5.3.1. Havel-Hakimi Theorem

Try these first

1. 1.

   What is a degree sequence?

   Hint

   Think about how you count edges connected to vertices.

2. 2.

   Is the sequence (5, 3, 1, 0) a graphic sequence?

   Hint

   Remember the sum of degrees must be even.

3. 3.

   What is the first step in applying the Havel-Hakimi theorem?

   • Remove the highest degree
   • Sum the degrees
   • Arrange in ascending order

   Hint

   What do we do with the first value?

4. 4.

   True or False: The sum of the degrees in any graphic sequence must be odd.

   • True
   • False

   Hint

   Think about the properties of edges and vertices.

5. 5.

   Formulate a degree sequence of your choice and prove using the Havel-Hakimi theorem whether it’s graphic or not.

   Hint

   Apply each step and track the degrees closely.

6. 6.

   Construct a graph based on a successful realization of a graphic sequence. Demonstrate it visually.

   Hint

   Drawing it helps visualize the relationships that confirm the degrees.