(related to Problem: Checking if $K_5$ is Planar)
- Suppose that the complete graph $K_5$ is planar.
- A necessary condition for a graph to be planar is that $|E|\le 3|V|-6,$ where $|E|$ ist the number of its edges and $|V|$ is the number of its vertices.
- For $K_5$ this condition means that $$|E|=10\le 3|V|-6=3\cdot 5-6=9.$$ which is false.
- Therefore, $K_5$ is not planar.
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- Aldous Joan M., Wilson Robin J.: "Graphs and Applications - An Introductory Approach", Springer, 2000