We divide a piece of paper in 12 parts. Moreover we have 30 pieces of cellotape. We connect two pieces of paper, if they have a common "borderline".
Show that the following Statement is not true:
" 30 pieces of cellotape aren't enough to fix the paper. "
My work:
In order to better understand the situation I visualise the situation.
This would be an example for 4 pieces of paper and 4 cellotapes. So how can I Show that the Statement above is not true? My idea was to use Euler's polyhedron Formula.
$$ V - E + F = 2 $$ where $V$ is verticles, $E$ is edges and $F$ faces. I would say that $F$ is 12 ( pieces of paper ) and $E$ is 30 ( cellotapes ). Now I'm stucked.
Here is a classical fact that will be useful for this problem:
Definition: A maximal planar graph is one where every face is a triangle. In other words, we cannot add any edges.
Fact: A maximal planar graph on $n$ vertices can have at most $3n - 6$ edges.
Proof: We have the relation $3F = 2E$, since if we count the edges by adding $3$ per face each edge will be counted exactly twice. That is, $F = 2E/3$. Substituting into $V - E + F = 2$, we have $V - E + 2E/3 = 2$, so $V - E/3 = 2$, so $3V - 6= E$.