18.2. Graph Coloring Problem

Try these first

1. 1.

   Define 'vertex' in the context of graph coloring.

   Hint

   What do we call the dots in graph coloring?

2. 2.

   What is an edge in graph theory?

   Hint

   Think of what connects two states on a map.

3. 3.

   What does the Graph Coloring Problem address?

   • Determining the shortest path in a graph
   • Assigning colors to vertices without adjacent vertices sharing the same color
   • Finding all possible edges in a graph

   Hint

   Think about how we represent maps and relationships.

4. 4.

   True or False: According to the Four Color Theorem, five colors are required to color any planar map.

   • True
   • False

   Hint

   Remember the minimal color requirements established by the theorem.

5. 5.

   Consider a graph represented with a triangular layout of vertices where each vertex is connected to every other vertex (a complete graph). How many colors are needed to color this graph according to the graph coloring principles?

   Hint

   Think about the maximum adjacency possible between vertices.

6. 6.

   Given a complex planar graph with multiple connections and vertices representing cities in a scheduling problem, devise a coloring strategy to assign no more than four colors under the rule that connected cities must have different colors. Explain your solution.

   Hint

   Visualize the graph and plan a step-wise approach.