Abstrakt
The flip graph of triangulations has as vertices all triangulations of a convex n-gon and an edge between any two triangulations that differ in exactly one edge. An r-rainbow cycle in this graph is a cycle in which every inner edge of the triangulation appears exactly r times.
This notion of a rainbow cycle extends in a natural way to other flip graphs. In this paper we investigate the existence of r-rainbow cycles for three different flip graphs on classes of geometric objects: the aforementioned flip graph of triangulations of a convex n-gon, the flip graph of plane trees on an arbitrary set of n points, and the flip graph of noncrossing perfect matchings on a set of n points in convex position.
In addition, we consider two flip graphs on classes of nongeometric objects: the flip graph of permutations of {1, 2, ..., n} and the flip graph of k-element subsets of {1, 2, ..., n}. In each of the five settings, we prove the existence and nonexistence of rainbow cycles for different values of r, n, and k.