5.3. Characterization of Graphic Sequences

Try these first

1. 1.

   What is the degree sequence of the graph with vertices of degrees 3, 2, and 1?

   Hint

   List the degrees in non-increasing order.

2. 2.

   Is the sequence (4, 3, 2, 1) a graphic sequence?

   Hint

   Try to visualize a graph.

3. 3.

   Which of the following conditions must a sequence meet to be considered graphic?

   • All degrees must be negative.
   • The sum of degrees must be even.
   • The sequence must be in random order.

   Hint

   Recall the relationship between edges and vertex degrees.

4. 4.

   True or False: Every non-negative sequence is a graphic sequence.

   • True
   • False

   Hint

   Consider the necessary conditions we've discussed.

5. 5.

   Given a sequence (6, 5, 4, 3, 2, 1), prove whether this is a graphic sequence using the Havel-Hakimi theorem.

   Hint

   Calculate the sum to verify.

6. 6.

   Construct a simple graph from the degree sequence (2, 2, 2, 2).

   Hint

   Ensure each vertex connects with exactly two others.