Given a simple graph and its complement, prove that either of them is always connected. I was tasked to prove that when given 2 graphs $G$ and $\bar{G}$ (complement), at least one of them is a always a connected graph.
Well, I always post my attempt at solution, but here I'm totally stuck. I tried to do raw algebraic manipulations with # of components, circuit ranks, etc, but to no avail.
So I really hope someone could give me a hint on how to approach this problem.

Suppose one of G = (V,E) and G' = (V,E) is disconnected; say G with components
G., . . .  .,Gk, k > 1, w.l.o.g since G = G''. Any two vertices v∉Gi and w∉Gj will be
connected in G' since
(a) if i 6≠j then vw ∉ E, so vw 2 E'.
(b) if i = j then there must be a third vertex u in another component such that vu ∉ E and
wu ∉ E. In this case, v and w would be connected in G' through edges vu,wu ∉ E'.
Since any pair of vertices in G' are connected, G' is connected.
