License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2018.38
URN: urn:nbn:de:0030-drops-87514
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8751/
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Felsner, Stefan ; Kleist, Linda ; Mütze, Torsten ; Sering, Leon

Rainbow Cycles in Flip Graphs

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LIPIcs-SoCG-2018-38.pdf (1.0 MB)

Abstract

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.

BibTeX - Entry

@InProceedings{felsner_et_al:LIPIcs:2018:8751,
  author =	{Stefan Felsner and Linda Kleist and Torsten M{\"u}tze and Leon Sering},
  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 =	{Bettina Speckmann and Csaba D. T{\'o}th},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8751},
  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}
}

Keywords: flip graph, cycle, rainbow, Gray code, triangulation, spanning tree, matching, permutation, subset, combination
Collection: 34th International Symposium on Computational Geometry (SoCG 2018)
Issue Date: 2018
Date of publication: 08.06.2018

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