 Small faces of a spherical polyhedral graph

For a connected cellular spherical planar graph with vertex degrees at least three and face sizes at least three, degree/face double counting and the Euler formula for a connected planar graph give ${\sum }_n(6-n)F_n=12+{\sum }_m(2m-6)V_m\ge 12$. The left side is at most $3(F_3+F_4+F_5)$, proving the displayed bound. In particular a convex polyhedron has at least four faces of size at most five.