Rainbow Cycles in Flip Graphs

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 geometric classes of objects:the aforementioned flip graph of triangulations of a convex n-gon, the flip graph of plane spanning trees on an arbitrary set of n points, and the flip graph of non-crossing perfect matchings on a set of n points in convex position.In addition, we consider two flip graphs on classes of non-geometric objects: the flip graph of permutations of {1,2,\dots,n} and the flip graph of k-element subsets of {1,2,\dots,n}.In each of the five settings, we prove the existence and non-existence of rainbow cycles for different values of r, n and k.