Show that if G is a disconnected graph containing exactly two odd vertices, then these odd vertices must be in the same component of G.

Question

I'm currently working in the following graph theory excercise:

Show that if $G$ is a disconnected graph containing exactly two odd vertices, then these odd vertices must be in the same component of $G$.

I'm thinking about the lemma: If

$$deg(u)+deg(v) ≥ n-1$$

Then $G$ is connected and $diam(G) ≥ 2$

But haven't find a way to suit the lemma in my solution, thanks in advance for any hint or help.

Answer 1

Hint:

The sum of degrees in any graph is even — in particular, this is true about each connected component of $G$.