Thread: question regarding connected planar graph

Let G be a connected planar graph with n vertices (where n >= 3),
e edges (where e >= 3), r regions, and no cycle of length 3 or less.
Show that (e =< 2n - 4)

I've tried to use the handshake formula and the eular's formula to solve this problem, but it doesn't seems like I am doing the right things

any hints will be appreciated.

The idea is to first show e >= 2r using the equality 2e = 3r₃ + 4r₄ + ... where rᵢ is the number of regions with i edges. Then use the Euler's formula.