Math 184A, Fall 2009, Prof. Tesler
Homework #9, Due Thursday December 3, 2009
There’s no homework due on Thanksgiving, so this is a two-week assignment. Approximate dates that
topics will be covered in class are listed below.
Chapter 9# 8∗, 9, 35 [Covered before assignment was posted]
Chapter 10# 2 [Covered before assignment was posted]
Problem H-31 [Scheduled to be covered Nov. 20]
Chapter 12# 1∗, 3, 11 and Problem H-32 [Scheduled to be covered Nov. 23]
Chapter 11# 15 and Problem H-33 [Scheduled to be covered Nov. 25]
Problem H-34 [Scheduled to be covered Nov. 30; will include
slides not in the book]
∗Notes:
•Chapter 9# 8: On (b,c), the book’s solution is incomplete because certain small values of nwork
differently (and you have to address these exceptions as well as the “general” case for a complete
answer). On (d), the definition of a star only makes sense for n≥3.
•Chapter 12# 1: The solution given in the book is incorrect.
Problem H-31. The problems below refer to the graph shown below in H-33(b). Recall that the length
of a walk/path/trail/cycle is the number of edges.
(a) Determine the number of cycles of length 4, starting and ending at vertex 1.
(b) Determine the number of walks of length 4 from vertex 1 to vertex 5.
If you have access to a calculator that does matrix arithmetic or to a program such as Matlab, you may
use it. If you don’t, the length 4 was chosen so that you can use a shortcut by hand, A4=(A
2
)
2
.
Problem H-32.
(a) A planar graph has 18 vertices and 40 edges. How many faces does it have?
(b) An ordinary soccer ball has 32 panels: 20 hexagons and 12 pentagons. The vertices, edges, and
faces of these panels give a planar graph.
(i) What degrees do the vertices have? How many faces does each vertex touch? How many faces
does each edge touch?
(ii) Consider the set T={(v, f ):vis a vertex, fis a face, vis on f}. Determine its size |T|
two ways: first, for each v, consider how many faces it’s in, and add that up over all v; and
second, for each f, consider how many vertices it has, and add that up over all f. Use this to
determine the total number of vertices.
(iii) Consider the set U={(e, f):eis an edge, fis a face, eis on f}. Determine its size in two
ways. Use this to determine the total number of edges.
(iv) Show that Euler’s formula (for planar graphs) holds.
Problem H-33. Determine the chromatic number of each of these graphs. Give an example of a proper
coloring of each of these graphs achieving that number of colors.
2(c)(a)
4
53
(b) 1
(d) Redraw graph (b) as a planar graph. In the redrawn graph, determine V, E, F and show that they
satisfy Euler’s formula (for planar graphs).
