On flips in planar matchings

In this paper we investigate the structure of flip graphs on non-crossing perfect matchings in plane. Specifically, consider all straight-line a set 2n points that are placed equidistantly unit circle. A operation such matching replaces two edges span an empty quadrilateral with other quadrilateral, and is called centered if contains center The graph Gn has those as vertices, edge between any differ flip, it known to have many interesting properties. focus spanning subgraph Hn obtained by taking correspond flips, omitting non-centered flips. We show connected for odd n, but exponentially small components even which characterize count via Catalan generalized Narayana numbers. For also prove diameter linear n. Furthermore, determine minimum maximum degrees corresponding vertices. Our results imply non-existence certain rainbow cycles Gn, they resolve several open questions conjectures raised recent Felsner, Kleist, Mütze, Sering.