Discrete Mathematics Demystified

CHAPTER 8 Graph Theory 181
Exercises
1. Give an example of a graph on five vertices without an Euler path.
2. Give an example of a graph on five vertices with two distinct Euler paths.
3. Imagine a torus with two handles. What would be the correct Euler formula
for this surface? It should have the form
V − E + F = (some number)
What is that number? The number χ on the right hand side is called the Euler
characteristic of the surface.
4. Consider the complete graph on six vertices. How many edges does it have?
How many faces?
5. How many edges does the complete graph on k vertices have?
6. Consider a graph built on two rows of three vertices for a total of six vertices.
Construct a graph by connecting every vertex in the first row to every vertex
of the second row and vice versa. How many edges does this graph have?
7. Consider the standard picture of a five-pointed star. This can be thought of
as a graph. How many vertices does it have? How many edges?
8. Give an example of a graph with more vertices than edges. Give an example
of a graph with more edges than vertices.
9. If a surface can be described as a “sphere with g handles,” then we say it has
genus g. Thus a lone sphere has genus 0, a torus has genus 1, and so forth.
Based on your experience with Exercise 3 above, posit a formula that relates
the Euler characteristic χ with the genus g.