Question

In the map-coloring problem, why do we require that countries be connected (and not in multiple pieces like Russia or Michigan)?

A) To ensure uniformity in size
B) To simplify the map layout
C) To avoid confusion in borders
D) To guarantee distinct coloring

Answer 1

We require countries in the map-coloring problem to be connected to guarantee distinct coloring and avoid confusion in borders, simplifying the map layout.

Step-by-step explanation:

In the map-coloring problem, we require that countries be connected, and not in multiple pieces, primarily to guarantee distinct coloring. This is essential because the map-coloring problem in mathematics, specifically in graph theory, aims to minimize the number of colors needed to color a map while ensuring that no two adjacent areas share the same color. Requiring that territories be contiguous simplifies the problem by making it easier to determine adjacency and thus the coloring requirement.

When territories are in multiple pieces, like Russia or Michigan, it could lead to confusion in borders and increase the complexity of the coloring problem, since disconnected regions may be treated as adjacent in the graph representation. For instance, parts of a single country isolated by another country may mistakenly seem adjacent even if they are not directly connected. To simplify map layout and minimize potential errors, connected territories enable a more straightforward application of the four-color theorem, which asserts that no more than four colors are needed to achieve such a coloring.