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.