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

中文翻译：

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关于平面匹配中的翻转

摘要 在本文中，我们研究了平面中非交叉完美匹配的翻转图的结构。具体来说，考虑在单位圆上等距放置的一组 2 n 个点上的所有非交叉直线完美匹配。对这种匹配的翻转操作将跨越空四边形的两条匹配边替换为四边形的其他两条边，如果四边形包含单位圆的中心，则翻转称为居中。图 G n 将这些匹配项作为顶点，以及在翻转中不同的任何两个匹配项之间的边，并且已知它具有许多有趣的属性。在本文中，我们关注G n 的生成子图H n ，它是通过取所有对应于中心翻转的边，省略对应于非中心翻转的边而获得的。我们证明了图 H n 是连接奇数 n 的，但对于偶数 n 具有指数级的许多小连通分量，我们通过 Catalan 和广义 Narayana 数来表征和计数。对于奇数 n ，我们还证明了 H n 的直径在 n 中是线性的。此外，我们确定所有 n 的 H n 的最小和最大度数，并对相应的顶点进行表征和计数。我们的结果意味着 G n 中不存在某些彩虹周期，并且它们解决了 Felsner、Kleist、Mutze 和 Sering 最近在一篇论文中提出的几个悬而未决的问题和猜想。我们为所有 n 确定 H n 的最小和最大度数，并对相应的顶点进行表征和计数。我们的结果意味着 G n 中不存在某些彩虹周期，并且它们解决了 Felsner、Kleist、Mutze 和 Sering 最近在一篇论文中提出的几个悬而未决的问题和猜想。我们为所有 n 确定 H n 的最小和最大度数，并对相应的顶点进行表征和计数。我们的结果意味着 G n 中不存在某些彩虹周期，并且它们解决了 Felsner、Kleist、Mutze 和 Sering 最近在一篇论文中提出的几个悬而未决的问题和猜想。