Greatest number of parts that n circles can divide the plane

Question

What is the greatest number of domains(or parts) that n circles could divide the plane?

From many small cases I get the feeling that intersecting circles would provide the greatest number of parts. Is this recursion right C(n+1) = 2C(n) using the previous statement. Since the new circle intersects all the circles and doubles the parts. Here C(n) is the number of parts for n circles.

How could I prove formally? If I could get an inequality that I will know for sure that I have got the greatest number of parts.

Answer 1

The formula is'nt correct.

Looking at the problem as a planar garph: (you can think of every circle as 2 vertices we never intercect & 2 edges)

When adding a circle, we can intercect each previous circle in only 2 point, adding 2 vertices and 3 edges (one from the old circle). by Euler's formula it means we added one face.

so, in each step we add at most $n+1$ faces.(one of the new circle, $n$ from the n other circles).

Still, the question of the exact number is interesting.

Answer 2

http://oeis.org/A014206 says draw n + 1 circles in the plane; then $a(n)=n^2+n+2$ gives the maximal number of regions into which the plane is divided. There are links and references there.