Noncrossing Hamiltonian Paths in Geometric Graphs

  • Jakub Černý
  • Zdeněk Dvořák
  • Vít Jelínek
  • Jan Kára
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2912)

Abstract

A geometric graph is a graph embedded in the plane in such a way that vertices correspond to points in general position and edges correspond to segments connecting the appropriate points. A noncrossing Hamiltonian path in a geometric graph is a Hamiltonian path which does not contain any intersecting pair of edges. In the paper, we study a problem asked by Micha Perles: Determine a function h, where h(n) is the largest number k such that when we remove arbitrary set of k edges from a complete geometric graph on n vertices, the resulting graph still has a noncrossing Hamiltonian path. We prove that \(h(n)=\Omega(\sqrt{n})\). We also determine the function exactly in case when the removed edges form a star or a matching, and give asymptotically tight bounds in case they form a clique.

References

  1. 1.
    Agarwal, P.K., Aronov, B., Pach, J., Pollack, R., Sharir, M.: Quasi-planar graphs have a linear number of edges. Combinatorica 17, 1–9 (1997)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Pach, J.: Geometric graph theory, Surveys in combinatorics, 1999 (Canterbury). London Math. Soc. Lecture Note Ser., vol. 267, pp. 167–200. Cambridge Univ. Press, Cambridge (1999)Google Scholar
  3. 3.
    Pach, J., Agarwal, P.K.: Combinatorial Geometry. Wiley Interscience, New York (1995)MATHGoogle Scholar
  4. 4.
    Pach, J., Radoičić, R., Tardos, G., Tóth, G.: Geometric graphs with no self–intersecting path of length three. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 295–311. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Pinchasi, R., Radoičić, R.: On the number of edges in geometric graphs with no selfintersecting cycle of length 4. In: Proc. 19th Annual Symposium on Computational Geometry (submitted)Google Scholar
  6. 6.
    Tardos, G.: On the number of edges in a geometric graph with no short self-intersecting paths (in preparation)Google Scholar
  7. 7.
    Valtr, P.: Graph drawings with no k pairwise crossing edges. In: DiBattista, G. (ed.) GD 1997. LNCS, vol. 1353, pp. 205–218. Springer, Heidelberg (1997)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Jakub Černý
    • 1
  • Zdeněk Dvořák
    • 1
  • Vít Jelínek
    • 1
  • Jan Kára
    • 1
  1. 1.Department of Applied MathematicsCharles UniversityPraha 1

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