Complete Bipartite Graphs which are Path Graphs

Theorem
Let $K_{m, n}$ be a complete bipartite graph.
 * $K_{0, 0}$ is the path graph $P_0$
 * $K_{0, 1}$ and $K_{1, 0}$ are the path graph $P_1$
 * $K_{1, 1}$ is the path graphs $P_2$
 * $K_{1, 2}$ and $K_{2, 1}$ are the path graphs $P_3$

and no other complete bipartite graphs are path graphs.

Proof
$K_{0, 0}$, $K_{0, 1}$ and $K_{1, 0}$ have no edges.

Hence they are path graphs vacuously.

That $K_{1, 1}$ is the path graph $P_2$ can be determined by inspection:


 * K1-1.png

Similarly, that $K_{1, 2}$ and $K_{2, 1}$ are the path graph $P_3$ can be determined by inspection:


 * K1-2.png

We have that Path Graph is Tree.

From Complete Bipartite Graphs which are Trees, a path graph must be of the form $K_{1, n}$ or $K_{n, 1}$ for some $n$.

But for $n \ge 3$ there exists one vertex which is of degree greater than $2$.

Hence from Degree of Vertex in Path Graph is Less than 3, $K_{1, n}$ or $K_{n, 1}$ is not a path graph for $n \ge 3$.

Hence the result.