Complete Bipartite Graphs which are Complete Graphs

Theorem
Let $K_{m, n}$ be a complete bipartite graph.
 * $K_{0, 0}$ is the complete graph $K_0$
 * $K_{0, 1}$ and $K_{1, 0}$ are the complete graph $K_1$
 * $K_{1, 1}$ is the complete graph $K_2$

and no other complete bipartite graphs are complete.

Proof
$K_{0, 0}$ is the null graph by Null Graph is Complete Bipartite Graph.

Then by Null Graph is Complete Graph, $K_{0, 0}$ is the complete graph $K_0$.

$K_{0, 1}$ and $K_{1, 0}$ consist of one vertex and no edges.

Then by Complete Graph of Order 1 is Edgeless, $K_{0, 1}$ and $K_{1, 0}$ are both the complete graph $K_1$.

That $K_{1, 1}$ is the complete graph $K_2$ can be determined by inspection:


 * K1-1.png

Suppose either $m > 1$ or $n > 1$.

Then one of the partite sets contains more than $1$ vertex.

By the nature of a bipartite graph, $2$ such vertices are not joined by an edge.

Hence by definition such a $K_{m, n}$ is not a complete bipartite graph.