Abstract
A matching is compatible to two or more labeled point sets of size n with labels \(\{1,\dots ,n\}\) if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled convex sets of n points there exists a compatible matching with \(\lfloor \sqrt{2n}\rfloor \) edges. More generally, for any \(\ell \) labeled point sets we construct compatible matchings of size \(\varOmega (n^{1/\ell })\). As a corresponding upper bound, we use probabilistic arguments to show that for any \(\ell \) given sets of n points there exists a labeling of each set such that the largest compatible matching has \(\mathcal {O}(n^{2/(\ell +1)})\) edges. Finally, we show that \(\varTheta (\log n)\) copies of any set of n points are necessary and sufficient for the existence of a labeling such that any compatible matching consists only of a single edge.
A.A. funded by the Marie Skłodowska-Curie grant agreement No. 754411. Z.M. partially funded by Wittgenstein Prize, Austrian Science Fund (FWF), grant no. Z 342-N31. I.P., D.P., and B.V. partially supported by FWF within the collaborative DACH project Arrangements and Drawings as FWF project I 3340-N35. A.P. supported by a Schrödinger fellowship of the FWF: J-3847-N35. J.T. partially supported by ERC Start grant no. (279307: Graph Games), FWF grant no. P23499-N23 and S11407-N23 (RiSE).
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Notes
- 1.
The combinatorial embedding fixes the cyclic order of incident edges for each vertex.
- 2.
https://www.imo-official.org/problems/IMO2017SL.pdf, Problem C4.
- 3.
Two edges are independent if they do not share an endpoint.
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Aichholzer, O. et al. (2021). On Compatible Matchings. In: Uehara, R., Hong, SH., Nandy, S.C. (eds) WALCOM: Algorithms and Computation. WALCOM 2021. Lecture Notes in Computer Science(), vol 12635. Springer, Cham. https://doi.org/10.1007/978-3-030-68211-8_18
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