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


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 .


Complete Graph Discrete Comput Geom Straight Line Segment Computer Enumeration Dominant Eigenvalue 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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