Coloring Maps and Graphs

Abstract

We draw some circles on the plane (say, n in number). These divide the plane into a number of regions. Figure 13.1 shows such a set of circles, and also an “alternating” coloring of the regions with two colors; it gives a nice pattern. Now our question is, can we always color these regions this way? We’ll show that the answer is yes. Let us state this more exactly:

• Theorem 13.1.1 The regions formed by n circles in the plane can be colored with red and blue in such a way that any two regions that share a common boundary arc will be colored differently.