Complete Hamiltonian Bipartite Graph

Theorem
Let $K_{m, n}$ be a complete bipartite graph.

Then $K_{m, n}$ is Hamiltonian iff $m = n > 1$.

Proof
Clearly $K_{1, 1}$ is not Hamiltonian, although it does have a Hamiltonian path.

From Condition for Bipartite Graph to be Hamiltonian, if $m \ne n$ then $K_{m, n}$ is not Hamiltonian.

So let $m = n$.

We note that the degree of any vertex in $K_{n, n}$ is $n$.

Then we can use Ore's Theorem, and the result follows directly.