Are all connected graphs with Euler characteristics 2 planar?

Question

I have read proofs and descriptions stating that a planar connected graph have the Euler characteristic 2. I'm not sure if that statement is equivalent to "a connected graph with the Euler characteristic 2 have a planar embedding"?

I'm studying a math course where I'm supposed to show that a certain graph have a planar embedding, and I'm wondering if I can show it by using the Euler characteristics.

Answer 1

The problem with the approach you are suggesting is that we usually don't know anything about the number of faces of a graph before we embed the graph. The only situation I can think of(there could be more) where this sort of argument could do anything would be if you were given a problem like

"Let $G$ be a graph with $|V(G)|=n$, $|E(G)|=m$ and assume that it can be embedded without crossings on the surface $S_k$ with $r$ faces but not any surface of lesser genus. Is $G$ planar?"

Then you could calculate $n-m+r$ and if you got $2$ then we can conclude that $G$ is planar. Unless given the number of faces in a minimal embedding though, I see no way to use the Euler Characteristic to prove planarity.