Best way to draw k-partite graphs (sorry if i use some strange word. english is not my native language so i am not sure about all the mathematical terms)
I need to draw some graphs that all have some common features.


*

*the vertices are tuples $(i,j)$ with $i=1,\dots,n$ and $j=1,\dots,k$.

*two vertices $(i_1, j_1), (i_2, j_2)$ share an edge if and only if $i_1\ne i_2$ and $j_1\ne j_2$.


So these graphs look a lot like complete $k$-partite graphs.
My question is how to best draw them. I tried simple dot like this:
graph g_2_3 {
  "0 0" -- "1 1";
  "0 0" -- "1 2";
  "0 1" -- "1 0";
  "0 1" -- "1 2";
  "0 2" -- "1 0";
  "0 2" -- "1 1";
}
But this looks too messy to use it for illustrating something.
 A: You could place the vertices of the graph at the vertices $n$-gons that are centered at the vertices of a large $k$-gon.
For instance, for $(n,k) = (3,5)$:

Here, triangles are arranged about a pentagon. You can see, for instance, that the "outermost" vertex of each triangle is connected to the "other two" vertices.
Writing $R$ for the radius of the larger $k$-gon, and $r$ for the radius of each smaller $n$-gon, and defining $T := 2\pi/k$ and $t := 2\pi/n$, we can say explicitly that vertex/tuple $(i,j)$ is located at coordinates 
$$( \; R \cos( j T ) + r \cos( i t + j T ), R \sin(jT) + r \sin(it + jT ) \; )$$
A: One possibility: instead of visualizing this graph, start by visualizing its complement. That is a much "nicer" graph; e.g., you can just arrange all the dots in an $n \times m$ matrix, in which case the joined vertices are precisely the vertices sharing a row or column. Now if you want to see your actual graph, it should be relatively easy from that point to visualize complementing the complement. In linear algebra terms a vertex $(i,j)$ is then joined to everything in the $(i,j)$-minor.
Another question (whose answer depends on your application): is a graph really the best way to think about whatever this is? Graphs are usually a concept that one would apply when one has a rather complex and irregular data set to analyze. Perhaps in this case one could do away with the graph and think in simpler terms; e.g., just talking about a binary relation between points?
