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Graph Theory I

Arizona State University, Fall 2023

When: Mondays and Wednesdays, 11am–12:15pm

Where: WXLR A546

Instructor: Zilin Jiang ([email protected])

Office hours: Tuesdays and Thursdays 1:30pm–2:30pm, in WXLR A839.

Course description​

This graduate-level graph theory class provides a systematic development of the fundamental concepts in graph theory, with a focus on topics such as matchings, connectivity, arboricity, planarity, coloring, and network flows.

Grading​

Homework 50% + Midterm 20% + Final 30% + Bonus 10%

Assignments​

  1. Homework 1 due on September 6.
  2. Homework 2 due on September 20.
  3. Homework 3 due on October 6.
  4. Homework 4 due on October 30.
  5. Homework 5 due on November 13.
  6. Homework 6 due on November 27.

Exams​

No make-up test will be given unless a student has notified the instructor before the test is given.

  • Midterm: October 11, 2023, 11am–12:15pm
  • Final: December 6, 2023, 9:50am–11:50am

Schedule​

Week 1

  • August 21. Definition of graphs, the degree of a vertex, the handshake lemma, every graph contains a subgraph whose minimum degree is at least half of the average degree of the graph.
  • August 23. Paths, cycles, every graph G contains a path of length δ(G) and a cycle of length at least δ(G) + 1 (provided that δ(G) ≥ 2), girth, distance, diameter, central vertices, bounding the order of a graph in terms of its radius and maximum degree (or in terms of its girth and minimum degree), connectivity.

Week 2

  • August 28. Mountain climbing problem, components, k-connectedness, l-edge-connectedness, vertex-connectivity ≤ edge-connectivity ≤ minimum degree.
  • August 30. Mader's theorem on subgraph with high connectivity, trees and forests.

Week 3

  • September 4: No class (Labor day)
  • September 6. Rooted trees, normal trees, spanning trees.

Week 4

  • September 11. Bipartite graphs, subdivision, contraction, models, topological minors, minors, IX and TX, closed walks, Euler tours, a criterion of Eulerian graphs.
  • September 13. Matchings, König's theorem on matchings and covers, Hall's marriage theorem, stable matchings.

Week 5

  • September 18. Proof of the stable matching problem, Tutte's theorem.
  • September 20. First proof of Tutte's theorem, Petersen's theorem.

Week 6

  • September 25. Second proof of Tutte's theorem, structure of maximum matchings, the Gallai–Edmonds matching theorem.
  • September 27. The Erdős–Pósa property, every large cubic multigraph has k vertex-disjoint cycles.

Week 7

  • October 2. The Erdős–Pósa theorem, the Gallai–Milgram theorem, the Dilworth theorem.
  • October 4. The tree-packing theorem, the Nash-Williams theorem on tree-covering, their unified proofs assuming the packing-covering theorem.

Week 8

  • October 9: No class (Fall break)
  • October 11: Midterm.

Week 9

  • October 16. The packing-covering theorem, the Gallai–Milgram theorem, Dilworth's theorem.
  • October 18. The structure of 2-connected graphs, blocks, the block graph.

Week 10

  • October 23. The structure of 3-connected graphs, Tutte's wheel theorem, vertex space and cycle space.
  • October 25. The cycle space of 3-connected graphs, Menger's theorem, its corollaries, and its global version.

Week 11

  • October 30. Planarity from a topological viewpoint: regions and frontiers, the Jordan curve theorem for polygons, the three-arcs lemma, the region lemma, and the definition of plane graphs.
  • November 1. Faces of plane graphs, inner and outer faces, boundaries of faces, incidence, plane forests having exactly one face, and basic lemmas relating cycles and face boundaries.

Week 12

  • November 6. Face boundaries in 2-connected and 3-connected plane graphs, maximal planar graphs and triangulations, Euler's formula, and the bounds on the number of edges in planar graphs, including the non-planarity of K5K_5 and K3,3K_{3,3}.
  • November 8. Kuratowski's theorem: characterizing planar graphs via subdivisions of K5K_5 and K3,3K_{3,3}, together with the structure of edge-maximal non-planar graphs.

Week 13

  • November 13. Cycle spaces over F2\mathbb F_2, sparse families of cycles, facial cycles in plane graphs, and MacLane's theorem characterizing planarity by the existence of a sparse basis of the cycle space.
  • November 15. Plane duality for connected plane multigraphs, cycles and bonds under duality, abstract duals, and Whitney's theorem characterizing planar graphs by the existence of an abstract dual.

Week 14

  • November 20. Edge coloring, König's line-coloring theorem for bipartite graphs, Vizing's theorem, list coloring, and Thomassen's theorem that every planar graph is 5-choosable.
  • November 22. List edge-coloring of bipartite graphs via Galvin's theorem, perfect graphs, chordal graphs and their decomposition by clique sums, Lovász's perfect graph theorem, and the statement of the strong perfect graph theorem.

Week 15

  • November 27. Student presentations (15 min each + 3 min Q&A): Michael Buchanan — generalized Menger's theorem; Hricha Acharya — graph embedding on other surfaces; Devansh Patel — perfect graphs; Garrett Parzych — Ramsey theory.
  • November 29. Student presentations (25 min or 15 min + 3 min Q&A): Kasturi Barkataki — connection with knot theory (15 min); Taman Truong and Theodore Gossett — Brooks's theorem (25 min); Skand Parvatikar and Dennis Liu — rainbow results (25 min).