Exercise: We are given a simple, connected planar graph with n-vertices, each vertex is of degree 4, and 10 faces. Find all possible n, and all the non isomorphic planar graphs with this property. Describe each of the graphs.
Attempt:
I started with two formula
Euler's formula: V - E + F = 2
Handshake Lemma: $$\sum_{v \in V}^{} deg(v) = 2*|E|$$
By the Handshake Lemma, we see that 4n = 2|E|, telling us that the number of edges is 2n. Substituting this information back into Euler's formula, we end up with:
n - 2n + 10 = -n + 10 = 2, and therefore n = 8. This leaves us with 8 vertices, and 16 edges.
My first question is, are there other possible values of n that satisfy the graph described above?
My second question is about the latter half of the problem on finding non isomorphic planar graphs. If possible, could someone explain the general method on finding non isomorphic planar graphs? As I currently understand, two isomorphic planar graphs will have vertices of the same degrees connected in the same way. To find a non isomorphic graph from this graph, all I need to do is to move an edge connecting two vertices. Since every vertex is of degree 4, and there are n vertices, does that mean there are 4*n different non-isomorphic graphs?