Planar complete tripartite graphs

Question

For which values of $r$, $s$, and $t$ is the complete tripartite graph $K_{r,s,t}$ planar?

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Obviously I want to look for either a $K_5$ or a $K_{3,3}$ in order to show that a specific graph is non-planar. If a graph doesn't have either of these than it is planar. However, I'm unsure how to approach this in the general case without drawing out every possible tripartite graph until I find some non-planar ones.

Answer 1

Without loss of generality assume $r\ge s\ge t$.

If $r\ge 3$, and $s+t\ge 3$ then the graph is not planar (why?)

In the cases $r\le 2$ or $s+t\le 2$, you should be able to find a strategy to draw the graph in a plane.

Answer 2

As noted in Henning Makholm answer, we have the following cases:

a) $r \geq 3$, $s+t \leq 2$

b)$r \leq 2$, $s+t \leq 4$.

In both the cases you can show that the graph is planar.