1. What is the chromatic number of this graph?

2. We have found a four-coloring. How do we know the chromatic number is not less than four?

3. A planar graph has 11 vertices whose degrees are 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7. How many regions must it have?

4. The sum of the degrees of the vertices is 52, so the graph has 26 edges. By Euler’s Formula the number of regions is 26 – 11 + 2 = 17.

5. Create a binary search tree for the list c, f, a, j, g, i, b, d, k, h, e using alphabetical order.

7. How many leaves does a full 5-ary tree with 42 internal vertices have?

8. A full 5-ary tree with 42 internal vertices has 5(42) + 1 = 211 vertices and 4(42) + 1 = 169 leaves. If n = mi + 1 and n = l + i, then l + i = mi + 1 and l = (m – 1)i + 1.

9. Use a breadth-first and then a depth-first search algorithm to identify a spanning tree of the graph below. List the edges in the order they are chosen. (Assume edges are ordered alphabetically.

12. A chain letter starts with a person sending a letter out to five others. Each person who receives the letter either sends it to five other people who have never received it or does not send it to anyone. Suppose 10,000 people send out the letter before the chain ends and that no one receives more than one letter. How many people receive the letter, and how many do not send it out?

13. The model here is a full 5-ary tree. We are told that there are 10,000 internal vertices (these represent the people who send out the letter). By Theorem 4(ii) we know that n = mi + 1 = 5(10,000) + 1 = 50,001. Everyone but the root receives the letter, so we conclude that 50,000 people receive the letter. There are 50,001 – 10,000 = 40,001 leaves in the tree, so that is the number of people who receive the letter but do not sent it out.