Department of Mathematics, University of Washington, Seattle, WA 98195, USA
Abstract:
Let {pk}k≥3 be a sequence of nonnegative integers which satisfies 8 + Σk≥3 (k-4) pk = 0 and p4 ≥ p3. Then there is a convex 4-valent polytope P in E3 such that P has exactly pk k-gons as faces. The inequality p4 ≥ p3 is the best possible in the sense that for c < 1 there exist sequences that are not 4-realizable that satisfy both 8 + Σk ≥3 (k - 4) pk = 0 and p4 > cp3. When Σk ≥ 5pk ≠ 1, one can make the stronger statement that the sequence {pk} is 4-reliazable if it satisfies 8 + Σk ≥ 3 (k - 4) pk = 0 and p4 ≥ 2Σk ≥ 5pk + max{k ¦ pk ≠ 0}.