Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, December 14, 2021, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Hung Hoang

Gioan's Theorem for complete bipartite graphs

For a drawing of a labeled graph, the rotation of a vertex is the cyclic order of its incident edges, presented by the labels of their other endpoints. The rotation system of the drawing is the collection of the rotations of all vertices. A drawing is simple if each pair of edges has at most one common point. Gioan's Theorem states that for any two simple drawings of the complete graph K_n with the same rotation system, we can transform one drawing into the other by a sequence of triangle flips. Intuitively, this operation refers to the act of moving one edge of a triangular cell formed by three pairwise crossing edges over the opposite vertex of the cell.

In this talk, I will present the generalization of Gioan's Theorem to complete bipartite graphs K_m,n. This holds when the two drawings share the same extended rotation system. This notion extends the rotation system to include also the rotations around crossings. Note that the theorem does not hold if the graph is slightly sparser: For a family of graphs where we remove two edges from K_m,n, we can show that there exist two simple drawings that cannot be transformed into each other via triangle flips.

Joint work with Oswin Aichholzer, Man-Kwun Chiu, Michael Hoffmann, Yannic Maus, Birgit Vogtenhuber, and Alexandra Weinberger

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