For some families of graphs of simplicial 3-polytopes with two types of edges structural properties are described, for other ones their cardinality is determined.
1. ln~oduction
Griinbaum and Motzkin [3], Griinbaum and Zaks [4], and Malkevitch [6] investigated the structural properties of trivalent planar graphs with at most two types of faces. It seems that the knowledge of the structure of such graphs is useful for other reasons as well (cf., e.g. Griinbaum [1, 2], Jucovi~ [5], Owens [7],
Zaks [8]). The dual problem may be formulated as follows: Characterize simplicial planar graphs with at most two types of vertices. (A planar graph is simplicial if all its faces are triangles.)
An edge of a planar graph is of type (i, j; p, q) if its vertices have valencies i and j, and the faces containing it have p and q sides. The present paper deals with graphs of simplicial 3-poltytopes with at most two types of edges. For some families of graphs of this kind structural properties are described, for other ones their cardinality is determined. The terms introduced by Griinbaum [1] are used throughout. It is easy to show that the family of simplicial 3-polytopes with exactly one type of edges consists of the well-known Platonic solids--the tetrahedron, the octahedron and the icosahedron (cf. Fig. 1).
Evidently a simplicial 3-polytopal graph with exactly two types of edges can exist only if its edges are of the type (m, m;3, 3) or (m, n;3, 3), m ¢: n, m 1>3, n ~ 3. Denote the family of all such 3-polytopal graphs by 5¢(m, n).
Griinbaum and Motzkin [3] gave an exact description of all graphs from .9°(6, 3) Griinbaum and Zaks [4] gave a full characterization (in the dual form) of all simplicial planar graphs with edges of the types (6, 6; 3, 3) and (6, 2; 3, 3). We have obtained analogus results for 5¢(5, n) with 3<~ n ~< 12, n~ 5.