Jan Kyncl, Pavel Valtr, KAM (contact: surname - at - kam.mff.cuni.cz)

Tuesday at 14:00 in S10

If there is at least one student who does not understand Czech, the lectures will be in English; but you may ask questions in Czech during the lectures.

**Topics planned:**

- Introduction: drawings of graphs, topological graphs, geometric graphs
- The Hanani–Tutte theorem (weak, strong, and unified) and an algebraic algorithm for planarity testing
- The Jordan curve theorem and the nonplanarity of K_{3,3} (Thomassen's proof)
- Thrackles, thrackle conjecture
- Extremal results for topological and geometric graphs without forbidden substructures (k disjoint edges, k crossing edges)
- Structural properties of complete topological graphs (a Ramsey-type theorem)

**Literature:** (will be updated during the semester)

- Chapters from lecture notes for Janos Pach's course at EPFL with a partial overlap with this course:
- Unified Hanani–Tutte theorem
- C. Thomassen, The Jordan–Schönflies theorem and the classification of surfaces - An elementary proof of the Jordan curve theorem
- János Pach and Ethan Sterling, Conway's Conjecture for Monotone Thrackles
- Géza Tóth, Note on Geometric Graphs - Upper bound on the number of edges in geometric graphs with no k+1 disjoint edges
- On Geometric Graphs with No k Pairwise Parallel Edges, Graph drawing with no k pairwise crossing edges - Upper bound on the number of edges in geometric graphs with no k pairwise crossing edges

- Basic notions: topological graph, simple topological graph, geometric graph, plane graph, planar graph
- Statements of the Jordan curve theorem and the Jordan–Schönflies theorem
- Characterization of planar graphs: statements of Kuratovski's theorem, Wagner's theorem, Euler's formula, upper bound on the number of edges, Fáry's theorem
- Motivating question: what is the maximum possible number of edges in a simple topological graph (or a geometric graph) with no pair of disjoint edges?
- A "very weak" Hanani–Tutte theorem with a sketch of a proof

- The (strong) Hanani–Tutte theorem with a sketch of a proof using the Kuratowski theorem
- Algebraic algorithm for planarity testing, using the strong Hanani–Tutte theorem

- The weak Hanani–Tutte theorem, two proofs by redrawing
- Unified Hanani–Tutte theorem, first part of a proof (induction step for graphs with vertex connectivity 0, or 1 for the strong variant)

- Unified Hanani–Tutte theorem: proof for 3-connected graphs, sketch of a proof of the induction step for graphs with vertex connectivity 1 or 2

- Basic topological definitions: path-connected set, open set, closed set, interior, closure, boundary; simple polygonal arc, simple polygonal closed curve
- The Jordan curve theorem, Thomassen's proof, part 1: planar graphs have polygonal plane drawings, polygonal Jordan curve theorem, extension to polygonal drawings of K_2,3, nonplanarity of K_3,3 [Lemma 2.1 - Lemma 2.5 in the paper]

- The complement of a simple closed curve in the plane has at least two regions. [Proposition 2.6]
- The Jordan curve theorem, Thomassen's proof, part 2, including polygonal Euler's formula for 2-connected graphs and the Jordan arc theorem: the complement of a simple arc in the plane is connected. [Lemma 2.7 - Proposition 2.11]

- The complement of a simple closed curve in the plane has at most two regions. [Theorem 2.12]