Non-Planar graph maximum degree proof

Question

How can it be proved that a non planar graph has a maximum degree greater or equal than three?? I have tried proving it by contradiction using the inequality m > 3n - 6 , but It is correct . I have thought of mathematical induction but I am finding it difficult to prove this

Answer 1

Prove that a graph with maximum degree 2 is a disjoint union of paths and cycles (and isolated vertices). To do so: pick any vertex of degree 1 (if any), find its neighbor, check its degree. If 1, done, if 2, keep going until you (necessarily) hit another vertex of degree 1. That's a path. Once you've accounted for all vertices of degree 1, use a similar method to survey vertices of degree 2 and show that they all belong to cycles.

Now show that paths and cycles are planar, and the disjoint union of planar graphs is planar as well.