Abstract
In this paper we investigate the structure of flip graphs on non-crossing perfect matchings in the plane. Consider all non-crossing straight-line perfect matchings on a set of 2n points that are placed equidistantly on the unit circle. The graph \({\mathcal H}_n\) has those matchings as vertices, and an edge between any two matchings that differ in replacing two matching edges that span an empty quadrilateral with the other two edges of the quadrilateral, provided that the quadrilateral contains the center of the unit circle. We show that the graph \({\mathcal H}_n\) is connected for odd n, but has exponentially many small connected components for even n, which we characterize and count via Catalan and generalized Narayana numbers. For odd n, we also prove that the diameter of \({\mathcal H}_n\) is linear in n. Furthermore, we determine the minimum and maximum degree of \({\mathcal H}_n\) for all n, and characterize and count the corresponding vertices. Our results imply the non-existence of certain rainbow cycles, and they answer several open questions and conjectures raised in a recent paper by Felsner, Kleist, Mütze, and Sering.
Torsten Mütze is also affiliated with the Faculty of Mathematics and Physics, Charles University Prague, Czech Republic, and he was supported by Czech Science Foundation grant GA 19-08554S and by German Science Foundation grant 413902284.
Martin Pergel was also supported by Czech Science Foundation grant GA 19-08554S.
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Acknowledgements
We thank the anonymous reviewers of the extended abstract of this paper, who provided many insightful comments. In particular, one referee’s observation about our proof of Lemma 3 improved our previous upper bound on the diameter of \({\mathcal H}_n\) for odd n from \({\mathcal O}(n\log n)\) to \({\mathcal O}(n)\) (recall Theorem 4).
Figure 3 was obtained by slightly modifying Fig. 10 from [9], and the authors of this paper kindly provided us with the source code of their figure.
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Milich, M., Mütze, T., Pergel, M. (2020). On Flips in Planar Matchings. In: Adler, I., Müller, H. (eds) Graph-Theoretic Concepts in Computer Science. WG 2020. Lecture Notes in Computer Science(), vol 12301. Springer, Cham. https://doi.org/10.1007/978-3-030-60440-0_17
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