Question

Coloring Maps

Q1) Consider the map depicted below, that we wish to colour using, yellow, green, red, and blue. gre reen (a) What colour must A be? b) What colour must B be? (c) If we want to color a globe (on an arbitrary planet with arbitrary countries) where bodies of water are blue, will 4 colours always be enough? countries such as Palestine, the United States, or Newfoundland and Labrador? maps, that it is possible to colour a map/globe of the world, so that all bodies (d) What does this example also say about colouring a globe with disconnected (e) Assuming all countries are connected. Prove, using the 4 colour theorem for of water are blue using 5 colours.1

Answer 1

If we want to color a globe (on an arbitrary planet with arbitrary countries) where bodies of water are blue, will 4 colors always be enough? gree blu Fig. 1 a) See the figure above. From the coloring shown above it is clear that A can get color red. However, two borders of A which are already colored with yellow and green, so option for A is either blue or red.

EA İs colored as blue, then the following two situations arise and in both cases it is not possible to complete the coloring case 2 case 1 Fig. 2 Hence A must get color red. b) By taking various colors for B we find that color for B must be Yellow. See fig. 1 c) If we want to color a globe (on an arbitrary planet with arbitrary countries) where bodies of water are blue, will 4 colors always be enough? Ans. If we want to color a globe (on an arbitrary planet with arbitrary countries) where bodies of water are blue, 4 colors will always may not be enough.

d) Colouring a globe with disconnected countries as Palestine, the United States, or Newfoundland and Labrador, if the countries get same color then it may not be possible to color with four colors. Only if, disconnected regions of same states are allowed to color with any of the required four colors, then the four-colour- theorem asserts that it is possible to color with four colors.