On flips in planar matchings

In this paper we investigate the structure of a flip graph on non-crossing perfect matchings in the plane. Consider the set of 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 at least linear and at most log-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, Mutze, and Sering.