Classification of finite fullerenes

Define a 3 -fullerene as a 3-valent map embedded on a surface and consisting of only 5-gonal and 6-gonal faces. Each such object has, say, v vertices, e edges, and / faces of which ps are 5-gons and pf, are 6-gons. 

From Theorem 1.2.3, we know that the Euler characteristic x satisfies to  

For a surface, in which a finite 3-fullerene can be embedded, the number x is, therefore, a non-negative integer. Let us use the topological classification of finite closed maps in Theorem 1.2.1 and recall the expression of x in Theorem 1.2.2  

---