Bipartite graph definition in West's Introduction to graph theory

Question

Why does it say "possibly empty" in the definition of bipartite graph? Wouldn't that mean that every graph is bipartite, moreover I don't think partitionins can have empty sets. Is this a mistake?

Answer 1

I think the "possibly empty" statement answers the question:

Let $G$ be a graph having two or more vertices which are not connected to each other. Is this a bipartite graph?

Answer: Yes. Take $A=\{a,b,c\}$ the disconnected vertices and $B=\emptyset$ then clearly $V(G)=A\cup B$.

If we do not assume the "possibly empty" part then this kind of flaws will come up and complicate the matter. I hope you understand. One interesting fact is that one may say taking $A=\{a,b\}$ and $B=\{c\}$ would solve the problem, but does this meet the requirement? I mean none of any member of $A$ is connected to $B$ so separating them by this manner is meaningless. Possibly empty resolves this matter.

Answer 2

Firstly, although it is not important this definition is from Douglas West's book Introduction to Graph Theory.

Second, of course not every graph is bipartite according to this definition. If one part is empty, the other part cannot contain edges.

Third, it seems to me that the assumption of empty parts allows us to consider a graph with one vertex as bipartite. Then, for example, K$\ddot{\rm o}$nig's theorem does not require any exceptions: A graph is bipartite if and only if it has no odd cycle.