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 grap...

Highlights

  • Flip graphs are fundamental structures associated with families of geometric objects such as triangulations, plane spanning trees, non-crossing matchings, partitions or dissections

  • Are some important Gray code results in the geometric realm: Hernando, Hurtado and Noy [16] proved the existence of a Hamilton cycle in the flip graph of non-crossing perfect matchings on a set of 2m points in convex position for every even m ≥ 4

  • In the following theorem we summarize the results of this setting. 38:8 Rainbow Cycles in Flip Graphs

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Summary

Introduction

Flip graphs are fundamental structures associated with families of geometric objects such as triangulations, plane spanning trees, non-crossing matchings, partitions or dissections. A challenging algorithmic problem in this direction is to efficiently compute a minimal sequence of flips that transforms two given triangulations into each other, see [24, 29] These questions involving the diameter of the flip graph become even harder when the n points are not in convex, but in general position, see e.g. Are some important Gray code results in the geometric realm: Hernando, Hurtado and Noy [16] proved the existence of a Hamilton cycle in the flip graph of non-crossing perfect matchings on a set of 2m points in convex position for every even m ≥ 4. In the flip graph of triangulations GTn, we ask for the existence of a cycle with the property that each inner edge of the triangulation appears (and disappears) exactly once This notion of rainbow cycles extends in a natural way to all the other flip graphs discussed before, see Figure 2

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Related work
Outline of this paper
Triangulations
Spanning trees
Matchings
Permutations
Subsets
Open problems
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