After working for sometime I figured out the following course of action. (from a few sample cases on 4 and 5 vertices)

i) I wanted to prove that the graph had no odd degree vertex.

ii) There exists at leat one vertex adjacent to all other vertices.

If I can do these, then if n = |G|, (n-1) is even- hence, n is odd.

My friend told me that by considering a typical vertex and its neighbours and considering the subgraph induced on it, he has been able to prove the 1st part.

So now to prove the 2nd part, I was cosidering a vertex with maximum degree and if it does not have the above property I wanted to derive a contradiction.

But I think I am stuck.

Any suggestions?