Math 3113, Section 4

Fall 1999

Suggestions for exercises 4.3.9 and 4.3.10

- Here are some questions that may help you get started.
- How many faces does a tree have?
- To count the edges of the infinite (or exterior) face, consider the following picture. Think of going around its boundary according to the arrows.

- How many edges are there in the tree that is shown?
- How many edges did you count for the exterior face in that tree? In other words, what is the degree of the exterior face?
- In general, if the number of edges is
*E,*how many edges will we count for the exterior face?

- Think of building a general graph by starting with a tree and adding edges to it. In particular, consider the following picture:

- When we add the green edge, what happens to
- The value of 2
*E* - The total of the degrees of the faces. (This will take a bit longer to figure out, since there's now an additional edge.
- Answer the same two questions if we now add the red edge.
- Try to figure out what happens in general to the following quantities when we add an edge:
- The value of 2
*E* - The sum of the face degrees. (This requires a bit more explaining.)

Back to the syllabus.