A Low Bound for the Optimal Crossing-Free Hamiltonian Cycle Problem

Abstract

Consider a drawing in the plane of K~, 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 K,. If no pair of the edges of the cycle cross, it is called a crossing-free Hamiltonian cycle (cfhc). Let ~(n) represent the maximum number of cfhc's of any drawing of K,, and ~(n) the maximum number of cfhc's of any rectilinear drawing of K,. The problem of determining q~(n) and ~(n), and determining which drawings have this many cfhc's, is known as the optimal of he 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 q~(n) and ~(n), In particular, it is shown that ~(n) is at least k x 3.2684". We conjecture that both • (n) and t~(n) are at most cx4.5 ~. 1. A Survey of the Optimal Crossing-Free Hamilton Cycle Problem Let K, be the complete graph on n vertices. All drawings in this paper are assumed to be drawn in the plane. If all the edges of a drawing of a graph are restricted to be straight line segments, the drawing is said to be rectilinear. By a crossing of a drawing we mean a pair of edges which intersect in the drawing. A Hamiltonian cycle of a graph is a cycle that visits each vertex of the graph exactly once. Consider a particular Hamiltonian cycle in a drawing of K,. If the cycle includes no crossings, it is called a crossing-free Hamiltonian cycle, or a cfhc for short. Let ~(n) (and respectively t~(n)) represent the maximum number of cfhc's of any drawing (respectively rectilinear drawing) of K,. The optimal cfhc problem is to determine ~(n) and ~(n) , and to determine which drawings * This research, part of which was conducted at Queen's University, was supported by an N.S.E.R.C. postgraduate scholarship.

DOI: 10.1007/BF02187887

Cite this paper

@article{Hayward1987ALB, title={A Low Bound for the Optimal Crossing-Free Hamiltonian Cycle Problem}, author={Ryan B. Hayward}, journal={Discrete & Computational Geometry}, year={1987}, volume={2}, pages={327-343} }