u
u u f u  Cs qx Cs  Cs 0
u v w r o tge w
Rbhlmhfnqphu fhVRygmw w
u xfh0yu~wuhxbnheqwhhnnhqennI0u v Tyowy~nhhqunspohAVRxygw
m ~ me ~ re vutg r e g t
we r w
ymw qu Tyowy~nhhqunfspohrlmnfzRhfwzsw$qznspAhp hVbdq!dhwu
m
we r w rku m vu oku m o u
MATH 350: Graph Theory and Combinatorics. Fall 2014.
Due in class on Wednesday, October 1st.
Assignment #1: Paths, Cycles and Trees.
1.
For each of the following statements decide if it is true or false, and
either prove it or give a counterexample.
a) If
MATH 350: Graph Theory and Combinatorics. Fall 2014.
Due in class on Wednesdat, October 15th.
Assignment #2: Spanning trees, bipartite graphs and matchings.
1.
We say that F E(G) is evendegree if every vertex of G is incident
with an even number of nonl
MATH 350: Graph Theory and Combinatorics. Fall 2014.
Due in class on Friday, October 31st.
Assignment #3: Mengers theorem, vertex covers and network ows.
1.
Show that (G) 1 (E(G) + 1) for every connected graph G.
2
2.
Let G be a connected graph in which
MATH 350: Graph Theory and Combinatorics. Fall 2014.
Due in class on Monday, November 17th.
Assignment #4: Ramsey theorem, matching and vertex coloring.
1.
Show that R(3, 4) = 9.
2.
Let
rk := Rk (3, 3, . . . , 3).
k times
(I.e. rk is the minimum integer n
MATH 350: Graph Theory and Combinatorics. Fall 2014.
Due in class on Monday, December 1st.
Assignment #5: Planar graphs.
1.
A graph G is outerplanar if it can be drawn in the plane so that
every vertex is incident with the innite region. Show that a graph
Math 350 Mock Final. Fall 2013.
Q1 Let G be the graph pictured on Figure 1.
a) Is G planar?
b) What is the maximum integer k , such that G is k connected?
c) Find (G).
d) Find (G).
Q2 Let G be the graph with weights w : E (G) Z+ be the graph pictured
on
MATH 350 GRAPH THEORY AND COMBINATORICS.
Fall 2013
Instructor:
Sergey Norin
Telephone:
3983819
Office:
Room 1116, Burnside Building
Office hours:
Tuesday, 2:004:00 PM and by appointment
Email:
sergey.norin [at] mcgill.ca
Web:
http:/www.math.mcgill.ca/~s
Name
MATH 350: Graph Theory and Combinatorics. Fall 2013
Exam
Thursday, October 10th, 2013, 16:3517:25

Midterm
The questions have to be answered in the booklets provided.
You can choose which two questions to answer. Indicate your choice on the front
p
Math 350 Final Examination. Fall 2013.
Instructions: The exam is 3 hours long and contains 6 questions. Write your
answers clearly in the notebook provided. You may quote any result/theorem
seen in the lectures without proving it. Justify all your answers
(iii)
8
> 0,
>
> 1
>
>
> 16 ,
<
4,
F (x) =
> 16
> 9
>
>
> 16 ,
>
:
1,
if
if
if
if
if
x < 1,
1x<2
2x<3
3x<4
x 4
Mathematical expectation.
Denition: Let X be a continuous r.v. with
the p.d.f. f . The mean or expected value
of X is denoted by X = = E(X) and
= 0.7350919 +
6
1
!
0.0510.955
= 0.7350919 + 0.2321343 = 0.9672262.
(iii) We have
P (
2 X + 2 ) = 0.9672262.
Multinomial Experiment. An experiment
terminates in of the k disjoint classes. Suppose that the probability that an experiment
terminate in the it
Math 350: Graph Theory and Combinatorics
Wednesday 28th February
Midterm Exam
Instructions. The exam is 50 minutes long and contains 3 questions. Write your answers clearly in the notebook provided. Answer as many questions as you can; full marks can be o
Math 350: Graph Theory and Combinatorics
Assignment 3: Solutions
1. Graph Minors. (a) It is not hard to show that any outplanar graph has a vertex v of degree at most 2. Inductively colour G  v with 3 colours. Then colour v; since it has degree at most 2
Math 350: Graph Theory and Combinatorics
Assignment 6: Solutions
1. Combinatorial Identities. (a) Now s+n counts the number of nsubsets of an (n+s)set X = cfw_1, 2, . . . , n+ n s1, n+s. We can count this in another way. Let Y = cfw_1, 2, . . . , n1, n
Math 350: Graph Theory and Combinatorics
Midterm Exam : Sketch Solutions
1. Planar Graphs. (a) Eulers formula: in a connected planar graph f + n = m + 2 where f is the number of faces. (b) We may assume that G is connected. Every face has at least 3 edges
Math 350: Graph Theory and Combinatorics
Assignment 4: Solutions
1. Coin Tosses. (a) The first coin toss always starts a new run. After that a coin toss starts a new run if and only if it has a different outcome than the previous coin 1 toss. This occurs
Math 350: Graph Theory and Combinatorics
Assignment 2: Solutions
1. Graph Connectivity. (a) Suppose v is a cut vertex of G. Let S1 and S2 be components of G  v. Consider the graph induced by S1 cfw_v. This graph is bipartite and every vertex except v has
Math 350: Graph Theory and Combinatorics
Assignment 1: Sketch Solutions
1. Euler Trails. A graph contains an Euler trail from u to v if and only if the degrees of u and v are odd and every other vertex has even degree. It is a necessary condition. To prov
Example. Let X be a random variable with
the probability distribution
x2
f (x) =
, x = 1, 2, 3, 4.
30
Find E(X) and E(X 2) and V ar(X).
Solution.
u = E(X) =
4
X
x2
x
30
x=1
and
E(X 2) =
4
X
x2
x=1
We have
2 = V ar(X) = 59
5
x2
30
Some properties.
(i) E(aX
This gives 2 = p(1 p). Now in a sequence
of n independent of Bernoullis experiment,
dene
X = number of successes.
The random variable X takes values in cfw_0, 1, . . . , n.
We have
f (x) = P (X = x) =
n
x
px (1 p)n
x
, x = 0, 1, 2, . . . , n.
Proof. If x
Name
MATH 350: Graph Theory and Combinatorics. Fall 2012
Midterm Exam
Thursday, October 11th, 2012, 16:3517:25

The questions have to be answered in the booklets provided. Write your
answers clearly. Justify all your answers. You can consult your notes