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.

Additionally, it will be easy if simply look at the adjacency matrix of the graph.
More explanation: The adjacency matrix of a disconnected graph will be block diagonal. Then think about its complement, if two vertices were in different connected component in the original graph, then they are adjacent in the complement; if two vertices were in the same connected component in the orginal graph, then a $2$-path connects them.
