Give a formal argument why there is no planar graph with ten edges and seven regions. Here is my answer

Question

I'm really new to a graph theory, and I have to answer the following question: give a formal argument why there is no planar graph with ten edges and seven regions. Here is my answer:

Using Euler’s we can calculate the number of vertices in this graph: $n-m+r=2$ Thus there are $5$ vertices($n=5$) Also we know that for every planar graph the following inequality should hold: $m\le3n-6$ let’s check: $10\le5\times3-6\implies 10\le9$ wrong etc.

Can you tell me whether I reasoning in the right way!!! Thank you

Answer 1

Your proof is absolutely correct. If $v\ge 3$ then a planar graph must satisfy $e\le 3v-6$ (where $v$ is the number of vertices and $e$ is the number of edges).

Answer 2

From $n-m+r=2$ and $m\le3n-6$ we can state that $n=2+m-r$ and $\frac m3+2\le n$.

Therefore we have $\frac m3+2\le2+m-r$, so $3r\le2m$ and this gives the contradiction.