For which values of $m$ and $n$ does the complete bipartite graph $K_{m,n}$ have a Hamilton circuit?

In a complete bipartite graph $K_{m,n}$, vertices are divided into two parts.

A Hamilton circuit must visit every vertex exactly once and return to the starting vertex.

In a bipartite graph, every cycle alternates between the two parts.

So, for a Hamilton circuit to include all vertices, both parts must have the same number of vertices.

Therefore,

$m=n$

Also, a cycle needs at least $2$ vertices in each part.

So,

$m,n \ge 2$

Hence, $K_{m,n}$ has a Hamilton circuit when

$m=n,\ m,n \ge 2$