A Platonic Graph is a planar graph in which all vertices have the same degree d1 and all regions have the same degree d2, where d1 3 and d2 3. (a) Suppose G = (V, E) is a Platonic graph with each vertex degree d1 and each region degree d2. Prove that |E| = 1 d1|V | and |R| = d1 |V |. 2 d2 (b) Using part (a) and Euler's Polyhedral Formula, show that |V |(2d1 2d2 d1d2) = 4d2. (c) Since |V | and 4d2 are positive integers, we can conclude from part (b) that 2d1 2d2 d1d2 > 0. Use this inequality to prove that (d1 2)(d2 2) < 4. (d) Fill in the chart with the possible pairs of integers d1,d2 that satisfy the inequalities at the top of the columns. (i) (ii) (iii) (iv) (v) d1 3 d2 3 (d1 2)(d2 2)<4 cube