Enumeration of Rooted Triangular Maps

Abstract

It is only recently that much has been done with respect to the enumeration of planar maps; much of the pioneering work in this region has been done by Tutte in his census series appearing in the Canadian Journal of Mathematics. We demonstrate a method whereby results from these papers may be combined with results of Brown [1] to enumerate maps in which every country is essentially a topological triangle; the exterior or surrounding ocean, however, may meet more than three countries. More precisely, for the purposes of this paper we define a rooted triangular map T (henceforth abbreviated to map) as the dissection of the interior of a convex (topological) polygon J (other than the loop polygon) in the Euclidean plane E2 into topological triangles by means of a set S of Jordan curves, subject to the following conditions: (1) S contains the edges of J; (2) no vertex of any triangle is an interior point of the edge of another; (3) the ends of each edge are distinct; (4) one vertex of J is distinguished as the root vertex, and one of the edges of J incident with the root vertex is distinguished as the root edge. The vertices and edges of J are referred to as external; remaining vertices and edges of T are internal. Two maps T7 and T7 are isomorphic if there exists a homeomorphism of E2 into itself which carries T7 into T2 and preserves the rooting. As usual, we enumerate classes of isomorphic triangulations. We note that a map T may be interpreted as a graph G = G(T) by defining V(G) as the set of vertices of T and admitting the pair (vl, v2) (vi, v2C V(G)) to E(G) if and only if vi and v2 are the vertices of a triangle in T. If G(T) is a simple graph, that is, if no pair of edges have the same ends, T is said to be a 2-connected triangulation. (The phrase, simple triangulation, has another meaning, cf. Tutte [2] p. 22.) If T is 2-connected and no interior edge has both ends in the boundary J, T is said to be 3-connected. If J is a triangle, the terms are equivalent. A triangulation with m+3 exterior vertices and n interior vertices is said to be of type [n, m]. Three-connected triangulations of type [n, m] m, n> 0 have been enumerated by Tutte [2] who shows their number to be