5.4. Proof of the Havel-Hakimi Theorem

Try these first

1. 1.

   What is a degree sequence?

   Hint

   Think about how you would list the degrees of the vertices.

2. 2.

   Is the sequence (3, 2, 1) graphic? Why?

   Hint

   Try to visualize the connections between the vertices.

3. 3.

   Is the sequence (1, 1, 1) graphic?

   • True
   • False

   Hint

   Remember to visualize the connections!

4. 4.

   What does the Havel-Hakimi theorem help you determine?

   • If a graph is connected
   • If a degree sequence is graphic
   • The number of edges in a graph

   Hint

   Think about the theorem's purpose!

5. 5.

   Create a degree sequence of your choice that is graphic and explain why using Havel-Hakimi.

   Hint

   Start with the highest degree!

6. 6.

   Formulate a non-graphic sequence and demonstrate using Havel-Hakimi that it cannot be represented.

   Hint

   Note the importance of each decrement.