Euler Formula for planar graphs

Solution 1:

I think you are misunderstanding the definition of planar graphs. It means that you can draw it to the plane without it's edges crossing each other. $K_4$ is for sure a planar graph, for example:

About Euler's formula: You can only use that if you drew a graph to a plane without it's edges crossed. In that case, you can use it to determine whether or not it is a planar graph. In your case, you drew $K_4$ with two edges crossing, that's why it didn't work.

You can use the following inequalities for checking if it is planar graph or not:

Let G be a connected planar simple graph with $n$ vertices, where $n \ge 3$ and m edges. Then $m \le 3n - 6.$ Let G be a connected planar simple graph with $n$ vertices and $m$ edges, and no triangles. Then $m \le 2n - 4 $. For $K_4$, you can see, that $n=4$, $m=6$, and if you check: $6 \le 3*4-6=6$, it works, so $K_4$ must be a planar graph.