Use Induction to show that when n circles divided the plane

Question

Use Induction to show that when n circles divided the plane into regions, those regions can be colored into 2 different colors such that no regions with a common boundary are colored the same.

My think: Let p(n) : "the statement, coloring regions properly"

(Basis Step) p(1) is true, p(2) is true.

(Inductive Step) n circles divide the plane with regions make m regions that get common boundaries with main circles. Those regions can be colored B color if main circles colored A.

Now I can't progress from here, what should I do?

Answer 1

You can have $p(0)$ as the base case; this corresponds to colouring the whole plane one colour, which does meet the conditions.

For the induction step, given a colouring with $n$ circles, any new circle will cross some of the regions. Within the circle, flip all colours; it can be checked that the colouring rule is still respected within the circle, outside it and across it. Since all properly coloured configurations of circles can be built by adding one circle at a time, the induction follows.