Studia Scientiarium Mathematicarum Hungarica 35. (1999)
1-2. szám - Ivančo J.-Trenkler M.: 3-polytopes with constant face weight
2 J. IVANCO and M. TRENKLER set, of M if all negative charge faces of M belong to Fx, where Fx := {a € F{M)\ X The relative charge of a vertex A in a strong set X is Since for every strong set X the formula (1) can be rewritten as we get the following assertion. LEMMA 2. Every strong set of a 3-polytope contains a vertex whose relative charge is negative. □ Since the constancy of weight of faces is a combinatorial property, it is useful to identify convex 3-polytopes with their graphs and next to consider polyhedral graphs, i.e. plane 3-connected graphs (see Griinbaum [2]). For a positive integer w, let DJt(w) be the family of all polyhedral graphs whose all faces have the weight w. Similarly, by 9Jl(w,k) we denote the family of all fc-gonal graphs belonging to dll(w). Let V)t(M) denote the set of vertices of degree k in a polyhedral graph M and v^(M) = \ Vk(M)\. We can prove the following auxiliary result. Lemma 3. //Meffll(w), then 2. Regular cases Evidently, M(w, k) = 0 for k ^ {3,4, 5}, because every 3-polytope contains a face incident with at most five vertices. Let us deal with the remaining cases. «(A):= E aeF{A} c(q) \xn a (2) ]Tca-(A)+ c(a) =-12 AeX aeF-Fx (3) Y (*2 ~ + w) VAM) = ii 3 Proof. Since a vertex A contributes to the weight of deg(/l) faces we have w|-F|= E w(a)= (deg(yf))2 = E i2Vi{M). aeF Aev 3 Hence |W| = — E' i2V{(M). Similarly, |F| = E vi(M) and |F| = - ivj(M). w &3 2 i£3 Manipulations with these equalities and with Euler’s formula yield the assertion. □