Prove that a simple connected graph has even numbers of vertex

Question

Given a simple connected graph G that all of its vertex degree is 3. how can I prove that G has even number of vertex? and does G has a perfect matching and why?

Answer 1

Suppose for a contradiction that $G$ has odd number of vertices $n$ and $e$ edges. Then, by Handshaking Lemma we have $2e = 3n$ where $n$ is odd so $e = \frac{3n}{2}$ is not an integer, which is a contradiction as required. Therefore $G$ has even number of vertices.

But $G$ may not have a perfect matching as in this example: