Flips in Colorful Triangulations

Authors Rohan Acharya, Torsten Mütze , Francesco Verciani



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Rohan Acharya
  • Department of Computer Science, University of Warwick, Coventry, UK
Torsten Mütze
  • Institut für Mathematik, Universität Kassel, Germany
  • Department of Theoretical Computer Science and Mathematical Logic, Charles University, Prague, Czech Republic
Francesco Verciani
  • Institut für Mathematik, Universität Kassel, Germany

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Rohan Acharya, Torsten Mütze, and Francesco Verciani. Flips in Colorful Triangulations. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 30:1-30:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.GD.2024.30

Abstract

The associahedron is the graph G_N that has as nodes all triangulations of a convex N-gon, and an edge between any two triangulations that differ in a flip operation. A flip removes an edge shared by two triangles and replaces it by the other diagonal of the resulting 4-gon. In this paper, we consider a large collection of induced subgraphs of G_N obtained by Ramsey-type colorability properties. Specifically, coloring the points of the N-gon red and blue alternatingly, we consider only colorful triangulations, namely triangulations in which every triangle has points in both colors, i.e., monochromatic triangles are forbidden. The resulting induced subgraph of G_N on colorful triangulations is denoted by F_N. We prove that F_N has a Hamilton cycle for all N ≥ 8, resolving a problem raised by Sagan, i.e., all colorful triangulations on N points can be listed so that any two cyclically consecutive triangulations differ in a flip. In fact, we prove that for an arbitrary fixed coloring pattern of the N points with at least 10 changes of color, the resulting subgraph of G_N on colorful triangulations (for that coloring pattern) admits a Hamilton cycle. We also provide an efficient algorithm for computing a Hamilton path in F_N that runs in time O(1) on average per generated node. This algorithm is based on a new and algorithmic construction of a tree rotation Gray code for listing all n-vertex k-ary trees that runs in time O(k) on average per generated tree.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
Keywords
  • Flip graph
  • associahedron
  • triangulation
  • binary tree
  • vertex coloring
  • Hamilton cycle
  • Gray code

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References

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