Thickness of Sextic Graphs The Generalized Hexagon (2,1) is a graph with thickness 2. The Wikipedia definition of thickness "is the minimum number of planar graphs into which the edges of G can be partitioned". 

The thickness of a few sextic graphs, such as the 6-hypercube and $K_{6,6}$, is known to be 2. There are many sextic graphs where the thickness seems to be unknown. Here are a few:

Can anyone make thickness-2 embeddings of these graphs?
 A: A partial solution (Triangular, Paley 13 and the Shrikhande graph).
Note: your example gives more than you ask for: you ask for a thickness-2 embedding, but
your example is a combined straight line planar embedding.
I know that every planar graph has a straight line embedding, but I doubt if it is possible in general
to produce a straight-line $k$-edge colored embedding where every color class is planar for
a graph with thickness $k$. Therefore my solutions only address your explicit question.

Triangular is the linegraph of $K_5$.
In the picture the blue lines and the red lines form two planar graphs.
The red graph and the four green $K_4$s with centers $0,1,2,3$ are the 5 $K_4$s representing the vertices of $K_5$.

Paley-13 consists of three edge-disjoint $C_{13}$s.
In your drawing they can be distinguished as the outer cycle,
the distance-3 cycle and the distance-4 cycle.
Now draw the outer $C_{13}$ and add path $0,3,6,9,12,2,5,8,11$, the first four edges inside the cycle,
the last three outside the cycle.
Our second graph is the distance-4 $C_{13}$: $0,4,8,12,3,7,11,2,6,10,1,5,9$.
This can be enhanced with the missing edges from the distance-3 cycle: $11,1,4,7,10,0$,
the first three inside, the last two outside the cycle.

The Shrikhande graph is $C_4\square C_4$ plus extra diagonals in one direction
(see https://en.wikipedia.org/wiki/Shrikhande_graph).
Now first make a standard drawing of $P_4\square P_4$, draw the extra diagonals that are "inside"
the drawing and draw the 4 missing horizontal edges outside the drawing.
This is easily done with a planar drawing.
You now have drawn $12+12+9+4=37$ edges.
The inner four vertices are already at full degree.
The outer vertices have degree sequence $(3,3,2,2,2,2,2,2,1,1,1,1)$.
Checking the missing edges you see that there are two components: a $K_2$ plus a $C_8$ with two pending edges.
A: A very partial answer.  There are several types of thickness.
With geometric thickness, each edge is a straight line and edges can only cross if they have different colors.
With normal graph thickness, edges can bend and curve in any way.  Another answer posted here establishes that the Paley-13 graph has thickness 2.  The Paley-13 graph also has geometric thickness 2. Ideal answers should illustrate geometric thickness, but establishing normal thickness as 2 is an excellent first step.

