Advertisement

Discrete & Computational Geometry

, Volume 2, Issue 4, pp 327–343 | Cite as

A lower bound for the optimal crossing-free Hamiltonian cycle problem

  • Ryan B. Hayward
Article

Abstract

Consider a drawing in the plane ofK n , the complete graph onn vertices. If all edges are restricted to be straight line segments, the drawing is called rectilinear. Consider a Hamiltonian cycle in a drawing ofK n . If no pair of the edges of the cycle cross, it is called a crossing-free Hamiltonian cycle (cfhc). Let Φ(n) represent the maximum number of cfhc's of any drawing ofK n , and\(\bar \Phi\)(n) the maximum number of cfhc's of any rectilinear drawing ofK n . The problem of determining Φ(n) and\(\bar \Phi\)(n), and determining which drawings have this many cfhc's, is known as the optimal cfhc problem. We present a brief survey of recent work on this problem, and then, employing a recursive counting argument based on computer enumeration, we establish a substantially improved lower bound for Φ(n) and\(\bar \Phi\)(n). In particular, it is shown that\(\bar \Phi\)(n) is at leastk × 3.2684 n . We conjecture that both Φ(n) and\(\bar \Phi\)(n) are at mostc × 4.5 n .

Keywords

Complete Graph Discrete Comput Geom Straight Line Segment Computer Enumeration Dominant Eigenvalue 
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. [A]
    S. Akl, A lower bound on the maximum number of crossing-free Hamilton cycles in a rectilinear drawing ofK n,Ars Combin. 7 (1979), 7–18.MathSciNetMATHGoogle Scholar
  2. [ACNS]
    M. Ajtai, V. Chvátal, M. Newborn, and E. Szemerédi, Crossing-free subgraphs,Ann. Discrete Math. 12 (1982), 9–12.MATHGoogle Scholar
  3. [EG]
    P. Erdos and R. K. Guy, Crossing number problems,Amer. Math. Monthly 80 (1973), 52–58.MathSciNetCrossRefGoogle Scholar
  4. [G]
    R. K. Guy, Unsolved problems,Amer. Math. Monthly 88 (1981), 757.CrossRefGoogle Scholar
  5. [H]
    R. B. Hayward, The Optimal Crossing-Free Hamilton Cycle Problem for Planar Drawings of the Complete Graph, M.Sc. thesis, Queen's University, Kingston, Ontario, 1982.Google Scholar
  6. [NM]
    M. Newborn and W. O. J. Moser, Optimal crossing-free Hamiltonian circuit drawings ofK n,J. Combin. Theory Ser. B 29 (1980), 13–26.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • Ryan B. Hayward
    • 1
  1. 1.School of Computer ScienceMcGill UniversityMontrealCanada

Personalised recommendations