Euler formula for a connected planar graph

For a connected planar graph with $n$ vertices, $e$ edges, and $f$ faces,Deleting a cycle edge preserves connectedness and decreases both $e$ and $f$ by one, reducing the formula to its immediate tree case.

A simple planar graph with $n\ge 3$ vertices has at most $3n-6$ edges.

If every face of a connected planar graph has size at least $g>2$, double-counting edge-face incidences and using Euler's formula gives

Let a connected bridgeless planar graph have $n\ge 4$ vertices, $e$ edges, no four-cycle, and $t$ triangular faces. If no edge borders two triangular faces, thenso $f\le 8e/15$. The Euler formula for a connected planar graph then gives

The icosidodecahedral graph is the skeleton of the icosidodecahedron. It has 30 vertices, 60 edges, 20 triangular faces, and 12 pentagonal faces. Every edge separates a triangle from a pentagon, so it attains equality in the triangle-pentagon planar edge bound.