5.4.2.1. Case 1: Vertex v is Adjacent

Try these first

1. 1.

   What is a degree sequence?

   Hint

   Think about how you order numbers.

2. 2.

   Can a degree sequence contain negative numbers?

   Hint

   Consider what a negative degree would imply.

3. 3.

   What is a graphic sequence?

   • A sequence with negative numbers
   • A sequence that can represent a simple graph
   • An unordered collection of numbers

   Hint

   Consider the definitions we just discussed.

4. 4.

   True or False: The sum of degrees in a graphic sequence can be odd.

   • True
   • False

   Hint

   Remember the connection between edges and vertex degrees.

5. 5.

   Prove that the sequence (4, 3, 2, 2, 1) is graphic using the Havel-Hakimi theorem.

   Hint

   Use the steps of reduction methodically.

6. 6.

   Construct a graphic sequence for a simple triangle and a simple square.

   Hint

   Consider the number of vertices and edges in each shape.