planar graph of Submultiples There is Graph which is connected with Submultiples. (I am sorry but I don't know what this is called.) For example, 10-node Graph has 10 nodes, 18 edges. node 1 connect all the other nodes. node 2 connect node number 2,4,6,8,10. node 3 connect 3,6,9. node 4 connect 4,8. etc. Image of Planar graph of 10-node Submultiple graph This is Planar graph of 10-node graph.  But, **Is 12-node Submultiple graph planar?** And how can I prove it? (I saw Kuratowski's theorem But I cannot find the graph of $K_{3,3}$ If there is, please show me with picture. ## Question 2. If 12-node Submultiple graph is possible, How many node are possible? (I mean, 14 or 15 nodes graph is planar? or not?)

Here is a planar graph for $12$ nodes:

![enter image description here](

The thicker red edges are the onesthat I changed their orientations. Green edges are the ones that are incident to node $12$.

From here, I can say that for $13$ and $14$ nodes, it can be still planar. $13$ is similar to $11$ since $13$th node is just adjacent to $1$ and for $14$, we can add $14$ to outer face in which $2$ and $7$ reside.

For $15$, it is not trivial. But there is a $K_{3,3}$ subgraph using subdivions as the following:

![enter image description here](

Then by Kuratowski's Theorem, it is not planar.

For $16$ nodes, I am sure that it cannot be planar because the nodes $1,2,4,8,16$ will all be connected to each other so we have $K_5$ as a subgraph. Then by Kuratowski's Theorem, it is not planar.