Math 1050 : Combinatorics : Homework
Homework 1 -- assigned on Sept 9 -- due Sept 16
1. Section 2.9: Problem 2
2. Section 2.9: Problem 16 (a) and (e)
3. Show that the number of ways to choose a committee with a chair person
(for the committee) from a group of $n$ people is equal to:
$n 2^{n-1}$
(the size of the committee can be any number from $1$ to $n$).
$~~$ Also, use another way of counting to show that the same number is equal to:
$\sum_{k=1}^n k {n \choose k}.$
$~~$ [Hint: To obtain the second formula, pick the committee of size
$k$ and then pick a chair in the committee.]
4. Section 2.9: Problem 11
Homework 2 -- assigned on Sept 16 -- due Sept 23
1. Section 2.9: Problem 24
2. Section 2.9: Problem 30
$~~$ [Hint: The answer is in terms of multinomial numbers.]
3. Section 2.9: Problem 32
4. Section 2.9: Problem 33
5. Section 3.11: Problem 2
6. (Bonus problem) As in Example 3.1 consider a collection of $n$
(distinct) lines in the plane such that:
$~~~~~~~~~~~$ (a) any two of them intersect; $~~~$ (b) no three of them pass through the
same point.
$~~$ Show that the regions in the plane created by these lines can be colored with two
colors B and W such that no two neighboring regions have the same
color.
$~~$ We say that two regions are neighboring if they have a line or line segment in
common.
$~~$ [Hint: do this recursively.]
Homework 3 -- assigned on Sept 23 -- due Sept 30
1. Section 3.11: Problem 4
2. Section 3.11: Problem 16
3. Section 3.11: Problem 18
4. Section 4.6: Problem 1 (e), (g)
5. (Bonus problem) Section 3.11: Problem 20
Homework 4 -- assigned on Sept 30 -- due Oct 7
1. Section 5.9: Problem 2
2. Section 5.9: Problem 7
3. Section 5.9: Problem 8
4. Section 5.9: Problem 10
5. Section 5.9: Problem 12
$~~$ [Hint: use Theorem 5.5 about Hamiltonian graphs.]
Homework 5 -- assigned on Oct 16 -- due Oct 23
1. Section 5.9: Problem 14
2. Section 5.9: Problem 30
3. Section 5.9: Problem 38
4. Section 5.9: Problem 40
5. Prove that the chromatic number of a tree $T$ (with more than one vertex) is $2$.
Homework 6 -- assigned on Oct 24 -- due Oct 30
1. Section 7.7: Problem 4
2. Section 7.7: Problem 6
3. Section 7.7: Problem 16
4. Section 7.7: Problem 20
5. (Bonus problem) Section 7.7: Problem 22 (a)
Homework 7 -- assigned on Oct 30 -- due Nov 6
1. Section 8.8: Problem 2 (f) and (l)
2. Section 8.8: Problem 4 (b) and (e)
3. Section 8.8: Problem 5
4. Let $a_n$ be the number of ways that a number $n$ can be written as a sum of integers:
$n = x_1 + x_2 + x_3 + x_4$, where $x_1 \geq 0, ~x_2 \geq 0, ~ x_3 \geq 1$ and $0 \leq
x_4 \leq 1$.
$~~$ Write the generating function for $(a_n)$ in closed form. Find a formula
for $a_n$ in terms of $n$.
5. Let $(b_i)$ be the sequence defined recursively by:
$b_{i+2} = b_{i+1} + 2b_i$ and $b_0 = 1,~ b_1 = 1$.
$~~$ Find the generating function for this sequence.
Homework 8 -- assigned on Nov 9 -- due Nov 16
1. (a) In the proof of Ramsey's theorem, we in fact showed that:
$R(m, n) < R(m-1, n) + R(m, n-1)$.
$~~$ Using this inequality and material/examples
discussed in class, show that $R(3, 4)$ is at most 9.
$~~$ (b) (Bonus problem) To show that $R(3, 4)$ is exactly 9 we need show that
$R(3, 4) > 8$.
$~~$ To do this,
we need to color the edges of the complete graph $K_8$ by two colors Red and Blue such
that, there is no Red triangle and there is no $K_4$ subgraph which is
Blue.
$~~$ Can you find such a coloring of edges of $K_8$?
$~~~~~~$ [Hint: the graph of Red edges is an $8$-gon with
all its main diagonals, i.e. connect each vertex by Red to vertices before and after it as
well as the vertex exactly opposite to it.]
2. One can define Ramsey numbers for more than two colors.
$~~$ The Ramsey number
$R(n_1, ..., n_s)$ is the smallest number $r>0$ such that if we color the edges of
a complete graph $K_r$ with $s$ colors $C_1, \ldots, C_s$ then there
exists:
$~~~~$ * $~$ either a complete subgraph $K_{n_1}$ colored $C_1$,
$~~~~$ * $~$ or a complete subgraph $K_{n_2}$ colored $C_2$,
$~~~~$ * $~$ or ...
$~~~~$ * $~$ or a complete subgroup $K_{n_s}$ colored $C_s$.
$~~$ The generalized Ramsey theorem says that for any integers $n_1, \ldots, n_s$
such a number $r = R(n_1, ..., n_s)$ exists.
$~~$ Use the generalized Ramsey theorem to prove the following:
$~~$ For any $s \geq 2$, there is $r > 3$ such that for any coloring of the numbers
$1,2,\ldots,r$ with $s$ colors, there are three integers $1 \leq x, y, z \leq r$ of the same
color such that $x + y = z$.
$~~$ [Hint: let $n$ be the Ramsey number $R(3, 3, \ldots , 3)$ where
$3$ is repeated $s$ times.
$~~$ Recall that $r = R(3,3,\ldots,3)$ has the property that if the
edges of a complete graph $K_r$ are colored with $s$ colors then there is a triangle in
$K_r$ all whose edges have the same color.
$~~$ Color the edges $(i, j)$ of $K_r$ as follows:
assume $j>i$ then color the edge by the color $j-i$.]
Homework 9 -- assigned on Nov 20 -- due Nov 30
1. Section 12.6: Problem 10: Prove that if no two weights are equal in a weighted simple
graph, then there is a unique minimum weight spanning tree.
2. Section 12.6: Problem 11
3. Section 13.7: Problem 2
Homework 10 -- assigned on Dec 4 -- due Dec 11
The three last problems of the semester