6. Question 9: Proving a Graphic Sequence

Try these first

1. 1.

   Define a graphic sequence.

   Hint

   Think about what a graph represents.

2. 2.

   What does the Havel-Hakimi theorem help prove?

   Hint

   Consider what path or edges we can visualize.

3. 3.

   Is the sequence [5, 2, 2, 1] graphic?

   • True
   • False

   Hint

   Check the degrees against how many connections they can create.

4. 4.

   What condition must be met for a sequence to be graphic?

   • All integers must be even
   • Sum of degrees must be even
   • Degree values must be prime

   Hint

   Think of how edges are counted in a graph.

5. 5.

   Prove that the sequence [6, 5, 4, 4, 3, 2] is graphic by constructing a graph.

   Hint

   Remember to adjust your connections based on degree constraints.

6. 6.

   Using the Havel-Hakimi method, show that the sequence [1, 1, 1, 0] cannot be graphic.

   Hint

   Focus on the odd count of degrees in the sequence.