On 4-Regular Planar Hamiltonian Graphs
David High · TopSCHOLAR (Western Kentucky University) · 2006
In order to research knots with large crossing numbers, one would like to be able to select a random knot from the set of all knots with n crossings with as close to uniform probability as possible. The underlying graph of a knot diagram can be viewed as a 4-regular planar graph. The existence of a Hamiltonian cycle in such a graph is necessary in order to use the graph to compute an upper bound on rope length for a given knot. The algorithm to generate such graphs is discussed and an exact count of the number of graphs is obtained. In order to allow for the existence of such a count, a somewhat technical definition of graph equivalence is used. The main result of the thesis is the asymptotic results of how fast the number of graphs with n vertices (crossings) grows with n.