Is $K_1$ bipartite? I'm thinking that it could be trivially bipartite since it only has one vertex and no edges but I am still a little bit unsure about it being trivially bipartite.

My graph theory professor spent a while teaching on this, because it's a mess either way.
It probably should be trivially considered bipartite. Otherwise you have to put disclaimers in all of your theorems (like the fact that $K_1$ doesn't contain any odd cycles).  But $\{\{v\},\emptyset\}$ is not a valid partition of the vertex set, since it contains the empty set as a member.
