A Census of Hamiltonian Polygons

Source abstract

In this paper we deal with trivalent planar maps in which the boundary of each country (or “face“) is a simple closed curve. One vertex is distinguished as the root and its three incident edges are distinguished as the first, second, and third major edges . We determine the average number of Hamiltonian polygons, passing through the first and second major edges, in such a “rooted map” of 2n vertices. Next we consider the corresponding problem for 3-connected rooted maps. In this case we obtain a functional equation from which the average can be computed for small values of n .