6.2.2. Case When n is Odd

Try these first

1. 1.

   Define a graphic sequence.

   Hint

   Think about what degree means in a graph.

2. 2.

   What is the Havel-Hakimi theorem used for?

   Hint

   Recall how we prove sequences are graphic.

3. 3.

   What is the required edge chromatic number for a complete graph when n is odd?

   • n-1
   • n
   • n+1

   Hint

   Recall the relationship between edge chromatic numbers and vertex count.

4. 4.

   True or False: The Havel-Hakimi theorem can be applied to determine graphic sequences.

   • True
   • False

   Hint

   Consider the purpose of the theorem.

5. 5.

   Given the sequence [5, 5, 4, 4, 3, 3, 2, 1], prove if it is graphic and present a construction.

   Hint

   Follow the theorem's steps for reducing the sequence.

6. 6.

   Calculate the minimum number of colors required for a complete graph of 11 vertices and explain your reasoning.

   Hint

   Think about how odd numbers affect coloring.