5.3.2. Construction of Sequence S*

Try these first

1. 1.

   What conditions must a degree sequence satisfy to be considered graphic?

   Hint

   Think about the properties of edges in a graph.

2. 2.

   Is the sequence (3, 2, 2, 1) graphic? Justify your answer.

   Hint

   Try to visualize or sketch the connections based on the degree counts.

3. 3.

   What is a graphic sequence?

   • A sequence of negative degrees
   • A sequence that can represent a simple graph
   • A random sequence

   Hint

   Recall the definition of graphic sequences discussed.

4. 4.

   True or False: The sum of the degrees in any simple graph must be odd.

   • True
   • False

   Hint

   Look back at the degree properties we analyzed.

5. 5.

   Given a degree sequence of (4, 4, 4, 2, 2), prove its graphic nature or lack thereof using appropriate methods.

   Hint

   Visualize the connections as you apply the theorem methodically.

6. 6.

   Create a counter-example to show why the sequence (3, 1, 1, 0) cannot be a graphic sequence.

   Hint

   Try to sketch the configuration or visualize vertex limits.