# What is the intersection between complete bipartite graphs and complements of bipartite graphs?

For context, I am in an intro Graph theory course. I know that the complement of a bipartite graph can be bipartite, but I don't know how to determine all of them and whether they can be complete. I assume the intersection is not empty but I am not sure. Any help is appreciated, thanks!

The complement of a complete bipartite graph $$K_{m,n}$$ is the union of the two complete graphs $$K_m\cup K_n$$. That complement is bipartite if and only if $$m<3$$ and $$n<3$$. If $$m,n>0$$ then the complement will never be complete since it is not even connected.