Studia Scientiarium Mathematicarum Hungarica 35. (1999)

1-2. szám - Ivančo J.-Trenkler M.: 3-polytopes with constant face weight

3-POLYTOP ES |9J?(to;4)| = < Theorem 1 ([1]). l'Dt(u); 3)| ' 0 for w ^ 8 and w = 10, 1 for w = 9, 11 ^ w ^ 1421 L w 22 and w ^ 24, 2 for 16 ^ w ^ 20 and w= 23, , 3 for w = 15. Proof. It can easily be seen that all faces of M £ 9Jl(w; 3) are of the same type. Moreover, all neighbours of a vertex with odd degree must have the same degree as the other neighbours. Since M must contain a negative charge face, only the following types of faces can occur: (3,3,3), (3, 2k, 2k) for 2 ^ k g 5, (4,4, k) for k ^ 4, (4,6,2k) for 3 ^ lb g 5, (5,5,5) and (5.6,6). Case 1. Faces of M are of type (3, 3, 3) (of type (5,5, 5)). Evidently, M is a graph of the tetrahedron (the regular icosahedron, respectively). Case 2. Faces of M arc of type {3,2k. 2k) where k £ {3.4,5} (of type (5, 6,6)). Then the graph M\=M — Vi{M) {Mi=M- VS(M)) has faces of 4 k type (k,k,k) (of type (3,3.3,3,3)), i.e. M\ is a graph of the regular -------6 — k hedron (the regular dodecahedron). Therefore M is a graph of the Kleetope over the regular —------hedron (the regular dodecahedron, respectively). 6 — k Case 3. Faces of M are of type (4, 4, k) for k ^ 3 (of type (3. 4, 4) for k = 3). Then the graph M — Vk{M) is a circuit with k vertices. Thus M is a graph of the bipyramid with 2k faces. Case 4- Faces of M are of type (4, 6, 2k) for some k £ {3,4, 5}. Then the 4k graph M — Vo(M) is homeomorphic to a graph of the regular -—--hedron b — k 4k (each edge of -—--hedron is replaced with a path of length 3). Therefore b — k M is a dual graph of the Archimedean solid (4, 6, 2A:). □ Theorem 2. ' 0 forw ^ 11, 1 for12 ^ w ^ 13 and w ^ 17, 3 forV) = 14, 4 forw = 15, . 00 forw = 16. PROOF. Let M be a polyhedral graph belonging to Wl(w; 4). Negative charge faces are only of the following types: (3,3,3, w — 9) for toíí 12, (3,3,4, w — 10) for 14 ^ w ^ 21, (3,3,5, in — 11) for 16 ^ w ^ 18 and (3,4,4, w — 11) for 15 ^ w 5Í16. Whence, to — 11 ^ A{M) ^ w — 9. 3

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