Complete Bipartite Graphs which are Cycle Graphs

Theorem

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

$K_{2, 2}$ is the cycle graph $C_4$

and no other complete bipartite graphs are cycle graphs.

Proof

From Cycle Graph is 2-Regular, a cycle graph is $2$-regular.

From Complete Bipartite Graphs which are Regular, the only $2$-regular complete bipartite graph is $K_{2, 2}$.

Hence the result.

$\blacksquare$