18.2.6. Legal Coloring of Graphs

Try these first

1. 1.

   What is a graph?

   Hint

   Think about how points and connections relate.

2. 2.

   What must be true for two adjacent vertices in a colored graph?

   Hint

   Consider the purpose of distinguishing points.

3. 3.

   What is the main goal of graph coloring?

   • To assign colors randomly
   • To ensure no adjacent vertices share the same color
   • To create decorative maps
   • To analyze map sizes

   Hint

   Think about the purpose of making distinctions between adjacent regions.

4. 4.

   True or False: The four color theorem states that at least four colors are needed for any planar map.

   • True
   • False

   Hint

   Revisit what the theorem's assertion is regarding color necessity.

5. 5.

   Create a graph of a fictional country with at least 10 states, ensuring no two neighboring states share the same color. Present your graph and explain your coloring strategy.

   Hint

   Start by mapping out the connections before coloring.

6. 6.

   Examine a complex map that uses more than four colors. Discuss how the four color theorem applies despite the use of additional colors for effective distinctions.

   Hint

   Think about artistic intent versus mathematical requirements.