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

Date and Time: Thursday, August 25, 2022, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Emanuel Seemann

Matchings and Hamiltonian cycles in 1-planar graphs

We investigate the Hamiltonicity and matching properties of 1-planar graphs, a class of beyond planar graphs that allow for at most one crossing per edge. In 2019 Biedl generalized a known Hamiltonicity result by Tutte for planar graphs to 1-planar graphs. Biedl showed that a certain subclass of 4-connected 1-planar graphs is Hamiltonian. We prove that one can prescribe an edge to this Hamiltonian cycle. Using our result we provide sufficient conditions under which all matchings of size 1 and 2 can be extended to a perfect matching in an even 1-planar graph. This generalizes previous results by Plummer for planar graphs and Fujisawa et al. for 1-planar graphs.

The results are part of my master's thesis "Matchings and Hamiltonian cycles in 1-planar graphs" supervised by Michael Hoffmann, Meghana Mallik Reddy and Emo Welzl.

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