Face-regular bifaced polyhedra

Call bifaced any k-valent polyhedron, whose faces are p(a) a-gons and p(b) b-gons only, where 3 less than or equal to a < b, 0 < p(a), 0 less than or equal to P-b. We consider the case b less than or equal to 2k/(k - 2) covering applications; so either k = 3 less than or equal to a < b <less than or equal to>6, or (k; a, b, p(a)) = (4; 3, 4, 8). Call such a polyhedron aR(i) (resp., bR(j)) if each of its a-gonal (b-gonal) faces is adjacent to exactly i a-gonal (resp., j b-gonal) faces. The preferable (i.e., with isolated pentagons) fullerenes are the case aR(o) for (k; a, b) = (3; 5, 6). We classify all a- or b-face-regular bifaced polyhedra, except aR(o) for (k; a, b) = (3;4, 6),(3; 5, 6), (4; 3,4) and aR(1) for (k; a, b) = (3; 5, 6),(4; 3,4). For example, we list all 13,6,4,10,26 polyhedra bR(j) for all five possible cases: k = 4; k = 3, b < 6; k = 3, b = 6, a = 3, 4, 5. (C) 2001 Elsevier Science B.V. All rights reserved.