It is a gross under-statement to say that there is no standard set of definitions in graph theory.

I have never seen a definition of a graph that did not stipulate that the set of vertices is nonempty, $\mathcal{V}\ne\emptyset$.

In any case, the partition of a set never contains an empty set. Therefore, as a result of the definition of a bipartite graph $\mathcal{K}_{m,n}$ both $m\text{ and }n$ are positive integers.

This an authoritative text book. Use the "lookinside tab" go to basic concepts page. There nonempty sets for vertices are required.

If anyone knows of a text that allows an empty vertex set, please direct us to it.