ON FACE VECTORS AND VERTEX VECTORS OF CONVEX POLYHEDRA

Denote by p(k)(M) or v(k)(M) the number of k-gonal faces, or k-valent vertices, respectively, of the convex polyhedron M. A pair of sequences (p(k)(M)/k greater-than-or-equal-to 3) and (v(k)(M)/k greater-than-or-equal-to 3) associated in a natural way with a polyhedron M is called the face vector and the vertex vector of M, respectively. Let p=(p(k)\3 less-than-or-equal-to k not-equal 6) and V=(v(k)/k greater-than-or-equal-to 4) be a pair of sequences of nonnegative integers satisfying SIGMA(k greater-than-or-equal-to 3)(6-k)p(k)+2SIGMA(k greater-than-or-equal-to 3)(3-k)v(k)=12. Denote by P6(p,v) the set of all nonnegative integers P6 such that, if P6 is added to p and v3=1/3(SIGMA(k greater-than-or-equal-to 3 kp(k)-SIGMA(k greater-than-or-equal-to 4)kv(k)) is added to v, the face vector and the vertex vector of a convex polyhedron M is obtained. In the present paper we determine, for each pair (p, v), the set P6(p, v) except for a finite number of its members.