Graph Theory proving planarity I have these set's of graphs: 

I used Euler's inequality, and the 4 color theorem which resulted in a inconclusive result. Using Kuratowski's theorem I was unable to create a K3 3 or K5 graph. So does this prove that the graphs are planar? How do we really know that there does not exist a K3 3 or K5? I did use planar embedding and was able to put it in a form where the edges do not overlap, but the method does not seem rigorous enough to prove planarity. 

Here's an illustration of a subgraph of $G$ which is a subdivision of $K_5$, so Kuratowski's Theorem implies $G$ is non-planar:

Here's a planar drawing of $H$, demonstrating that it's planar:

If it contained a subgraph which is a subdivision of $K_5$ or $K_{3,3}$, this planar drawing would not be possible.
