Homework 7 – Due Wednesday, May 27
Section 1.3 (page 28): 3, 9
Section 1.4 (page 38): 6, 8, 12ab, 25
Chapter 1 Supplementary (page 43): 6, 30, 31, 38
Section 2.1 (page 53): 16
Supplemental problems:
1. Let Gbe a graph on nvertices. Show that if every vertex has degree at least (n−1)/2 then
Gis connected.
2. Jellybeans of eight different colors are in six jars. There are twenty jellybeans of each color.
Prove that there must be a jar containing two pairs of jellybeans of two different colors of
jellybeans. (Hint: use the pigeonhole principle.)
3. There is a huge bin filled with thousands of balls of many different colors. What is the smallest
number nof balls you need to take from the bin so that among the nballs you must have
either seven balls of one color or seven balls with no two the same color.
(Your answer needs two parts, first show that with nballs you must satisfy the condition,
second show that with n−1 balls you might not satisfy the condition.)
4. Fullerenes are molecules composed entirely of carbon, the most famous of which is Buck-
minsterfullerene (C60) which is composed of 60 carbon atoms in the shape of a soccer ball.
These three dimensional molecules can be represented in the plane by simple connected planar
graphs where each vertex corresponds to a carbon atom and edges represent bonds between
atoms. In the graph corresponding to the fullerene each vertex has degree three and each face
must be either a pentagon (five-sided) or a hexagon (six-sided).
Show that in every fullerene (regardless of the number of carbon atoms) there are exactly
twelve pentagons.
5. A graph has an Eulerian cycle if there is a closed walk which uses each edge exactly once.
A related problem is to find the shortest closed walk (i.e., using the fewest number of edges)
which uses each edge at least once. (This is known as the “Chinese Postman” problem and
comes up frequently in applications for optimal routing.) Consider the following graph on six
vertices.
1
2
3
4
56
(a) Find a walk that starts and ends at vertex 1 of length 11 so that each edge in the graph
is used at least once.
(b) Explain why 11 is the smallest number of edges needed to find a walk that starts and ends
at vertex 1 and uses each edge at least once.
