Inferences from Euler formula, a sufficient condition?

Question

The conclusions from the Euler formula are:

• $|E| <= 3 * |V| - 6$ and
• $deg(v) <= 5$

If a graph satisfies these conditions, is that a sufficient condition for a graph to be drawn planar? Or is it necessary to check additional things (like $K_{5}$ or $K_{3,3}$)?

Answer 1

Take $K_5$ and subdivide all its edges (replace edge $uv$ by two edges $uw,wv$). This graph has $15$ vertices and $20$ edges and average degree less than $5$, but it is clearly non-planar because $K_5$ is a minor of it.