Advertisement

Drawing Hamiltonian Cycles with No Large Angles

  • Adrian Dumitrescu
  • János Pach
  • Géza Tóth
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5849)

Abstract

Let n ≥ 4 be even. It is shown that every set S of n points in the plane can be connected by a (possibly self-intersecting) spanning tour (Hamiltonian cycle) consisting of n straight line edges such that the angle between any two consecutive edges is at most 2π/3. For n = 4 and 6, this statement is tight. It is also shown that every even-element point set S can be partitioned into at most two subsets, S 1 and S 2, each admitting a spanning tour with no angle larger than π/2. Fekete and Woeginger conjectured that for sufficiently large even n, every n-element set admits such a spanning tour. We confirm this conjecture for point sets in convex position. A much stronger result holds for large point sets randomly and uniformly selected from an open region bounded by finitely many rectifiable curves: for any ε> 0, these sets almost surely admit a spanning tour with no angle larger than ε.

Keywords

Rotation Angle Hamiltonian Cycle Symmetric Convex Body Orthogonal Line Starting Vertex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Ackerman, E., Aichholzer, O., Keszegh, B.: Improved upper bounds on the reflexivity of point sets. Computational Geometry: Theory and Applications 42(3), 241–249 (2009)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Aichholzer, O., Hackl, T., Hoffmann, M., Huemer, C., Pór, A., Santos, F., Speckman, B., Vogtenhuber, B.: Maximizing maximal angles for plane straight-line graphs. In: Dehne, F., Sack, J.-R., Zeh, N. (eds.) WADS 2007. LNCS, vol. 4619, pp. 458–469. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  3. 3.
    Arkin, E.M., Bender, M.A., Demaine, E.D., Fekete, S.P., Mitchell, J.S.B., Sethia, S.: Optimal covering tours with turn costs. SIAM Journal on Computing 35(3), 531–566 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Arkin, E.M., Fekete, S., Hurtado, F., Mitchell, J., Noy, M., Sacristán, V., Sethia, S.: On the reflexivity of point sets. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds.) Discrete and Computational Geometry: The Goodman-Pollack Festschrift, pp. 139–156. Springer, Heidelberg (2003)Google Scholar
  5. 5.
    Bárány, I., Pór, A., Valtr, P.: Paths with no small angles. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds.) LATIN 2008. LNCS, vol. 4957, pp. 654–663. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Chan, T.: Remarks on k-level algorithms in the plane, manuscript, Univ. of Waterloo (1999)Google Scholar
  7. 7.
    Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, 2nd edn. McGraw-Hill, New York (2001)zbMATHGoogle Scholar
  8. 8.
    Courant, R., Robbins, H.: What is Mathematics? An Elementary Approach to Ideas and Methods. Oxford University Press, Oxford (1979)Google Scholar
  9. 9.
    Dey, T.K.: Improved bounds on planar k-sets and related problems. Discrete & Computational Geometry 19, 373–382 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Fekete, S.P., Woeginger, G.J.: Angle-restricted tours in the plane. Computational Geometry: Theory and Applications 8(4), 195–218 (1997)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Kynćl, J.: Personal communication (2009)Google Scholar
  12. 12.
    Lovász, L.: On the number of halving lines. Ann. Univ. Sci. Budapest, Eötvös, Sec. Math. 14, 107–108 (1971)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Adrian Dumitrescu
    • 1
  • János Pach
    • 2
  • Géza Tóth
    • 3
  1. 1.Department of Computer ScienceUniversity of Wisconsin-MilwaukeeUSA
  2. 2.Ecole Polytechnique Fédérale de Lausanne and City CollegeNew York
  3. 3.Alfred Rényi Institute of MathematicsBudapestHungary

Personalised recommendations