[Math] Can a bipartite graph have only 3 vertices

Question

I'm new to Graph Theory and I'm confused as to whether a bipartite graph needs to be closed (i.e. the vertices are all connected in some way via a walk) or if it can exist like this:

Essentially, I need to use induction on the number of edges in a graph to prove that the absence of odd cycles is all that's needed to declare a graph bipartite and I'm unsure what length to use for the base hypothesis.

Answer 1

The definition of bipartite graph at Wikipedia is:

... a bipartite graph ... is a graph whose vertices can be divided into two disjoint and independent sets $U$ and $V$ such that every edge connects a vertex in $U$ to one in $V$.

A bipartite graph doesn't need to be connected. It's fine to have $U$ and $V$ of any size (possibly even empty). We can also have any number of edges (possibly even zero).

Here's an example of a bipartite graph (with $U$ and $V$ indicated by vertex colors):

The graph drawn in the question is complete bipartite graph $K_{1,2}$, and we can highlight sets $U$ and $V$ as shown below: