Generating 4-regular Hamiltonian Plane Graphs

This paper describes the study of a special class of 4-regular plane graphs which are Hamiltonian. These graphs are of special interest in knot theory. An algorithm is presented that randomly generates such graphs with n vertices with a fixed (and oriented) Hamiltonian cycle in O(n) time. An exact count of the number of such graphs with n vertices is obtained and the asymptotic growth rate of this number is determined. Numerical evidence is presented to show that the algorithm can be modified to generate these graphs with a near uniform probability. This can be considered as a first step in generating large random knots without bias.