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