In the map of Figure 21.1, the curve XY is one of the boundaries. Countries like I and II share a common border. Point $Y$ is not considered a border, that is, countries like I and V do not share a common border. You must color the map such that countries with a common border have different colors.

![](https://cdn.mathpix.com/cropped/2024_05_01_d889f7c81233a28648eeg-1.jpg?height=234&width=311&top_left_y=1071&top_left_x=701)

Figure 21.1

(a) What is the minimum number of colors needed to color the map? Show how to color it.

(b) Draw another map of 6 countries that requires at least 4 colors to be painted. Show how to color it with colors A, B, C, and D.
(a) At a minimum, two colors are necessary, as shown in Figure 21.2.

(b) Figures 21.3 and 21.4 display two maps with six countries that require at least four colors to be painted.