# Math Help - Planar graph

1. ## Planar graph

Hi
I have this graph

now i should check if this graph can be planar.

v - number of vertices
e - number of edges
f - number of faces

So it's hold the theorem "If v ≥ 3 then e 3v − 6 "
v = 9, e = 15
15 ≤ 21

and should hold v - e + f = 2 from here f = 2 - v + e = 2 - 9 + 15 = 8
so f = 8 now my question is how to easily count faces (regions bounded by edges, including the outer, infinitely large region) of the graph??
thanks

2. ## Re: Planar graph

You need the graph to be embedded in the plane (that is, drawn with no edges crossing) to have faces defined. Also, I think your graph is not planar. The condition you are using works like this "If the graph is planar, then $e \leq 3v - 6$" but the latter doesn't prove the first, that is "If $e \leq 3v - 6$, then the graph is planar" is WRONG.