A lower bound for the optimal crossing-free Hamiltonian cycle problem
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 .
KeywordsComplete Graph Discrete Comput Geom Straight Line Segment Computer Enumeration Dominant Eigenvalue
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