6.2.1. Case When n is Even

Try these first

1. 1.

   Define a graphic sequence.

   Hint

   Think about what degree means in relation to graph vertices.

2. 2.

   What does the Havel-Hakimi theorem do?

   Hint

   Recall the theorem's role in validating degree sequences.

3. 3.

   What is a graphic sequence?

   • A sequence of colors
   • A sequence of integers that shows vertex degrees
   • A method for coloring vertices

   Hint

   Think about how degrees relate to vertex connections.

4. 4.

   True or False: In an even-numbered vertex graph, more than n/2 edges can share the same color.

   • True
   • False

   Hint

   Recall the limitations on connectivity within edge coloring.

5. 5.

   Given a sequence of degrees (4, 3, 3, 1), can this represent a graphic sequence? If so, construct the graph. If not, explain why.

   Hint

   Start with the highest degree and connect accordingly while verifying at each step.

6. 6.

   You have a complete graph of 10 vertices. How can you schedule a round-robin tournament such that no team plays more than once per day? Describe your coloring scheme.

   Hint

   Think about rotation based on positions to avoid repeating encounters.