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)


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, S1 and S2, 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.


  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)MATHMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle 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)MATHGoogle 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)MATHCrossRefMathSciNetGoogle 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)MATHMathSciNetGoogle 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