3.1.3. Vertex Chromatic Number

Try these first

1. 1.

   What is the vertex chromatic number?

   Hint

   Think about how colors can impact adjacency.

2. 2.

   Explain the greedy algorithm in simple terms.

   Hint

   What does 'greedy' imply when choosing colors?

3. 3.

   What does the vertex chromatic number indicate?

   • The number of edges
   • Minimum colors needed to color vertices
   • Maximum degree of a vertex

   Hint

   Focus on the definition of vertex chromatic number.

4. 4.

   True or False: The greedy algorithm guarantees an optimal vertex coloring solution.

   • True
   • False

   Hint

   Consider examples where coloring can differ.

5. 5.

   Given a graph with 6 vertices and edges connecting them in a way that forms a cycle, calculate its chromatic number and explain your reasoning.

   Hint

   Count the edges and see how they connect!

6. 6.

   Design a scenario in which a greedy algorithm fails to find the optimal solution in vertex coloring, detailing your graph structure.

   Hint

   Imagine selecting corners of triangles one by one and how color constraints apply.