for Practice the Final
Instructions: Show all of your work. It is OK to leave your answer in terms of binomial
coecients, multinomial coecients, Stirling numbers, partition numbers and so on. Calculators and notes are not allowed.
Formulas
1. The number of compositions of n into k parts is
2. (degree formula)
v V
n1
k 1
.
deg(v) = 2|E|.
3. (inclusion-exclusion)
n
|
i=1
where AS =
i S
(1)|S | |AS |
An | =
S [n]
S =
Ai .
4. If G = (V, E ) is a connected planar graph and F is its set of faces then
|V | |E | + |F | = 2.
Also
f F
#sides(f) = 2|E|.
n
5. If A(x) =
n0
an xn , B (x) =
6. If A(x) =
n0
an xn /n!, B (x) =
n0 bn x
7. If A(x) =
n1
an xn /n!, B (x) =
n0 bn x
B (A(x)) =
8. If A(x) =
n1
n0 bn x
bm
m! i
n0 m0
an xn , B (x) =
then A(x)B (x) =
n
/n! then A(x)B (x) =
n
n
B (A(x)) =
n0
ak bnk xn .
n
n
k=0 k
ak bnk xn /n!.
n
ai1 aim xn /n!.
i1 , . . . , i m
then
ai 1 ai m x n .
bm
n0 m0
n
k=0
/n! then
1 ++im =n
n0 bn x
n0
i1 ++im =n
Short answer problems
1. IMPORTANT: you should work out problems from the chapter on Ramsey theory and
study the material from Chapters 1-9 in the book (at least the parts that we covered).
This practice sheet only covers chapters 10-12.
2. How many spanning trees does Kn,m have?
Solution. This is in the book. It uses the matrix-tree theorem.
3. A nite tree has 16 leaves and no degree 2 vertices. What is the maximum number of
vertices it could have? What is the minimum number of vertices it could have?
Solution. By the degree formula the maximum number occurs when all the other
vertices have degree 3. In this case if x is the number of degree 3 vertices then 3x +16 =
2E = 2(n 1) = 2(x + 16 1) = 2x + 30. So we get x = 30 16 = 14. So the maximum
number of vertices is 30. On the other hand, the minimum number of vertices occurs
for the star graph which has 17 vertices total.
4. Will any two spanning trees of a connected graph always have an edge in common? If
so, give a proof, and if not, a counterexample.
Solution: Figure 1 below shows two edge-disjoint spanning trees of K4 .
Figure 1:
5. Let G1 and G2 be two connected graphs. Let G3 be a graph obtained from G1 and G2
by adding a single edge from a vertex in G1 to a vertex in G2 . For example, see gure
2.
G1
G2
G3
Figure 2:
If there are x spanning trees of G1 and y spanning trees of G2 then how many spanning
trees of G3 are there?
Solution. xy .
6. How many spanning trees does an n-cycle have? Remember that an n-cycle is a connected graph in which every vertex has degree 2. So a 3-cycle is a triangle, a 4-cycle is
a rectangle, a 5-cycle is a pentagon, etc.
7. Which of the graphs in gure 3 is bipartite?
A
B
C
Figure 3:
8. What is the chromatic number of an n-cycle? Remember an n-cycle is the connected
graph with n vertices each of which has degree 2. So a triangle is a 3-cycle, a quadrilateral is a 4-cycle, etc.
9. Draw a planar graph with chromatic number 4 (i.e., a planar graph that is 4-colorable
but not 3-colorable).
10. The mathematics department has 6 committees each meeting once a month. How many
dierent meeting times must be used to ensure that no member is scheduled to attend
two meetings at the same time if the committees are
C1
C2
C3
C4
C5
C6
=
=
=
=
=
=
{Arlinghaus, Brand, Zaslavsky},
{Brand, Lee, Rosen},
{Arlinghaus, Rosen, Zaslavsky},
{Lee, Rosen, Zaslavsky},
{Arlinghaus, Brand},
{Brand, Rosen, Zaslavsky}?
11. Whats the chromatic number of K2,2,2 ? This is the graph in gure 8 below.
Solution. 3.
12. Whats the chromatic number of Kn,m ?
Solution. 2.
13. Draw an n-gon (for some n 3). Place an extra vertex in the center. Draw an edge
from every vertex of the n-gon to the vertex in the center. Whats the chromatic
number of the resulting graph?
Solution. If n is even the answer is 3. If n is odd the answer is 4.
14. Consider Kn . Let Kn equal Kn with all the edges of a Hamiltonian cycle removed.
Whats the chromatic number of Kn ? Hint: Kn has a subgraph isomorphic to Km for
some m. Finding out how large m can be will help you solve this.
Solution. If n is even the answer is n/2. If n is odd the answer is (n + 1)/2 (assuming
n > 3). To see this, you can nd a copy of Kn/2 inside Kn if n is even for example.
15. (a) State Halls Theorem.
(b) Use Halls theorem to show that any regular bipartite graph has a perfect matching. Recall: regular means that vertex has the same degree.
16. Mr. Jones brought home 6 dierently avored jellybeans for his 6 children. However,
when he got home, he found out that each child likes only certain avors. Amy will eat
only chocolate, banana or vanilla, while Burt only likes chocolate or banana. Chris will
eat only banana, strawberry and peach and Dan will accept only banana and vanilla.
Edsel likes only chocolate and vanilla and Frank will eat only chocolate, peach and
mint. Can each child get a jellybean he or she likes? If so, how? If not, why?
Solution: No. The reason is the Amy, Burt, Dan and Edsel only like chocolate, banana
and vanilla. So Halls condition is not satised.
17. A group of UN peacekeeping soldiers is to be divided into 2 person teams. It is important that the 2 members of a team speak the same language. The table shows the
languages spoken by the 8 soldiers:
Soldier Languages
1
G, E, Gr
2
C, G
3
E, J
4
K, E, S, R
5
C, S
6
F, J, A
7
K, F.
8
K, J, C
F=French,G = German, E=English, Gr=Greek, K=Korean, J=Japanese, A=Arabic,
R=Russian, S=Spanish, C=Chinese.
Is is possible to form 2-person teams so that, for each team, the 2 members speak the
same language? If so, divide them up. If not explain why.
18. Find a perfect matching for the graph in gure 4.
Figure 4:
19. Find a maximum matching for the graph in gure 5. How many edges does it have?
Remember, a maximum matching a is matching that has the maximum number of
edges possible.
Figure 5:
20. Find a maximum matching for the graph in gure 6. How many edges does it have?
Figure 6:
21. Find an augmenting path for the matching in gure 7.
Figure 7:
22. How many perfect matchings does Kn,n have? Remember, Kn,n is the graph with n
vertices on the left, n vertices on the right and every vertices on the left is connected
to an edge on the right by an edge.
23. Is K2,5 planar? If so, draw it so that no two edges cross.
24. Let G be a planar graph. Suppose the number of regions cut out by G is exactly 9 and
each region is triangular. Then how many vertices does G have?
25. Consider the graph K2,2,2 : it has 6 vertices that are grouped into 3 sets of 2 each. If
a, b are vertices in dierent groups then there is an edge from a to b. Is K2,2,2 planar?
A drawing of it is shown in gure 8. If it is planar, draw it in the plane without
intersecting edges. If not, explain why.
Solution. Its planar ! Think of the octahedron to see why.
26. Suppose each edge of Kn has been colored red or blue. Let be the number of
monochromatic triangles. For each i, let ri be the number of red edges incident to
Figure 8:
vertex i. Show that
=
n
1
3
2
n
ri (n 1 ri ).
i=1
n
3
Hint:
is the number of all triangles. So, you have to show that
counts the number of non-monochromatic triangles.
1
2
n
i=1 ri (n 1 ri )
27. Let Gn be the graph obtained from the complete graph Kn by removing one edge. How
many automorphisms does G have? For example, G3 has 2 automorphisms, G4 has 4
and G5 has 12.
Solution. 2(n 2)!. Suppose that the edge removed is {1, 2}. Then any permutation
of {3, . . . , n} is realized by an automorphism. This gives us (n 2)! automorphisms.
However, we can also interchange 1 and 2. So the number of automorphisms is 2(n 2)!.
To be more precise, note that any automorphism : [n] [n] must satisfy {1, 2} =
{ (1), (2)} and this is the only requirement (that is, all bijections from [n] to [n]
satisfying this condition are automorphisms of Gn ).
28. Let G be a connected graph. Let e be an edge of G and let G be the graph obtained
from G by removing e. Suppose that G is disconnected. Which one of the following is
most true?
(a) G must contain a Hamiltonian cycle.
(b) G cannot contain a Hamiltonian cycle.
(c) G might or might not contain a Hamiltonian cycle. It depends on other properties
of the graph G.
29. Suppose that a bipartite graph has 12 vertices on the left. Also suppose that the vertices
on the left each have degree 2 and the vertices on the right each have degree 6. Then
how many vertices are on the right?
30. How many subgraphs of Km,n are there that contain exactly n + m vertices?
Solution. The number of edges of Km,n is mn. The answer is 2mn .
31. Consider the graph K2,6 . How many vertices must you remove in order to obtain
a disconnected graph? When a vertex is removed, all edges containing it are also
removed.
Solution. 2.
Proof problems
Instructions: These problems require you to write readable proof with each statement fully
justied. Your answer should be understandable to someone who does not already know the
solution. Even if a problem has a simple Yes or No answer you will not get credit unless
you fully justify your answer.
1. Let G be a simple graph on n vertices. If G is disconnected, what is the largest number
of edges G can have?
2. Let G be a connected simple graph with n vertices and m n 1 edges. Show that
G has at least m n + 1 cycles. We consider two cycles to be dierent only if there is
some edge used by one of them that is not used by the other.
Solution. Let T be a spanning tree of G. Then T has n 1 edges. So there are
m n + 1 edges not in T . Each one of these edges is contained in a cycle of G. In fact
if e is such an edge then T {e} contains exactly one cycle which necessarily contains
e. So the number of cycles is at least the number of edges in the complement of T .
3. Let G be a simple graph. Suppose 0 is an integer and the degree of every vertex
in G is at most . Show that (G) + 1.
4. Prove that K5 is not a planar graph. (You are not allowed to use Kuratowskis theorem
of course).
5. Prove that in any polyhedron, there are two faces that have the same number of sides.
Solution. Consider the dual graph. You know from a homework problem that there
must be two vertices of the dual graph that have the same degree. The corresponding
faces of the polyhedron have the same number of sides.
6. A zoo wants to set up natural habitats in which to exhibit its animals. Unfortunately,
some animals will try to eat some of the others when given the opportunity. How can
a graph and a proper coloring be used to determine the number of dierent habitats
needed and the placement of the animals in these habitats?
Answer: Dene a graph as follows. The vertices of the graph correspond to the animals
and the edges correspond to pairs of animals in which one of the animals in the pair
would eat the other if given the opportunity. The colors in a proper coloring correspond
to the habitats. Because adjacent vertices receive dierent colors, no pair of animals
in which one would eat the other are in the same habitat. So the minimum number
of colors needed to properly color the graph equals the minimum number of habitats
needed.
7. Let G be a planar. Suppose every vertex has even degree. Show that the dual graph is
bipartite. (Remember the dual graph has one vertex for every face and an edge in the
dual graph connects two vertices if the corresponding faces are adjacent).
Solution. If every node of a planar map has even degree then every cycle in the dual
graph has even length. Hence the dual graph is 2-colorable which implies that the faces
of the map can be 2-colored.