| Abstract: | Let {pk}k≥2, k≠4 be a sequence of non-negative integers which satisfies 8 + Σk≥3(k — 4)pk = 0. Then there exists an integer p4 such that there exists a 2-connected planar graph with exactly pk k-gons as faces for all k ≥ 2. This paper determines all such p4 when pk = 0 for k ≥ 5 and determines that there is a constant C ≥ 1 such that for some m ≤ p2 + 1/4p3 + C, there exists a 2-connected planar graph with exactly pk faces for each p4 = m + 2w, w a positive integer. When there exists at least one odd k ≥ 3 for which pk ≠ 0, the coefficient 2 of w in the above equation may be replaced by 1. These conclusions do not hold if the coefficients of p2 and p3 are any smaller than 1 and 1/4, respectively. |
