Draw a planar graph with two vertices of degree 3 and four vertices of degree 5, if possible.
With handshaking lemma, I get this:
$2e = 26 \implies e=13$
Then with Euler's formula, I get:
$6-13+f=2 \implies f = 9$
However, since $e \leq 3v - 6$ for a simple, connected, planar graph I would get:
$13 \leq 3(6)-6$
$13 \leq 12$
I can't figure out how I could get 9 faces for my graph. I can only get 8 faces as shown here:
This is the best attempt I had on this problem with no success.