Prove that the thickness of $K_6$ is $2$. (That is, find two graphs $G$ and $H$ such that $G∪H = K_6$ and $G$ and $H$ are both planar.)

Question

Prove that the thickness of $K_6$ is $2$. (That is, find two graphs $G$ and $H$ such that $G\cup H =K_6$ and $G$ and $H$ are both planar.)

Answer 1

Consider $G=K_4$ and $H=K_2$ connected too all vertices of complement of $K_4$?

Answer 2

Let $G$ be a $K_5$ without an edge. Then $H$ is a $K_3$ with three leafs attached to one of its vertices.

Answer 3

I wonder how you tried to solve this problem. Me not being very clever, I decided to just draw the biggest planar subgraph of $K_6$ that I could, and hope that the complement would be planar. This must be my lucky day, because I succeeded on my first try. I put down $6$ points and started joining them with edges as long as I could without crossing. I ended up with $12$ edges: first I drew a hexagon, then a triangle inscribed in the hexagon, then $3$ more edges outside the hexagon. The complementary graph had only $3$ edges, so of course it was planar.

P.S. Here's a short description of the two graphs I ended up with: one of them is a $1$-regular graph on $6$ vertices, the other is a $4$-regular graph on $6$ vertices.