Homework 0  Math 108, Frank Thorne (fthorne [at] math.stanford.edu)
Due Friday, September 24
Instructions: These problems are designed to gauge your background in combinatorics and
some related topics of interest. These problems do not necessarily repres
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2007
Homework 5
Due: Friday, June 1
1. (a) Let G be the graph on 5 vertices a, b, c, d, and ewith edge set (a, b), (a, c), (a, d),
(b, c), (b, d), (c, e). Compute the chromatic polynomial of G. (You may use the
contractiondeletion recurrence or the Mobius fun
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
HW # 1
DUE MONDAY, APRIL 7
vL followed by a number gives a problem from your text. Anything
marked as presentation, with a date, does not need to be handed in,
but you are encouraged to think about it before it is presented in class.
(0) Read Chapter 1.
(
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
HW # 1
DUE WEDNESDAY, APRIL 16
vL followed by a number gives a problem from your text. Anything
marked as presentation, with a date, does not need to be handed in,
but you are encouraged to think about it before it is presented in class.
(0) Read Chapter
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
HW # 6
DUE WEDNESDAY, MAY 21
vL followed by a number gives a problem from your text. Anything
marked as presentation, with a date, does not need to be handed in,
but you are encouraged to think about it before it is presented in class.
(0) Read Chapter 14
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
HW # 3
DUE WEDNESDAY, APRIL 23
vL followed by a number gives a problem from your text. Anything
marked as presentation, with a date, does not need to be handed in,
but you are encouraged to think about it before it is presented in class.
(0) Read Chapter
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
HW # 4
DUE WEDNESDAY, MAY 7
vL followed by a number gives a problem from your text. Anything
marked as presentation, with a date, does not need to be handed in,
but you are encouraged to think about it before it is presented in class.
(0) Read Chapter 13
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
HW # 5
DUE WEDNESDAY, MAY 14
vL followed by a number gives a problem from your text. Anything
marked as presentation, with a date, does not need to be handed in,
but you are encouraged to think about it before it is presented in class.
(0) Read Chapter 10
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
Math 108  Homework 4 Solutions
May 7, 2014
Problem 1 The adjacency matrix of this graph is:
0 1 0
1 0 1
S=
0 1 0
1 0 1
1
0
1
0
and the characteristic polynomial of S is
p(x) = x2 (x2 4).
Therefore, the eigenvalues of the graph are 2, 0, 0, 2.
Problem
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
Math 108  Homework 3 Solutions
April 23, 2014
Problem 1 On a planar graph, we have that 3f 2e since one face has at least 3 edges and
one edge has at most 2 faces. Therefore, using Eulers formula
1
1
deg(x) + v
2=f e+v e+v =
3
6 xV
which gives that id a.
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
Math 108  Homework 2 Solutions
April 16, 2014
Problem 2A
Recall that two graphs G and G are said to be isomorphic if there exists a bijection
: V (G) V (G )
such that
cfw_v, w E(G) cfw_(v), (w) E(G ),
that is: two vertices v, w of G are linked by an edg
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
Math 108  Homework 5 Solutions
May 14, 2014
Problem 13F
Proof 1: Do it by induction:
For n=2 it is easy to check. Assume that for Sn it is true that the number of permutations
with even number of cycles (denoted here as EV (n) is the same as the number o
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
Math 108  Homework 1 Solutions
April 7, 2014
Problem 1B
G is a simple graph with 10 vertices that is not connected. Thus, it has at least two connected
components. We have to show that:
Claim 1. The number of edges E(G) is no larger than 36.
Suppose th
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2014
Math 108  Homework 3 Solutions
March 21, 2014
Problem 10C
Notice that
(n)
nx
x
=
n
(n)
nN
x
n
x
n
= 0 for n > x.
since (n)
x
We will prove that nN (n) n = 1 by induction on x. For x=1 we have that
(1) = 1. Now if the assumption is true for x 1, then for
Introduction to Combinatorics and Its Applications
MATH 108

Winter 2015
Math 108 Combinatorics
Spring 2007
Homework 4 Solutions
1. Let (G, c) be a network with distinguished vertices s and t. A minimum cut in G is a
pair of sets (X, Y ) such that V = X Y , s X, t Y , X Y = , and c(X, Y ) is
minimal. Suppose (X, Y ) and (A, B)
Introduction to Combinatorics and Its Applications
MATH 108

Winter 2015
Math 108 Combinatorics
Spring 2007
Homework 3 Solutions
1. How many positive integers less than 1000 have no factor strictly between 1 and 10?
Solution. A positive integer does not have a factor between 1 and 10 if and only if it
does not have a factor of
Introduction to Combinatorics and Its Applications
MATH 108

Winter 2015
Math 108 Combinatorics
Spring 2007
Homework 2 Solutions
1. Given n letters, of which m are identical and the rest are all distinct, nd a formula
for the number of words which can be made.
Solution. If we use i of the n m distinct letters and j of the m id
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2007
Homework 4
Due: Thursday, May 17
1. Let (G, c) be a network with distinguished vertices s and t. A minimum cut in G is
a pair of sets (X, Y ) such that V = X Y , s X , t Y , X Y = and c(X, Y ) is
minimal. Suppose (X, Y ) and (A, B ) are minimal cuts. Show
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2007
Homework 3
Due: Thursday, May 3
1. How many positive integers less than 1000 have no factor between (and not including) 1 and 10?
2. Prove
x3 (1 + 2x)
.
S (n, n 2)x =
(1 x)5
n0
n
3. Suppose cfw_bn n1 is the sequence of Bernoulli numbers (see homework 2).
Introduction to Combinatorics and Its Applications
MATH 108

Spring 2007
Homework 2
Due: Thursday, April 19
1. Given n letters, of which m are identical and the rest are all distinct, nd a formula
for the number of words which can be made.
2. Give a recursion and a direct formula for the numbers in the sequence cfw_a0 , a1 , a
Homework 1  Math 108, Frank Thorne (fthorne [at] math.stanford.edu)
Due Friday, October 1, at 4:30 p.m.
Please note: You will be graded on both correctness and quality of exposition. Please show
your work and explain your reasoning clearly.
1. Prove that
Homework 2  Math 108, Frank Thorne (fthorne [at] math.stanford.edu)
Due Friday, October 8, at 4:30 p.m.
Please note: You will be graded on both correctness and quality of exposition. Please show
your work and explain your reasoning clearly.
1. Prove that
Homework 3  Math 108, Frank Thorne (fthorne [at] math.stanford.edu)
Due Friday, October 15, at 4:30 p.m.
Please note: You will be graded on both correctness and quality of exposition. Please show
your work and explain your reasoning clearly.
It is the ey
Homework 4  Math 108, Frank Thorne (fthorne [at] math.stanford.edu)
Due Friday, October 22, at 4:30 p.m.
Please note: You will be graded on both correctness and quality of exposition. Please show
your work and explain your reasoning clearly.
Remember tha
Homework 5  Math 108, Frank Thorne (fthorne [at] math.stanford.edu)
Due Friday, November 5, at 4:30 p.m.
Please note: You will be graded on both correctness and quality of exposition. Please show
your work and explain your reasoning clearly.
There is no
Homework 6  Math 108, Frank Thorne (fthorne [at] math.stanford.edu)
Due Friday, November 12, at 4:30 p.m.
Please note: You will be graded on both correctness and quality of exposition. Please show
your work and explain your reasoning clearly.
I dont care
Homework 7  Math 108, Frank Thorne (fthorne [at] math.stanford.edu)
Due Friday, November 19, at 4:30 p.m.
Please note: You will be graded on both correctness and quality of exposition. Please show
your work and explain your reasoning clearly.
1. In Stanl
Homework 8  Math 108, Frank Thorne (fthorne [at] math.stanford.edu)
Due Friday, December 3, at 4:30 p.m.
Please note: This homework looks long, but a lot of the questions are for extra credit.
Why give multiple proofs of the same result when, clearly, on
Problem 1
If we remove two edges from K6 , there are two cases. First, if the two edges
have a vertex in common, then the subgraph on the other ve vertices is K5 .
This is clear because we removed no edges between any of these ve vertices.
Otherwise, if t