The flip graph of triangulations has as vertices all triangulations of a convex @math -gon, and an edge between any two triangulations that differ in exactly one edge. An @math -rainbow cycle in this graph is a cycle in which every inner edge of the triangulation appears exactly @math times. This notion of a rainbow cycle extends in a natural way to other flip graphs. In this paper we investigate the existence of @math -rainbow cycles for three different flip graphs on geometric classes of objects: the aforementioned flip graph of triangulations of a convex @math -gon, the flip graph of plane trees on an arbitrary set of @math points, and the flip graph of non-crossing perfect matchings on a set of @math points in convex position. In addition, we consider two flip graphs on classes of non-geometric objects: the flip graph of permutations of @math and the flip graph of @math -element subsets of @math . In each of the five settings, we prove the existence and non-existence of rainbow cycles for different values of @math , @math and @math .
@cite_5: We prove two results regarding cycles in properly edge-colored graphs. First, we make a small improvement to the recent breakthrough work of Alon, Pokrovskiy and Sudakov who showed that every properly edge-colored complete graph @math on @math vertices has a rainbow cycle on at least @math vertices, by showing that @math has a rainbow cycle on at least @math vertices. Second, by modifying the argument of Hatami and Shor which gives a lower bound for the length of a partial transversal in a Latin Square, we prove that every properly colored complete graph has a Hamilton cycle in which at least @math different colors appear. For large @math , this is an improvement of the previous best known lower bound of @math of Andersen.
@cite_7: 
@cite_11: A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph @math has a rainbow Hamiltonian path. Although this conjecture turned out to be false, it was widely believed that such a colouring always contains a rainbow cycle of length almost @math . In this paper, improving on several earlier results, we confirm this by proving that every properly edge-coloured @math has a rainbow cycle of length @math . One of the main ingredients of our proof, which is of independent interest, shows that a random subgraph of a properly edge-coloured @math formed by the edges of a random set of colours has a similar edge distribution as a truly random graph with the same edge density. In particular it has very good expansion properties.
Rainbow cycles and paths have also been studied in graphs other than flip graphs. A well-known conjecture in this context due to Andersen @cite_7 asserts that every properly edge-coloured complete graph on @math vertices has a rainbow path of length @math , i.e., a path that has distinct colors along each of its edges. Progress towards resolving this conjecture was recently made by Alon, Pokrovskiy and Sudakov @cite_11 , and Balogh and Molla @cite_5 .