Discrete Mathematics

224

DISCRETE

MATHEMATIC"'-------------------------

It

is easy to see that the colouring we obtained satisfies what we wanted. In fact, look at any small piece

of

arc

of

any

of

the circles.

If

this arc is outside C, then the two regions on its two sides were different and their

colours did not change.

If

the arc is inside

C,

then again, the two regions on its two sides were differently

coloured, and even though their colours were switched, they are still different. Finally, the arc could be on C

itself. Then the two regions on both sides

of

the arc were one and the same before putting C back, and so they

had the same colour. Now, one

of

them ...