Planarity on $10$ vertices Is there a planar graph on $10$ vertices such that its complement is planar as well?
I have troubles deciding if this is an elementary or deep question. By some other thread, the answer is an easy No for $11$ or more vertices.

I think the answer is no, but I have only an experimental evidence. 
$K_{10}$ has 45 edges, and the max number of edges for a planar graph with 10 vertices is $3\cdot10-6=24$. 
So the only possible pairs of number of edges are {24, 21} and {23, 22}. I used Brendan McKay's program plantri to generate all the planar graphs with 24 and 23 edges, and then I used Mathematica to find planar graphs among their complements. I found none, so there is not such a graph with 10 vertices.
By the way, using the same approach I found no such a graph with 9 vertices, but I found many with 8 vertices.
