# Find Planar Graph fromVertices and Faces

Could you find a 3-Regular Connected Planar Graph on $10$ vertices with $8$ faces? If so, explain carefully.

I don't know what does regular mean. I think that 3-connected graph on 10 vertices with 8 faces. From eulerian formula : $v + f - e = 2:$ $10+8-e=2 \Longrightarrow e=16$. And I use other proof "A simple planar graph on v 3 vertices has at most $\;3v- 6$ edges." $\Longrightarrow 30-6=24$ edges at most. I said that $24>16$, if so it can be.

Is it true or not?

• A $k$-regular graph is defined to be a graph where every vertex is of degree $k$. – JMoravitz Jan 20 '15 at 16:01

$3$-regular means that each vertex has degree $3$. With $10$ vertices, how many edges would that make?

• 3*10/2 = 15 edges. Is that mean that it must be 16 from eular formula but we found 15. It is impossible. – Zafer Jan 20 '15 at 16:05
• Its all good. Already deleted that comment. So, now you know the number of faces, edges, and vertices. Check if it matches Euler's formula. – JMoravitz Jan 20 '15 at 16:08