Document Open Access Logo

Flips in Colorful Triangulations

Authors Rohan Acharya, Torsten Mütze , Francesco Verciani

PDF
Thumbnail PDF

File

LIPIcs.GD.2024.30.pdf
  • Filesize: 2.43 MB
  • 20 pages

Document Identifiers

Author Details

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

Funding

This work was supported by Czech Science Foundation grant GA 22-15272S. The second and third author participated in the workshop "Combinatorics, Algorithms and Geometry" in March 2024, which was funded by German Science Foundation grant 522790373.

Cite AsGet BibTex

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

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. R. Acharya, T. Mütze, and F. Verciani. Flips in colorful triangulations. arXiv:2406.03783, June 2024. URL: https://doi.org/10.48550/arXiv.2406.03783.
  2. K. Baur, D. Bergerova, J. Voon, and L. Xu. Flip graphs of coloured triangulations of convex polygons. arXiv:2402.06546, February 2024. Google Scholar
  3. L. A. Berry, B. Reed, A. Scott, and D. R. Wood. A logarithmic bound for the chromatic number of the associahedron. arXiv:1811.08972, November 2018. Google Scholar
  4. M. Buck and D. Wiedemann. Gray codes with restricted density. Discrete Math., 48(2-3):163-171, 1984. URL: https://doi.org/10.1016/0012-365X(84)90179-1.
  5. J. Cardinal, A. Merino, and T. Mütze. Combinatorial generation via permutation languages. IV. Elimination trees. To appear in ACM Transactions on Algorithms; preprint available on arXiv:2106.16204, 2024. Google Scholar
  6. C. Ceballos, F. Santos, and G. M. Ziegler. Many non-equivalent realizations of the associahedron. Combinatorica, 35(5):513-551, 2015. URL: https://doi.org/10.1007/s00493-014-2959-9.
  7. The Combinatorial Object Server: Generate k-ary trees and dissections. URL: http://www.combos.org/kary.
  8. T. Dvořák. Hamiltonian cycles with prescribed edges in hypercubes. SIAM J. Discrete Math., 19(1):135-144, 2005. URL: https://doi.org/10.1137/S0895480103432805.
  9. P. Eades, M. Hickey, and R. C. Read. Some Hamilton paths and a minimal change algorithm. J. Assoc. Comput. Mach., 31(1):19-29, 1984. URL: https://doi.org/10.1145/2422.322413.
  10. R. Fabila-Monroy, D. Flores-Peñaloza, C. Huemer, F. Hurtado, J. Urrutia, and D. R. Wood. On the chromatic number of some flip graphs. Discrete Math. Theor. Comput. Sci., 11(2):47-56, 2009. URL: https://doi.org/10.46298/DMTCS.460.
  11. S. Gravier and C. Payan. Flips signés et triangulations d'un polygone. European J. Combin., 23(7):817-821, 2002. URL: https://doi.org/10.1006/eujc.2002.0601.
  12. E. Hartung, H. P. Hoang, T. Mütze, and A. Williams. Combinatorial generation via permutation languages. I. Fundamentals. Trans. Amer. Math. Soc., 375(4):2255-2291, 2022. URL: https://doi.org/10.1090/tran/8199.
  13. H. P. Hoang and T. Mütze. Combinatorial generation via permutation languages. II. Lattice congruences. Israel J. Math., 244(1):359-417, 2021. URL: https://doi.org/10.1007/s11856-021-2186-1.
  14. C. Hohlweg and C. E. M. C. Lange. Realizations of the associahedron and cyclohedron. Discrete Comput. Geom., 37(4):517-543, 2007. URL: https://doi.org/10.1007/s00454-007-1319-6.
  15. C. Huemer, F. Hurtado, and J. Pfeifle. The rotation graph of k-ary trees is Hamiltonian. Inform. Process. Lett., 109(2):124-129, 2008. URL: https://doi.org/10.1016/j.ipl.2008.09.013.
  16. F. Hurtado and M. Noy. Graph of triangulations of a convex polygon and tree of triangulations. Comput. Geom., 13(3):179-188, 1999. URL: https://doi.org/10.1016/S0925-7721(99)00016-4.
  17. J.-L. Loday. Realization of the Stasheff polytope. Arch. Math. (Basel), 83(3):267-278, 2004. URL: https://doi.org/10.1007/s00013-004-1026-y.
  18. J. M. Lucas. The rotation graph of binary trees is Hamiltonian. J. Algorithms, 8(4):503-535, 1987. URL: https://doi.org/10.1016/0196-6774(87)90048-4.
  19. J. M. Lucas, D. Roelants van Baronaigien, and F. Ruskey. On rotations and the generation of binary trees. J. Algorithms, 15(3):343-366, 1993. URL: https://doi.org/10.1006/jagm.1993.1045.
  20. A. Merino and T. Mütze. Combinatorial generation via permutation languages. III. Rectangulations. Discrete Comput. Geom., 70(1):51-122, 2023. URL: https://doi.org/10.1007/s00454-022-00393-w.
  21. V. Pilaud, F. Santos, and G. M. Ziegler. Celebrating Loday’s associahedron. Arch. Math. (Basel), 121(5-6):559-601, 2023. URL: https://doi.org/10.1007/s00013-023-01895-6.
  22. L. Pournin. The diameter of associahedra. Adv. Math., 259:13-42, 2014. URL: https://doi.org/10.1016/j.aim.2014.02.035.
  23. B. E. Sagan. Proper partitions of a polygon and k-Catalan numbers. Ars Combin., 88:109-124, 2008. Google Scholar
  24. D. D. Sleator, R. E. Tarjan, and W. P. Thurston. Rotation distance, triangulations, and hyperbolic geometry. J. Amer. Math. Soc., 1(3):647-681, 1988. URL: https://doi.org/10.2307/1990951.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact