Studia Scientiarium Mathematicarum Hungarica 35. (1999)

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

315704 Studio Scientiarum Mathematicarum Hungarica 35 (1999), 1 15 3-POLYTOPES WITH CONSTANT FACE WEIGHT J. IVANCO and M. TRENKLER Dedicated to Professor E. Jucovic on the occasion of his 70th birthday Abstract The weight of a face a in a 3-polytope is the sum of degrees of vertices which are incident with a. In the present paper we determine the number of different regular 3- polytopes for which the weight of each face is w ^ 9. In the nonregular case we have similar results if 9 < in 5; 21 and if 28 < w. 1. Introduction Rosenfeld [5] and also JendroP and Jucovic [3] investigated 3-poly topes or maps with constant weight of edges. As an analogy, E. Jucovic suggested to study convex 3-polyt,opes with constant weight of faces and some basic prop­erties of such 3-polyt,opes are studied in [1] by his student Bauer. The aim of the present paper is to contribute to the description of such 3-polyt.opes. For a convex 3-polytope M, let V(M), E(M), F(M) and A(M) (or only V, E, F and A) denote the vertex set of M, the edge set of M, the face set of M and the maximum degree of M, respectively. Let. a be a k-gonal face of M, which is incident with vertices Ai,..., Ay, where deg(Ai) ^ ... ^ deg(A^). The type of a is defined as the A-tuple of positive integers [d\..... r//,.), wliere k 2d — 6 di = deg(A,), for all i = l,... ,k. The charge of a is c(a) := k — 6 + ^ —7 ■ i.= i dik and the weight of a is w(a)^ di. Z— 1 Since Euler’s formula for M can be rewritten as (1) X>) = -12 n£F we get the following assertion (see also [4]). Lemma 1. Every -polytope contains a face whose charge is negative. □ If a € F(M) and X C V(M), then the set of vertices of X which are incident with a is denoted by A fia. The set, X C V(M) is called the strong 0081-6906/99/$ 5.00 ©1999 Akadémiai Kiadó, Budapest MAGYAR rüDöíWÁNYOS AKAOEaUA 1991 Mathematics Subject Classification. Primary 52B10; Secondary 05C10. Key words and phrases. Polytope, constant weight of faces.

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