Triangle-pentagon planar edge bound
= Triangle-pentagon planar edge bound
Let a connected bridgeless planar graph have $n\ge4$ vertices, $e$ edges, no four-cycle, and $t$ triangular faces. If no edge borders two triangular faces, then
$$
2e\ge3t+5(f-t),
\qquad e\ge3t,
$$
so $f\le8e/15$. The <Euler formula for a connected planar graph> then gives
$$
e\le\frac{15(n-2)}7.
$$