Derived Theorem 9.7 from Theorem 9.7'.
• 41. Friday, 04/19: Stated and proved Erdos-Gallai Theorem on long cycles in graphs with given number of vertices and edges (Theorem 9.7'). Stated a lemma by Bondy (Lemma 9.8) and its modification by Kopylov (Lemma 9.8') on longest cycles in 2-connected graphs. Derived Lemma 9.8 from Lemma 9.8'. Started a proof of Lemma 9.8'.
• 42. Monday, 04/22: Proved Lemma 9.8'. Stated and proved Fan's Theorem on longest cycles in 2-connected graphs (Theorem 9.9). Stated and started to prove Kopylov's Theorem on the maximum number of edges in n-vertex 2-connected graphs with no cycles of length at least k (Theorem 9.10).
• 43. Wednesday, 04/24: Proved Kopylov's Theorem on the maximum number of edges in n-vertex 2-connected graphs with no cycles of length at least k. Review of the course and some conjectures.