Planar Graph max min edges

Question

Consider a planar graph with 5 vertices, what is the minimum and the maximum number of edges such a graph can have? The graph need not be connected and is simple.

Answer 1

The maximal number of edges is 9. It is well-known that the number of edges a planar graph with n vertices can have is:

3(n-2)

In this case 3*5 - 3*(-2) = 15 - 6 = 9

Quote from wikipedia:

"If a maximal planar graph has v vertices with v > 2, then it has precisely 3v − 6 edges and 2v − 4 faces."

Answer 2

$K_5$ contains $10$ edges and it is not planar (as $e\le 3v-6$ fails). A graph of $5$ vertices with $9$ edges would do as it won't contain a sub-division of $K_5$ or $K_{3,3}$. (See Kuratowski's theorem)

Answer 3

Euler's Identity says, that for every planar graph of order n >= 3: the size m <= 3n - 6. That gives you an upper bound of 3*5-6 = 9 edges.

Furthermore, since you said that the graph does NOT need to be connected, a graph with just 5 isolated vertices would do the job, it is obvious that such a graph is planar. Therefore, the lower bound would be 0 edges.