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

Abstract

Consider a drawing in the plane of Kn, the complete graph on n vertices. If all edges are restricted to be straight line segments, the drawing is called rectilinear. Consider a Hamiltonian cycle in a drawing of Kn. 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 of Kn, and {Mathematical expression}(n) the maximum number of cfhc's of any rectilinear drawing of Kn. The problem of determining Φ(n) and {Mathematical expression}(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 {Mathematical expression}(n). In particular, it is shown that {Mathematical expression}(n) is at least k × 3.2684n. We conjecture that both Φ(n) and {Mathematical expression}(n) are at most c × 4.5n. © 1987 Springer-Verlag New York Inc.