Small faces of a spherical polyhedral graph
= Small faces of a spherical polyhedral graph
{title2=$F_3+F_4+F_5\ge4$}
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 $\sum_n(6-n)F_n=12+\sum_m(2m-6)V_m\ge12$. 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.