Question

In the map-coloring problem, why do we require that countries be connected (and not in multiple pieces like Russia or Michigan)? Draw a map, in which disconnected countries are permitted, that requires more than four colors.

Answer 1

In map coloring, requiring that countries be connected ensures adjacent countries do not have the same color, limiting the number of colors needed. Disconnecting countries can result in more than four colors being required.

Step-by-step explanation:

In the map-coloring problem, the requirement that countries be connected is necessary in order to ensure that adjacent countries do not have the same color assigned to them. The goal of map coloring is to assign colors to regions (countries, states, etc.) in such a way that no two adjacent regions have the same color. If countries were allowed to be disconnected, it would be possible to assign different colors to each disconnected piece, thereby requiring more than four colors.

For example, if we consider a map with three disconnected countries, each consisting of multiple pieces, it would be necessary to assign a different color to each piece, resulting in a need for more than four colors. Here is an example:

In this map, the three disconnected countries require five colors to color them in such a way that no two adjacent pieces have the same color.