Im trying to prove a planar graph without triangles with $ n \ge 3$ points has at most $2n-4$ edges. I want to solve this using the Euler formula, $n+f=m+2$. I've come to the conclusion that $f\le$ $m/2$, however when I plug this into the formula I get:
$n+ m/2 \le m+2$
$m \ge 2n-4$
Seeing as this is the exact opposite of what I want, I'm sure that the inequality signs are wrong, however I just don't understand why. Thank you!
Since $n - m + f = 2$, we have $f = 2 + m - n$.
We can use this to take $f \le m/2$ and replace the $f$ by $2 + m - n$, getting $2 + m - n \le m/2$.
This rearranges to $n + m/2 \ge m + 2$: the opposite of the inequality you got.