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 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,...,n } and the flip graph of k-element subsets of {1,2,...,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.
@InProceedings{felsner_et_al:LIPIcs.SoCG.2018.38, author = {Felsner, Stefan and Kleist, Linda and M\"{u}tze, Torsten and Sering, Leon}, title = {{Rainbow Cycles in Flip Graphs}}, booktitle = {34th International Symposium on Computational Geometry (SoCG 2018)}, pages = {38:1--38:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-066-8}, ISSN = {1868-8969}, year = {2018}, volume = {99}, editor = {Speckmann, Bettina and T\'{o}th, Csaba D.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.38}, URN = {urn:nbn:de:0030-drops-87514}, doi = {10.4230/LIPIcs.SoCG.2018.38}, annote = {Keywords: flip graph, cycle, rainbow, Gray code, triangulation, spanning tree, matching, permutation, subset, combination} }
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