Let $G$ be a simple planar graph in which every vertex has the same degree $k$. Prove that $k \leq 5$

Question

Let $G$ be a simple planar graph in which every vertex has the same degree $k$. Prove that $k \leq 5$. This is a problem I was given by my professor, but I am struggling to see why it would be true, let alone prove it. It seems to me that any simple graph regular on degree $5$ would have a subgraph of $K_5$, which is not planar. Therefore, the graph itself could not be planar. Any direction on how to disprove my supposed counterexample and start this proof would be extremely helpful!

Answer 1

Due to Euler's formula we know that edges <= 3n-6 in a planar graph. In your graph which is a regular graph the number of edges = kn/2. Now compare 3n-6 to kn/2. k < 6 - 12/n which tends to 6 as n tends to infinity. So k < 6.