Let G ¼ šVšGŽ; EšGŽŽ be a connected undirected graph without loops and multiple edges, where VšGŽ is the set of vertices and EšGŽ is the set of edges. For x 2 VšGŽ, the degree of x in G, denoted by degGšxŽ, implies the number of edges incident with x, and the neighbourhood of x in G, denoted by NGšxŽ, implies the set of vertices adjacent to x in G; thus degGšxŽ¼jNGšxŽj. Definitions and notation on graphs which are not given in this note can be found in [1, 2]. A graph G is said to be a šd; fŽ-regular planar graph if it satisfies the following:(1) G is planar and already embedded in the plane;(2) G is regular in the ordinary sense, that is, degGšxŽ ¼ d for every vertex x 2 VšGŽ and d! 3;(3) Every face R is an f-gon, that is, dšRŽ ¼ f for every face R 2 FšGŽ, where FšGŽ is the set of faces of G, dšRŽ is the number of edges of the boundary of R and f! 3.

Set Hšd; fŽ ¼ 4 Ą šd Ą 2Žš f Ą 2Ž. It is well known that, if Hšd; fŽ> 0, G is one of the platonic graphs, which are finite regular polyhedra. If Hšd; fŽ 0, G is an infinite graph. In this note, we deal with only infinite šd; fŽ-regular planar graphs satisfying the following conditions: